Imaging neutral particle analyzer for fast-ion transport measurements in the ASDEX upgrade tokamak

Imaging Neutral Particle Analyzer for fast-ion transport measurements in the ASDEX Upgrade Tokamak José Rueda Rueda Plasma Science and Fusion Technology Department of Atomic, Molecular and Nuclear Physics PhD program in Physical Sciences and Technologies Supervisors: Prof. M. García-Muñoz Prof. E. Viezzer Seville, January 2024 To Hidalgo and Oyola, who made the journey a pleasant one Abstract In future fusion reactors, suprathermal particles (fast ions, FI) will play a key role in the generation of fusion power as they are an important source of energy (heating) and momentum (current drive). A loss of their confinement will lead to a decrease in reactor performance, and, when localized and intense, to damage in the first wall components. Understanding the mechanisms behind the suprathermal particle transport and losses is capital for achieving a future fusion power plant. One of the main observed causes for the FI transport and eventual loss is their interaction with a wide range of electromagnetic fluctuations. An accurate understanding of the fast-ion behavior in the presence of magnetohydrodynamic fluctuations is required for achieving a good fast-ion confinement. To this end, new diagnostics are being developed to measure the fast-ion distribution over a broad region of the phase space with high resolution. In this PhD thesis, an Imaging Neutral Particle Analyzer (INPA) has been installed and operated at the ASDEX Upgrade (AUG) tokamak, located at the Max Planck Institute for Plasma Physics in Garching (Germany). INPA employs the operational principles of both fast-ion loss detectors (FILD) and neutral particle analyzers (NPA) to measure the fast-ion distribution in energy and radius. This diagnostic system analyses fast neutrals that emerge from charge exchange (CX) reactions between fast ions and neutral particles. These fast neutrals are ionized through an ultra-thin carbon foil located within the in-vessel optical head and are deflected towards a scintillator using the local magnetic field of the tokamak. From the impinging location of a particle on the INPA scintillator, its energy and velocity projection along the magnetic field lines can be deduced. The use of an active source of neutrals enables the direct correlation of this velocity projection with the radial position of the fast ion. The FILDSIM code, which facilitates the calculation of synthetic signals for the FILD diagnostic, has undergone a major upgrade to handle the INPA diagnostic. This upgrade includes a model for simulating the scattering and energy loss of fast neutrals within the carbon foil. Additionally, it encompasses a model for estimating the scintillator yield and the capacity to conduct tomographic reconstructions. This updated code has been benchmarked against experimental data during the 2021-2022 campaign, showing an excellent agreement between simulations and measurements. Tomographic inversions also agree with neoclassical calculations during MHD quiescent phases. Fast-ion acceleration during second harmonic ion cyclotron resonance heating has been characterized and compared to simulations. The agreement found serves as validation of these codes for their extrapolation to future machines. Fast-ion flows driven by Alfvén eigenmodes have been measured for the first time at ASDEX Upgrade. The observed flows align well with the theoretical models and with full-orbit simulations. vi Table of contents List of figures ix List of tables xiii 1 Introduction 1 2 Physics background 5 2.1 Definitionofaplasma ........................... 5 2.2 Movement of a charged particle in an electromagnetic field . . . . . . . 6 2.3 Thetokamakconcept............................ 8 2.4 Charge exchange reactions . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Sources of neutral particles . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Fast-ion phase space and dynamics . . . . . . . . . . . . . . . . . . . . 15 3 Fast-ion diagnostics in the ASDEX Upgrade tokamak 23 3.1 Neutronspectroscopy............................ 23 3.2 Scintillator fast-ion loss detector . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Neutral particle analyzer . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Fast-ion Dαdiagnostic ........................... 27 3.5 Imaging neutral particle analyzer . . . . . . . . . . . . . . . . . . . . . 27 4 Synthetic INPA diagnostic 29 4.1 TheTRANSPcode............................. 29 4.2 TheASCOTcode.............................. 31 4.3 TheFIDASIMcode............................. 32 4.4 TheFILDSIMcode............................. 32 5 Design and simulations of the ASDEX Upgrade INPA diagnostic 43 5.1 Phase-spacematch ............................. 43 5.2 Design of the diagnostic 3D head . . . . . . . . . . . . . . . . . . . . . 46 5.3 Theoretical diagnostic resolution . . . . . . . . . . . . . . . . . . . . . . 47 5.4 Characteristics of INPA signals . . . . . . . . . . . . . . . . . . . . . . 50 Table of contents 6 Commissioning of the ASDEX Upgrade INPA diagnostic 53 6.1 Resolution of the INPA diagnostic . . . . . . . . . . . . . . . . . . . . . 53 6.2 Detector characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.3 INPA response to different fast-ion distributions . . . . . . . . . . . . . 56 6.4 Passive signal characterization . . . . . . . . . . . . . . . . . . . . . . . 57 6.5 Evolution of the INPA signal during a discharge . . . . . . . . . . . . . 58 6.6 Absolute calibration of the synthetic diagnostic . . . . . . . . . . . . . 59 7 Tomographic reconstruction of INPA measurements 65 7.1 The inverse problem formulation . . . . . . . . . . . . . . . . . . . . . . 65 7.2 INPAweightfunction ........................... 67 7.3 Inversion of synthetic signals . . . . . . . . . . . . . . . . . . . . . . . . 73 7.4 Inversion of experimental signals . . . . . . . . . . . . . . . . . . . . . . 82 8 Fast-ion transport in phase space 85 8.1 ICRH induced energy diffusion . . . . . . . . . . . . . . . . . . . . . . . 85 8.2 TAE induced fast-ion transport . . . . . . . . . . . . . . . . . . . . . . 89 Summary and outlook 99 Appendix A List of publications and contributions 101 A.1 Publications as main author . . . . . . . . . . . . . . . . . . . . . . . . 101 A.2 Publications as coauthor . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.3 Contributions to conferences and seminars as main author . . . . . . . 103 Appendix B Parameters of the ASDEX Upgrade tokamak 105 B.1 Generalparameters............................. 105 B.2 NBIparameters............................... 106 Acknowledgments 107 Nomenclature 109 References 115 viii List of figures 2.1 Gyroradius and gyrofrequency of hydrogen isotopes and electrons . . . 6 2.2 Decomposition of particle gyromotion. . . . . . . . . . . . . . . . . . . 7 2.3 Drift in a purely toroidal magnetic device . . . . . . . . . . . . . . . . . 8 2.4 Drift suppression in tokamaks . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Basic overview of a tokamak fusion device . . . . . . . . . . . . . . . . 9 2.6 Fluxsurfaces ................................ 11 2.7 Illustrated representation of the charge exchange reaction . . . . . . . . 12 2.8 Cross sections and reaction rates for electron loss . . . . . . . . . . . . 12 2.9 Toroidal view of the density of neutrals injected by NBI#3 . . . . . . . 14 2.10 Birth and slowing down fast-ion distributions . . . . . . . . . . . . . . 15 2.11 Typical fast-ion orbit trajectories . . . . . . . . . . . . . . . . . . . . . 18 2.12 Trapped-passing boundary . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Main components of the FILD head . . . . . . . . . . . . . . . . . . . . 25 3.2 Relation between NPA LOS and explored pitch . . . . . . . . . . . . . 26 3.3 Schematic representation of the INPA diagnostic . . . . . . . . . . . . . 28 4.1 Basic workflow of the suite of codes used in this PhD thesis . . . . . . . 30 4.2 FILDSIM model for the ionization yield . . . . . . . . . . . . . . . . . . 34 4.3 Photon yield of the scintillator for deuterium . . . . . . . . . . . . . . . 34 4.4 Reference system to initialize markers for the INPA mapping calculation 35 4.5 Radial distribution of mapping markers . . . . . . . . . . . . . . . . . . 36 4.6 Distribution of strike position from the mapping markers . . . . . . . . 37 4.7 INPA calibration frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.8 INPA sensitivity in λand zspace..................... 40 4.9 Weight function of the INPA diagnostic . . . . . . . . . . . . . . . . . . 41 4.10 INPA synthetic signal comparison . . . . . . . . . . . . . . . . . . . . . 42 5.1 Target fast-ion distribution functions . . . . . . . . . . . . . . . . . . . 44 5.2 Scanned locations for the INPA diagnostic . . . . . . . . . . . . . . . . 44 5.3 Explored phase space for each possible INPA position. . . . . . . . . . 44 ix Introduction The Lawson condition is naturally fulfilled in the stars. On Earth, the challenge of satisfying this criterion is attempted via two distinct approaches: inertial and magnetic confinement. In inertial confinement fusion, a D-T pellet is subject to intense heating and compression using laser or heavy ion beams, resulting in extremely high densities for a short duration (about 300 g cm−3 during the order of a fraction of a nanosecond) [ 3 ]. Via this approach, Q > 1was achieved for the first time in history in 2022 [ 7 ]. In magnetic confinement fusion, densities are approximately five orders of magnitude smaller than that of air, with confinement on the order of a second [ 6 , 8 ]. For the scope of this thesis, only magnetic confinement fusion will be considered. Within the magnetic confinement approach, two magnetic configurations stand out: tokamaks and stellarators. Both are toroidal devices where the fuel exists in a plasma state. Tokamaks exploit both an external set of coils and the plasma itself to generate the magnetic field responsible for the confinement of the fuel, while stellarators exclusively rely on an external set of coils. This thesis is centered around the tokamak concept. As of yet, experimental reactors remain non-operational for commercial use, with one of the primary hurdles being the effective confinement of supra-thermal ions (also called fast ions, FI). This confinement is essential for two reasons: firstly, as a crucial source of plasma energy and momentum, these particles must be retained long enough to facilitate the transfer of their energy and momentum to the thermal plasma through collisions. Secondly, suboptimal confinement leads to an elevated energy flux directed towards the wall of the reactor, which, if localized, holds the potential to damage the first wall [9]. The initial step towards achieving optimal confinement of the fast ions, and thereby moving closer to a functional fusion reactor, involves thoroughly characterizing the distribution function of these particles and understanding how they respond to, and drive, electromagnetic perturbations. For this purpose, several diagnostics such as fast-ion D α (FIDA) [ 10 ], neutral particle analyzers (NPA) [ 11 ] and neutron detectors [ 12 ] have been developed throughout the last decades. Yet, all present intrinsic limitations: NPAs only give integrated measurements of a localized region of the phase space, FIDA can not distinguish between different values of energies and velocity orientation of the fast ions, and techniques based on neutron detectors have limited temporal resolution. Only the fast-ion loss detector (FILD) can obtain detailed measurements of the velocity space of the fast ions with MHz temporal response [ 13 ], but it only gives information about the lost fast-ion population. This thesis will close this information gap, as the main objective is to provide high spatio-temporal resolution of the confined fast-ion population by means of the new imaging neutral particle analyzer (INPA). The INPA diagnostic combines the already working principles of NPA and FILD to provide the energy and location of the confined fast-ion population with unprecedented resolution [14]. 2 In this thesis, the INPA diagnostic has been developed for the ASDEX Upgrade tokamak. The detector was installed, commissioned and operated. The first measurements confirm and validate the synthetic diagnostic. This validation of the simulation codes enables the possibility of performing signal inversions via tomographic techniques to unravel the true fast-ion distribution function. Furthermore, it enables a direct comparison of prediction and measurements of complex scenarios like the second harmonic beam ion heating due to ion cyclotron resonant heating. To conclude, for the first time in ASDEX Upgrade, fast-ion transport along streamlines caused by Toroidal Alfven Eigenmodes was observed and compared against linear modeling. The document is structured as follows: chapter 2 introduces the fundamental physics concepts to understand the INPA diagnostic and its measurements. Chapter 3 gives an overview of the different fast-ion diagnostics used during this thesis to characterize the fast-ion population. Chapter 4 presents the set of codes used to obtain a synthetic INPA signal and the concept of the weight function. Chapter 5 introduces the design of the AUG INPA diagnostic, its resolution and expected performance. Chapter 6 shows the first INPA measurements at the AUG tokamak. It contains a detailed characterization of the diagnostic, the scaling of its signal with plasma profiles and the benchmark and calibration of the synthetic diagnostic against experimental data. Chapter 7 explores how to extract the fast-ion distribution function from the INPA measurements via regression techniques. Finally, chapter 8 presents two examples of redistribution in phase space of the fast-ion population. Firstly, it shows the modification of the fast-ion population due to an externally applied wave, ion cyclotron resonance heating. Secondly, it shows the redistribution of fast ions due to a plasma electromagnetic perturbation, a toroidal Alfven eigenmode. 3 Chapter 2 Physics background This chapter presents the fundamental physics concepts necessary to understand the INPA diagnostic and its measurements. It starts with the definition of a plasma. Next, the introduction to the movement of charged particles in an electromagnetic field is shown, in order to present the drift motions and to understand how the magnetic configuration of a tokamak can confine a plasma. Afterwards, a description of the tokamak magnetic configuration and its coordinates is given. Subsequently, charge exchange reactions, one of the basic ways to obtain information about tokamak plasmas, are presented. Finally, the sources of neutral particles and fast ions are discussed, together with a presentation of the needed coordinates to describe an orbit in a tokamak. 2.1 Definition of a plasma As said in [ 15 ]: A plasma is a quasineutral gas of charged and neutral particles which exhibits collective behavior.Quasineutrality implies that strict neutrality is only kept over a given scale, the Debye length [ 6 ]. Collective behavior implies that motions depend not only on local conditions but on remote regions, which interact via long-range electromagnetic forces. The Debye length, λD, can be written as [6]: λD=sϵ0Te nee2,(2.1) where ϵ0 stands for the vacuum permittivity; and Te, ne and e are the electron temperature, density and charge, respectively. Notice that the Debye length is solely dependent on the electron temperature and density. As electrons possess a smaller mass, they exhibit greater mobility compared to ions. Therefore, they generally play the key role in maintaining neutrality. 5 Physics background 1071091011 Gyrofrequency [rad/s] 10−2100102 Energy [keV] 10−5 10−3 10 (a) (b) −1 Gyroradius [m] H D Te Fig. 2.1 Gyroradius and gyrofrequency of hydrogen isotopes and electrons. (a) Gyroradius of the particles as a function of their energy. As a reference, the minor radius of ASDEX Upgrade is included as a dashed black line. (b) Gyrofrequency of the particles, the color code is the same as in subplot (a). The black cross represents the typical frequency of a shear Alfvén wave (more precisely a Toroidal Alfvén eigenmode, see section 2.6.4) in the ASDEX Upgrade tokamak. In both cases a magnetic field of 2.5 T has been assumed. 2.2 Movement of a charged particle in an electromagnetic field In the presence of an electromagnetic field, the movement of a charged particle is governed by the Lorentz equation of motion: dv dt =q m E+v × B,(2.2) where v , q and m stand for the velocity, the charge and the mass of the particle, respectively, and  Eand  Bare the electric and magnetic fields. Assuming constant and homogeneous electric and magnetic fields, and  E ×  B = 0, equation (2.2) yields a helix as the solution. The axis of this helix is aligned with the magnetic field and its radius is given by the so-called Larmor radius: rL=mv⊥ |q|B,(2.3) where v⊥ represents the modulus of the normal component of the velocity, with respect to the magnetic field 1 . If instead of using the perpendicular velocity, the total velocity is inserted in equation (2.3) a normalized momentum is obtained: g=mv |q|B.(2.4) This is denoted as gyroradius across the present literature related to fast-ion diagnostics [ 13 , 16 – 18 ]. The frequency of gyration, the so-called cyclotron frequency, ωc , is 1 In this document, unless explicitly mentioned, the parallel and perpendicular direction will refer to the local magnetic field. 6 2.3 The tokamak concept Gyrocenter (a) (b) Particle rL B B rL Fig. 2.2 Decomposition of particle gyromotion. (a) 3D illustration of the movement of a charged particle in an arbitrary magnetic field. (b) Gyromotion of the particle in the plane perpendicular to the magnetic field. The angle α is the so-called gyrophase. Adapted from [ 19 ]. given by [6]: ωc≡|q|B m.(2.5) Electrons gyrate clockwise while ions gyrate counter-clockwise, with respect to the magnetic field. Due to this, the magnetic field created by the gyrating charge opposes the imposed field,  B . Hence, plasmas are diamagnetic [ 15 ]. The typical scales of the Larmor radius (gyroradius) and gyrofrequency for hydrogen isotopes and electrons is shown in figure 2.1 for a typical magnetic field at the ASDEX Upgrade tokamak (2.5 T). In the presence of either non-uniform or non-constant fields, finding a solution to the equation of motion becomes complex. However, if the scales of the field variations are significantly larger than the scale of the movement |∇B/B| ≪ r−1 Land |∂tB/B| ≪ ωc , the problem can be broken down into two fundamental components: a circular and periodic motion around a gyrocenter (with radius rL and frequency ωc ), along with a gyrocenter drift [ 6 ]. This is represented in figure 2.2. The primary drift motions include: ■vE =  E×  B B2 : the  E ×  B -drift. It is caused by the changes of gyroradius along the orbit due to the electric field. It does not depend on the charge. ■v∇B = −mv2 ⊥ 2qB3∇B× B : the ∇B -drift. It originates from the variations in the gyroradius due to the different strength of the magnetic field at each point of the gyromotion. ■vc = −mv2 ∥ qB3∇B× B : the curvature drift. It is present where the field lines are curved. It is important to observe that the last two drifts share the same direction. There exists no possible cancellation between these drifts. As a result, even when an electric field is absent, a non-uniform magnetic field will inevitably trigger particle drifts. 7 Physics background (a) (b) Fig. 2.3 Drift in a purely toroidal magnetic device. Particle drifts encountered by electrons are depicted in red, while those experienced by ions are shown in blue. (a) Within a purely toroidal magnetic apparatus (cross-section indicated by the black line), the drift arising from field curvature and gradient in field strength induces charge separation, thus generating an electric field, (b), and subsequently, an  E ×  B drift. Since the drift resulting from the electric field is not influenced by charge, the whole plasma will undergo a displacement and ultimately be expelled toward the reactor wall. (a) (b) Fig. 2.4 Drift suppression in tokamaks. Particle drifts encountered by electrons are depicted in red, while those experienced by ions are shown in blue. (a) Within a tokamak, the existence of a toroidal current gives rise to a poloidal magnetic field. As particles initiate drifting due to curvature and gradient in the magnetic field, the poloidal field forces them to alter their course, (b), preventing charge separation and the subsequent emergence of an electric field that induces losses. 2.3 The tokamak concept As explained in section 2.2, a static and uniform magnetic field can only confine particles in the transverse direction, allowing charges to move freely along the field lines. Consequently, any configuration employing a uniform magnetic field will encounter losses at the ends. To mitigate these losses, magnetic configurations featuring closed field lines, such as toroidal arrangements, come into play. At first glance, such a device might promise perfect confinement. However, a pure toroidal setup would produce curvature and ∇B -drifts, introducing charge separation. This charge separation generates an electric field which triggers an  E ×  B -drift, initiating particle transport that eventually 8 2.3 The tokamak concept Transformer coils Vertical (poloildal) field coil Toroidal field coil Plasma Magnetic field line Plasma current Fig. 2.5 Basic overview of a tokamak fusion device. The helical field created by the toroidal field coils and the plasma current (induced by the transformer) confines the plasma. Vertical field coils help to position and shape the plasma. Adapted from [20]. expels particles towards the vessel wall (see the schematic cross-section 2 of the toroidal device depicted in figure 2.3). In a tokamak, this charge separation is short-circuited by the presence of a poloidal field, as shown in figure 2.4. This field, typically an order of magnitude weaker than the toroidal field, arises from the plasma current, generated by a central solenoid as depicted in figure 2.5, where a schematic representation of a tokamak can be seen. In addition to the already mentioned toroidal field coils and the central solenoid, a set of vertical coils is used to shape and position the plasma. The helical field configuration inherently displays a series of nested flux surfaces, i.e., toroidal surfaces on which the magnetic field lines lie. These surfaces confine charged particles as their gyrocenters trace the magnetic field lines. Drifts, collisions and magnetohydrodynamic (MHD) instabilities can lead to movement across these surfaces. Nonetheless, before significantly drifting beyond a flux surface, particles tend to complete numerous cycles around the torus. This repeated cycling ensures an equilibrium is established on these surfaces, with pressure, temperature, and plasma density remaining uniform across the entire surface. Therefore, these surfaces serve as the natural reference system for the equilibrium of the plasma. One of the most common methods to define the flux surface reference frame is the normalized poloidal flux radius [21], ρp: ρp≡sΨ−Ψa Ψs−Ψa ,(2.6) where Ψis the poloidal flux and the indices s and a refer to the separatrix (last closed flux surface) and the magnetic axis (the single field line at the center of the set of nested 2 To avoid confusion with the engineering term, in this document cross section will refer to the physical quantity associated with the reaction probability while cross-section will be used for the cuts done in a 3D computer model to visualize the interior of a piece. 9 Physics background surfaces), respectively. With this definition, ρp = 0 is located at the magnetic axis and ρp = 1 at the separatrix. The poloidal flux is defined as the magnetic flux traversing a surface oriented such that its normal vector lies on the z-axis: Ψ = ZS  B·ds. (2.7) Due to the toroidal symmetry of the tokamak, this expression reduces to [21]: Ψ(R, z)=2πZR 0Bz(R′, z)R′dR′.(2.8) An example of these flux surfaces and the relation between ρp and plasma major radius can be seen in figure 2.6 for an ASDEX Upgrade equilibrium. Inside the surfaces of constant ρp , the magnetic field lines follow helical paths as they go round the torus. For a given toroidal angle, ϕ (see figure 2.6(b)), the magnetic field line will have a certain position in the poloidal plane (see figure 2.6(a)). After a given change of toroidal angle, ∆ ϕ , the magnetic field line will return to its initial poloidal position. The larger the value of ∆ϕis, the more the field line will advance toroidally before closing poloidally. This is generally expressed via the ratio [6]: q=∆ϕ 2π.(2.9) Hence, if a magnetic field line returns to its initial poloidal position after exactly one turn along the torus, q = 1. This q -value is usually called safety factor because several instabilities might be excited at the positions where q takes rational values [ 6 , 21 ]. All magnetic field lines belonging to the same flux surface have the same value of q. In a conventional tokamak, the magnetic field is mainly toroidal and thus scales as 1 /R , making it so that the modulus of the field is not constant along the flux surface. The region of major radius which is smaller than the magnetic axis (highlighted in yellow in figure 2.6) is called the high-field side (HFS) while the region from the magnetic axis outwards is called the low-field side (LFS) (highlighted in green in figure 2.6). 2.4 Charge exchange reactions Charge exchange reactions (CX) are reactions in which an ion and a neutral particle exchange an electron. These interactions are a primary source of information for plasma diagnosis (refer, for instance, to [ 22 , 23 ]). In the context presented here, the focus is put on reactions involving particles that belong to the same atomic species, and have a charge +1. Consequently, the ion undergoes neutralization and is no longer confined by the tokamak magnetic field. Taking the case of deuterium as a reference: D+D+→D++ D∗,(2.10) 10 2.5 Sources of neutral particles (a) 0.0 0.5 z [m] -0.5 -1.0 1.0 1.5 R [m] 2.0 1.0 (c) #37513 HFS LFS HFS LFS R [m] 2.0 1.2 1.6 0.8 1.0 0.6 0.4 0.2 0.0 (b) 0.0 0.0 y [m] x [m] 2.0 2.0 -2.0 -2.0 ϕ θ Fig. 2.6 Flux surfaces. (a) Poloidal representation of the AUG first wall and flux surfaces for a typical magnetic equilibrium of the AUG tokamak. The separatrix and the magnetic axis are indicated in blue. Dashed red lines represent a series of flux surfaces. (b) Top-down (toroidal) representation of the AUG wall and flux surfaces. As before, in blue the magnetic axis and separatrix and in dashed red a set of magnetic flux surfaces. (c) Relation between ρp and major radius, for z = 5 cm, the height of the magnetic axis for this discharge. In all subplots, the HFS is highlighted in yellow and the LFS in green. For convenience, only the region inside the separatrix is colored in the toroidal plot. where the asterisk denotes an excited state. The process does not involve any change in energy of the bound electron other than the translational energy change associated with the different atomic velocities [ 24 ]. Due to the substantial mass disparity between ions and electrons, this energy exchange is negligible. In terms of momentum, the angular deviation connected to this exchange remains below 1 o [ 25 ]. Consequently, the formed neutral conserves the energy and momentum of the initial ion. A sketch of this sequence is available in figure 2.7. Note that the excited CX neutral decays to its ground state, emitting a photon, shortly after the CX reaction takes place. Charge exchange reactions compete with alternative mechanisms through which a neutral particle can be transformed into an ion within a plasma; i.e., the ionization processes due to either electron or ion collisions. The cross sections, and reaction rates, associated with these three processes are presented in figure 2.8. CX reactions are dominant at lower energies (< 50 keV). A comprehensive comparison of these cross sections and their integration into the FIDASIM code (presented in section 4.3) is available in the appendix of [27]. 2.5 Sources of neutral particles Within a tokamak, the temperature of the thermal plasma generally reaches several keV, far above the ionization energy of hydrogen. Consequently, the natural state of the system entails full ionization. Nonetheless, significant concentrations of neutrals can 11 Physics background -1.0 -0.5 0.0 0.5 1.0 (a) (b) 0.2 0.4 0.6 0.8 1.0 λ = 0.7 λ = 0.5 1.0 1.5 R [m] z [m] 2.0 Fig. 2.11 Typical fast-ion orbit trajectories. Ions were followed only during one poloidal cycle. The starting position is marked with a magenta star. (a) 3D representation of two typical orbits, in blue a passing orbit, in red a trapped one and in shaded gray a third of the AUG vessel. (b) Poloidal representation of both orbits. The legend indicates the pitch angle at the position where the ion was initialized. Even if three is the minimum number of variables to label a fast-ion orbit, four variables will be used to describe the fast-ion phase space. This selection is done to simplify the calculation of the INPA instrument response and the analysis of its signals (see chapters 4 to 6). Imposing only the conservation of µ and the toroidal symmetry, two variables suffice to describe the configuration space and two for the velocity space: ( R, z, E, λ ), being major radius, height over the mid-plane, energy and pitch angle, respectively. 2.6.3 Orbit topology In a tokamak magnetic configuration, the presence of a field varying like 1 /R allows for the existence of particle trapping, which arises from the conservation of energy and magnetic moment. The kinetic energy of a charged particle can be written as: E=1 2mv2 ∥+1 2mv2 ⊥=1 2mv2 ∥+µB. (2.14) Hence, when the particle moves to an area with a stronger magnetic field, its parallel velocity will reduce. Eventually, this parallel motion can even be reversed. It is straightforward to prove that a particle will undergo a bounce-back ( v∥ = 0) under the condition [6]: |λ(r)|<s1−B(r) Bmax .(2.15) Here, Bmax represents the peak magnetic field strength of the flux surface that traverses through r ; i.e., the highest magnetic field accessible to the particle, neglecting 18 2.6 Fast-ion phase space and dynamics (a) 0.3 0.2 0.1 (b) 0.6 0.6 0.5 0.4 0.3 0.2 0.1 0.5 0.4 R [m] 1.7 1.8 1.9 2.0 R [m] z [m] 1.0 1.5 2.0 2.5 λBoundary λBoundary 1.0 0.5 0.0 -0.5 -1.0 Fig. 2.12 Trapped-passing boundary. Only the region inside the separatrix has been represented. (a) Particles initiated in a given ( R, z )position with pitch larger than the one indicated by the colour code would be passing, otherwise, trapped. (b) Cut along z = 8 cm, the height of the magnetic axis for this shot. This height is indicated in (a) with a dashed green line. drift motions across magnetic flux surfaces. Assuming a magnetic field that scales proportionally to the inverse of the major radius: |λ(r)|<s1−Rmin R≡ |λTP (r)|,(2.16) where Rmin is the minimum R of the flux surface where the particle is located. This criterion establishes the boundary known as the trapped-passing boundary [ 3 ]. Particles with a pitch (in absolute value) smaller than |λTP ( r ) | undergo repetitive oscillation, following banana (trapped) trajectories, whereas those with a pitch greater than this boundary engage in rotational motion around the torus along passing orbits. Both kinds of orbits are illustrated in figure 2.11, while the pitch associated with the trapped-passing boundary is depicted in figure 2.12. Other kinds of orbits like stagnated or ripple trapped exist, but their description is out of the scope of this introduction. The reader is referred to [ 33 , 40 ] for details on those orbits. 2.6.4 Transport A complete description of the theory of fast-ion transport is out of the scope of this work. Here, only a brief introduction to support the experimental findings will be presented. Detailed descriptions of the interaction of fast ion with Alfvénic modes can be found in [ 41 ], models for the diffuse transport in [ 42 ] and a complete overview in [ 33 , 40 ] and the references therein. 19 Physics background Neoclassical transport Although fast ions are collisionless particles in an orbit time scale, they collide with the thermal plasma on longer time scales, as explained before. In a homogeneous magnetic field, these Coulomb collisions would lead to a diffusive transport, similar to the so-called drunk sailor problem [ 43 ]. This is termed as classical transport. For the magnetic configuration of a tokamak, changes in velocity due to collisions can result in a transition between different orbit types, therefore enhancing transport. This enhanced transport is called neoclassical. It can be shown that the neoclassical diffusion scales with 1 /√E [ 44 ]; therefore, neoclassical diffusion of fast ions is smaller than the one for thermal ions. The neoclassical transport is well validated against experiments under MHD quiescent conditions [45]. In this thesis, the term neoclassical fast-ion distribution will be used for fast-ion distributions calculated including only neoclassical effects. Resonant transport and E′conservation Given a perturbation in the plasma with electric field  E , the change of kinetic energy, ∆E, along an orbit can be written as: ∆E=Iq E ·v dt=qI E ·d l. (2.17) When the phase changes of the electric field perturbation and the fast-ion orbit do not match, the integral will average to zero. But if the particle and the perturbation return to the same phase after cycling through the periodic motion, the integral can be non zero and the so-called resonance occurs. A generic electric perturbation can be written as [46]: E(ρp, ϕ, θ)≡X n,m En,m(ρp)ei(ωt−nϕ−mθ),(2.18) where ρp labels the flux surface, θ and ϕ are the two Boozer angles which define the position inside a given flux surface [ 47 ], ω is the mode frequency and n and m are the toroidal and poloidal mode numbers, respectively. With the electric perturbation written in this form, the resonance condition for trapped particles reads as [41]: ω=jcωc+jθωθ+nωϕ,(2.19) where j are integers, and ωc , ωθ and ωϕ are the cyclotron, poloidal and toroidal frequencies, respectively. For passing ions, it reads as [48]: ω=jcωc+ (jθ+m−nq)ωθ+nωϕ,(2.20) 20 2.6 Fast-ion phase space and dynamics where q is the safety factor. The change of energy produced by such a resonant interaction is associated with a change in toroidal canonical angular momentum because the quantity E′ = E−ωPϕ/n is conserved [ 41 ]. As the variation of the toroidal angular momentum is related to radial transport [ 49 ], the radial transport is also related to an energy exchange. One of the most important perturbations which can interact with the fast-ion population are the shear Alfvén waves. These waves are incompressible fluctuations ( ∇·v = 0) which have the magnetic field line tension as the only restoring force [ 21 ]. They normally appear as a damping term for instabilities, as their excitation is a sink of energy. However, in a toroidal device, due to the coupling between eigenmodes caused by toroidicity, some discrete non-damped frequencies can exist, the so-called Toroidal Alfvén eigenmodes (TAEs). These particular modes are key for the fast-ion population as they have been observed to produce large transport and redistribution of fast ions [50]. Diffusive transport due to plasma instabilities Under certain conditions, such as the overlapping of multiple resonances [ 40 ], the redistribution of fast ions can be modeled as a diffusive phenomenon. Hence, the corresponding continuity equation must be fulfilled. This equation can be written as: ∂n ∂t =−∇· Γ + S,(2.21) where n stands for the particle density, S for the sources and sinks, while  Γ represents the particle flux, which can be modeled as:  Γ = −∇(Dn) + nu, (2.22) where u is the fluid velocity and D the diffusion coefficient. Notice that the particle flux is composed of two terms, the convective contribution, nu , and the diffusive one, ∇ ( Dn ). As of yet, there is no universally accepted theory which is able to predict a value for the diffusion coefficient for fast ions in agreement with the measured experimental values. The previously introduced neoclassical transport can differ an order of magnitude from the experimental data when plasma instabilities are present [ 6 ]. Transport caused by these phenomena is termed as anomalous. 21 Chapter 3 Fast-ion diagnostics in the ASDEX Upgrade tokamak This chapter introduces the diagnostics used in this thesis to study the fast-ion population. It begins by introducing neutron detectors, one of the most used diagnostics to measure the total number of fast ions in a tokamak. The chapter continues by introducing the fast-ion loss detector and the neutral particle analyzer, to present the basic principles that exploits the concept of the imaging neutral particle analyzer. Afterwards, the fast-ion D α diagnostic, one of the most used diagnostics to study fast-ion transport, will be presented. The chapter is concluded with a detailed explanation of the imaging neutral particle analyzer working principle and parameters. A complete overview of the different diagnostics to measure the plasma parameters such as density, temperature or the magnetic equilibrium and fluctuation is out of the scope of this document, the reader is referred to [16, 51, 52] and the references therein. 3.1 Neutron spectroscopy Neutron measurements have been used to obtain information of the fast-ion distribution for several decades [ 53 ]. Neutrons are produced in the fusion reactions which take place within the plasma. For deuterium plasmas, the most notable fusion reaction is: 2D + 2D→3He [0.82 MeV] + n [2.50 MeV],(3.1) where the numbers in square brackets represent the nominal kinetic energy of the products in the center-of-mass frame. Although the nominal kinetic energy for the neutron is 2.5 MeV, the emitted neutron can have different kinetic energies in the reference system of the diagnostic. This is due to the energy and momentum conservation. 23 Fast-ion diagnostics in the ASDEX Upgrade tokamak In general, the energy of the emitted neutron can be written as [12]: En=1 2mnv2 CM +m3He mn+m3He (Q+K) + vCM cosα2mnm3He mn+m3He (Q+K)1/2 ,(3.2) where mn and m3He are the masses of the emitted neutron and 3He ion, respectively; vCM is the velocity of the center of mass of the two initial deuterium particles, in the detector frame; Q is the mass difference between reactants and products; K is the kinetic energy of the deuterium particles in the center of mass frame; and α is the angle between the center of mass velocity and the velocity of the produced neutron. This angle is fixed by the geometry of the diagnostic view. If vCM ≈ 0and K≈ 0, the expression simplifies to: En0≡En(vCM = 0, K = 0) = m3He mn+m3He Q= 2.5 MeV.(3.3) Hence, the energy difference En−En0 encodes the information of the movement of the reactants. At the AUG tokamak, the neutron production is dominated by beam-target fusion reaction, during NBI heated phases. Thus, the neutron rate and energy distributions are footprints of the fast-ion distribution function [ 54 ]. The AUG neutron spectrometer uses a liquid scintillator. This liquid scintillator is located outside the reactor wall, at roughly 10 m from the plasma. When neutrons arrive to the scintillator and interact with it, light is emitted. The emitted photon flux is proportional to the neutron energy. This flux is collected and guided to a photomultiplier tube, where the photon flux pulse is converted into an electric pulse which is then digitized. Further information of the AUG neutron spectrometer and its calibration can be found in [ 54 , 55 ]. Beside spectroscopy, the total number of neutrons and its profile is a reliable source of information to determine the spatial location of fast ions [56]. 3.2 Scintillator fast-ion loss detector The scintillator fast-ion loss detector (FILD) [ 13 ] is capable of determining the velocity distribution of fast-ion losses with a resolution of less than 2 ◦ in pitch and 20 keV in energy, for 100 keV deuterium ions within a 2.5 T magnetic field. Its temporal response can reach up to hundreds of kHz (when using its fast acquisition system, which has low spatial resolution [ 9 , 57 ]). Positioned within the reactor, along the vessel wall, the detector works like a magnetic spectrometer, using the local magnetic field to guide fast ions onto a scintillator. The scintillator emits light at the strike position of the fast ion, which depends on its gyroradius and pitch. Hence, FILD cannot resolve different particles with the same charge-to-mass ratio, for instance, D and 4He. 24 3.3 Neutral particle analyzer (a) (b) (c) Fig. 3.1 Main components of the FILD head. (a) Side and (b) front sketches of the collimator and scintillator, including some FI orbits. The velocity of the FI at the pinhole determines the strike position on the scintillator. Extracted from [ 35 ]. (c) 3D view of the probe head. Notice the purple orbit, which is stopped at the collimator. Adapted from [57]. The emitted light from the scintillator plate is guided by a series of lenses and directed towards the acquisition system: a high-resolution camera and a fast acquisition system. For the AUG setup, the high-resolution cameras are a charge-coupled device (CCD), or a complementary metal oxide semiconductor (CMOS), while the fast systems are photomultipliers (PMTs), or an avalanche photodiode (APD) camera, depending on the particular FILD [ 17 , 18 , 35 ]. A representation of the operational principle, along with a 3D computer-aided design (CAD) of the essential components within the detector head, is depicted in figure 3.1. At the AUG tokamak, there are four operational FILDs located in different positions along the vessel wall. This allows for the characterization of the spatial asymmetries in the fast-ion losses. TG-Green ( SrGa2S4: Eu2+ ) is used as scintillator material in these detectors due to its short decay time (490 ns), elevated yield, and resilience against radiation [58, 59]. FILD detectors have also been installed in many other fusion experiments such as TFTR [ 60 ], DIII-D [ 61 ], MU [ 62 ], TCV [ 63 ], KSTAR [ 64 ] or JET [ 65 ] and are being designed to be installed at the JT60-SA [ 66 ] and ITER [ 67 ] tokamaks. Their unique capabilities make them a key player in the study of fast-ion transport and losses due to a wide variety of plasma instabilities [34, 68–72]. 3.3 Neutral particle analyzer Neutral particle analyzers (NPA) [ 11 ] diagnose CX neutrals (refer to section 2.4) to obtain information about the confined ion population. Among the earliest tools for studying confined ion properties [ 11 ], NPAs can be used to study both the thermal ion population [ 24 , 73 ] or the fast ions [ 74 ]. Typically, these diagnostics provide lineintegrated measurements, although it is possible to obtain localized measurements by using a localized source of neutrals [75]. 25 Fast-ion diagnostics in the ASDEX Upgrade tokamak Fig. 3.2 Relation between NPA LOS and explored pitch. Extracted from [75]. The operational principle of a neutral particle analyzer is as follows: within the plasma, an ion undergoes a CX reaction with a neutral particle. Since the magnetic field does not confine the newly created CX neutral particle, it can escape the plasma unless it is re-ionized. If its trajectory falls within the acceptance range of the collimator, it enters the detection region. Here, its energy and/or momentum are analyzed, depending on the specific setup employed by the NPA. Various mechanisms exist for determining these properties. For instance, ionization on a stripping foil, followed by a deflection using electromagnetic fields, directing particles toward an array of channeltrons 1 [ 76 , 77 ]. In these kinds of setups, the combination of the electric field and channeltron array allows for the separation of particles by mass and energy. An alternative method involves pulse height analysis using a silicon diode [ 23 , 75 ]. While this approach sacrifices mass separation capability, it eliminates the need for externally applied electromagnetic fields and offers flexibility on the line-of-sight (LOS) as the volume of the detector is considerably smaller. For all NPA systems, if the initial neutrals involved in the CX reaction originate from an active source, such as an NBI, the signal is called active; otherwise, it is termed passive. Regardless of the specific operational approach, the CX neutrals detected by NPAs travel in straight lines towards the diagnostic and maintain the velocity orientation of the plasma ion at the moment of the CX reaction, as no significant deflection occurs during these reactions (see section 2.4). The detector location and its collimator determine its view; therefore, they determine the range of velocity orientations which can be measured. Hence, there is a relationship between the geometry of the NPA detector and the range of velocity orientations (pitch) of fast ion which can be measured. A graphical representation of this is illustrated in figure 3.2. See also figure 8 of [ 75 ] for a deeper insight. 1A channeltron is a vacuum-tube structure acting as a charge multiplier. 26 3.4 Fast-ion Dαdiagnostic 3.4 Fast-ion Dαdiagnostic The fast-ion D α (FIDA) diagnostic [ 78 ] analyze CX photons (refer to section 2.4) to obtain information about the confined fast-ion population. FIDA measures the wavelength of the photon emitted by the excited CX neutral when it decays from its third to its second excited state. In the absence of a Doppler shift, the wavelength of this radiation is 656.1 nm in air, which corresponds to the visible part of the electromagnetic spectrum. Due to the movement of the CX neutral, the detected photon will be Doppler shifted by ∆ΛDoppler [27]: ∆ΛDoppler = Λ0vFI ·uLOS c,(3.4) where Λ 0 is the wavelength in the absence of a Doppler shift, vFI the velocity of the fast ion before the CX reaction, uLOS the unit vector along the LOS and c the speed of light in vacuum. Hence, the information about the velocity space of the fast ions is encoded in the measured Doppler shift. The interpretation of the FIDA signal is not straight forward because different values of energy and pitch angle can yield the same values of Doppler shift. Therefore, it is not possible to directly obtain phase-space resolved measurements of the confined fast-ion distribution. Nonetheless, tomographic inversions [ 10 , 79 – 82 ] allow this information to be obtained from the FIDA signal once the detector sensitivity is fully characterized. FIDA detectors have been widely used to determine properties of the fast-ion dynamics and to quantify the transport caused by several instabilities, see for example [78, 83–86] and the references therein. 3.5 Imaging neutral particle analyzer The imaging neutral particle analyzer (INPA) [ 14 ], merges the operational principles of FILD and NPA detectors to extract information about the energy and position of fast ions within the plasma: as an NPA, it employs CX neutrals to diagnose fast ions; as a FILD, it uses a scintillator as active component. By combining both approaches, the INPA diagnostic achieves exceptional results: spatial resolution below 10% of the plasma minor radius in the LFS region, and energy resolution of around 10% for 100 keV fast deuterium ions for the AUG setup [ 87 ]. The diagnostic also covers a broad section of the plasma, spanning from mid-radius in the HFS to the LFS separatrix. The innovation behind the INPA diagnostic lies in the use of an ultra-thin carbon foil (20 nm for the case of the AUG setup [ 88 ]), placed within the reactor vessel, to ionize incoming neutral particles. This arrangement facilitates coverage of a large plasma region that would otherwise remain unobserved by individual LOSs. The ionized CX particles are then deflected onto a scintillator (TG-Green) by the local magnetic field, as illustrated in figure 3.3. From the strike point on the scintillator, the particle energy and pitch can be precisely determined. Consequently, the trajectory of the particle can 27 Synthetic INPA diagnostic Reference FILDSIM model Fig. 4.2 FILDSIM model for the ionization yield. The reference data (black dots) is extracted from [108], in red the model implemented in FILDSIM. The energy loss of the markers follows the empirical relation found during the first measurements of the INPA diagnostic at the AUG tokamak. The reader is referred to section 6.2.2 for full details on this relation. 4.4.3 Modeling of the scintillator yield The scintillator yield is modeled using Birk’s law, as in previous studies at the AUG tokamak [9, 59]: dY dx =SBirk dE dx 1 + kBirk dE dx ,(4.5) where dY dx is the yield of photons per unit of length traveled by the particle inside the material, dE dx the stopping power and SBirk and kBirk two constants of the material. For TG-Green these constants are: 0.1 γ/(ion ·eV) and 3 · 10 13 atoms/(cm2·eV) , respectively. The yield, for deuterium, can be seen in figure 4.3. In this energy interval, the yield depends almost linearly on energy. Fig. 4.3 Photon yield of the scintillator for deuterium. Number of emitted photons per impinging ion. Calculated for deuterium using the parameters found in [59]. 34 4.4 The FILDSIM code u3 β α u1 u2 Fig. 4.4 Reference system to initialize markers for the INPA mapping calculation. 4.4.4 Mapping of the scintillator plate To map the INPA scintillator plate, FILDSIM launches MC markers at the pinhole, with a determined energy and a random velocity direction inside the collimator acceptance. Using a right-handed reference system, {u1, u2, u3} , where u3 is normal to the pinhole, markers are initialized with random velocities inside cones with aperture β and the axis in the plane formed by {u1, u3} , see figure 4.4. β is the acceptance angle of the collimator. Ideally, the collimator would be aligned with the NBI line ( u1·uNBI = 0) and β→ 0for an infinite collimator, so each of these cones will correspond to a single point along the NBI line. When considering the finite acceptance of the collimator ( β∼ 0 . 1rad for the AUG setup) and the misalignment of the collimator with the NBI line (caused by tolerances in the installation process, see [ 109 ]) this does not longer hold exactly. Hence, the code calculates the closest point between the initial trajectory of the marker and the NBI line and uses the average radial position of these points for all the markers launched inside the cone as the coordinate for the strike map. An example of the distribution of radial locations and distance to the central NBI line for the AUG setup can be seen in figure 4.5. Notice that the width of the radial distribution of markers directly determines the INPA radial resolution due to the detector geometry. For a full calculation of the radial resolution, please see section 5.3.2 on the detector characterization. In a zeroth order approach, the active signal dominates over the passive component and the origin of the signal is located in the closest point of the LOS towards the NBI central line. Hence, there is a one-to-one relation between the LOS and the radial birth position [ 88 ]. Therefore, the set of centroids of the strikes distribution of the MC markers can be used as a map to translate from the strike position on the scintillator plate to energy and position of the fast ions inside the plasma. The validity of this approximation is checked in section 5.3.2 where the simulated radial birth position of the measured particles is presented. The strike distribution of the mapping markers and the map is shown in figure 4.6. Each of the squared structures corresponds to a set of markers launched with a given pair of energy (gyroradius) and LOS ( α in figure 4.4). The same number of markers were launched for each pair of values. While moving 35 Synthetic INPA diagnostic 1.70 1.72 1.74 1.76 1.78 1.80 0.0 0.2 0.4 0.6 0.8 1.0 Markers [a.u.] (a) g = 2.25 cm R = 1.75 m 0.02 0.04 0.06 0.08 0.10 0.0 0.2 0.4 0.6 0.8 1.0 Markers [a.u.] (b) g = 2.25 cm R = 1.75 m dmin [m]R [m] Fig. 4.5 Radial distribution of mapping markers. (a) Distribution of radial location of the closest point of the neutral trajectory with respect to the NBI line. The dashed line represents a Gaussian fit to the data. There is no a priori reason for this behavior, it has been included just as a guideline for the eye. (b) Distribution of the distance between the closest point of the marker trajectory to the NBI line. Both correspond to the set of markers launched with gyroradius g= 2.25 cm and assigned to the radial location R= 1.75 m. towards higher energies and major radius, the distribution extends over a larger area of the scintillator; hence the number of markers per units of area is smaller. 4.4.5 Monte-Carlo technique to remap the signal As detailed before, due to the finite collimator and pinhole sizes, markers with similar energy and radial birth location can strike in different points of the scintillator, which are then observed at different points of the camera sensor. Let D ( x1, x2|E, R )be the distribution of markers on the camera sensor (defined by the variables x1 , x2 ) conditioned on the marker energy, E , and radial birth location, R . There exists a mapping that relates the 2D camera space ( x1, x2 )with the 2D phase space ( E, R ). This map is defined by two functions [110]: ψE:ED(x1,x2|E,R)(x1, x2)7−→ E, (4.6) ψR:ED(x1,x2|E,R)(x1, x2)7−→ R. (4.7) In this mapping, the center of mass of the strike points distribution D ( x1, x2|E, R ), denoted as ED(x1,x2|E,R) , corresponds to the point ( E, R ). This one to one mapping is the so-called strike map presented in section 4.4.4. The strike map can be used to translate the camera signal, which is a distribution of counts per pixels, into a gyroradius (energy) - major radius distribution. To this end, a MC approach is used to ease the calculation of the Jacobian of the coordinate transformation. The signal for each pixel, Sij , is written as a collection of MC markers 36 4.4 The FILDSIM code 0.10 0.05 0.00 0.00 0.05 0.150.10 x1s[m] x2s[m] Markers [#/m2] 3 2 1 0 109 0.05 1.50 0.05 1.60 1.70 1.90 2.00 2.20 Radius [m] 1.0 1.5 2.0 2.5 3.0 Gyroradius [cm] 1.80 2.10 Fig. 4.6 Distribution of strike position from the mapping markers. Each of the squares corresponds to a pair of energy and LOS. Calculated for a constant and homogeneous magnetic field. as: Sij =Cij N N X k=1 δ(x1−xk 1, x2−xk 2),(4.8) where Cij is the total number of counts in the camera pixel ( i, j ), N the number of launched MC markers, and ( xk 1, xk 2 )the position of the marker, which is selected randomly along the pixel size via a uniform distribution. Employing the strike map, this distribution along the pixel is directly transformed to the phase space: Sij(E, R) = Cij N N X k=1 δ(E−ψE(xk 1, xk 2), R −ψR(xk 1, xk 2)),(4.9) from where the total signal per volume of the phase space can be written as: S(E, R) = X i,j Sij(E, R) = X i,j Cij N N X k=1 δ(E−ψE(xk 1, xk 2), R −ψR(xk 1, xk 2)).(4.10) If the process is done for a series of delta like camera frames, i.e., ideal camera frames such that Sij = 0 for all (i, j)except one pixel of the camera, the phase space covered by that particular pixel can be determined. Using this information, it is direct to define a 4D matrix, Tij αβ, the translation matrix, such that: Sαβ =Tij αβSij,(4.11) where Sαβ is the signal per unit of gyroradius (energy) and major radius (the discretization on a uniform grid of S ( E, R )from equation (4.10)), Sij is the camera frame and αβ run over the gyroradius-radius space while ij over the pixels. Einstein summation criteria is assumed. These matrices can be pre-computed and stored for each strike 37 Synthetic INPA diagnostic Fig. 4.7 INPA calibration frame. A chess-pattern image was placed in the scintillator plane. The scintillator and carbon foil contours are overlaid in white. In green, the chess-pattern image generated with the calculated optical parameters. Adapted from [87]. map, reducing the problem of transforming from the camera space to the phase space to a simple product of matrices. This new technique allows to eliminate the jitters introduced in the remapping when just considering the coordinates of the pixel centroid, an example of which can be seen in figure 12(a) of [ 57 ]. As the INPA code was combined with the FILD code, this feature is now also readily available for FILD data analysis. 4.4.6 Scintillator to camera relation To relate points in the scintillator with points in the camera sensor, a one parameter distortion model is applied: rCCD|D=rc+ (1 + D)(rCCD −rc),(4.12) where rCCD|D is the distorted position in the camera sensor (in pixels, px), rc is the position of the optical axis in the camera sensor (in pixels), and D and rCCD are the distortion factor and the position in the camera when there is no distortion, respectively. The distortion factor is calculated as: D=D1|rCCD −rc|,(4.13) rCCD =M←→ R(α)rS+r0,(4.14) with D1 being the parameter to model the distortion magnitude, M the magnification factor at the optical axis (in px/m), ←→ R a rotation matrix to take into account the rotations of the image caused by the mirrors and camera sensor, rS the position in the scintillator plate (in m) and r0 the position of the reference point of the scintillator in the camera (in px). The constants M , α and D1 are estimated from the optical modeling and then fine tuned to fit the calibration images. These images are taken during the installation/post-campaign calibration of the detector. An example of a calibration image is shown in figure 4.7. The field-of-view does not include the corners of the scintillator, but no signal is expected in that region of the plate. 38 4.4 The FILDSIM code 4.4.7 Forward problem: Signal calculation The calculation of the synthetic signal, the so-called forward problem, can be done via two procedures: direct tracking of the FIDASIM markers or a matrix multiplication of the fast-ion distribution function and the INPA weight function. In the former procedure, FIDASIM markers are tracked until they collide with the collimator or carbon foil. The weights and energy of the markers colliding with the carbon foil are adjusted following the relations presented in sections 4.4.2 and 4.4.3, and they are tracked until they impinge on the scintillator. At this point, their strike position is translated from the scintillator to the camera sensor using the relations presented in section 4.4.6. As an alternative to the carbon foil and scintillator yields models, the weights of the FIDASIM markers can be adjusted based on an energy-dependent absolute calibration derived from the first INPA experimental signals (see section 6.6) Finally, a histogram on the pixel space is performed and the finite focus of the camera is applied as a convolution of the histogram in the pixel space with the appropriate Gaussian kernel. The latter procedure for the calculation of the synthetic signal in each camera pixel, Sij, is based on the expression: Sij ≡ZWij(ζ)F(ζ)dζ, (4.15) where Wij is the weight function of the diagnostic, F the fast ion distribution function and ζ stands for the set of variables sampled in the phase space. Assuming toroidal symmetry and averaging over the gyrophase, just four coordinates suffice to span the fast-ion phase space. For example, as introduced in section 2.6: ( R, z, E, λ ). However, the dependency of the weight function on λ and z is solely originating in geometry. Neglecting the dependency of this geometric factor on energy (which comes from the defocussing of the optics), it is possible to decouple this dependency as: Wij(R, z, E, λ)≡˜ Wij(R, E)G(z, λ|R),(4.16) where G ( z, λ|R )represents the diagnostic sensitivity in the ( z, λ )space for a given R and satisfies the normalization condition: ZG(z, λ|R)dzdλ= 1.(4.17) Despite of the exact analytic form of G, it is direct to write equation (4.15) as: Sij =Z˜ Wij(R, E)ZG(z, λ|R)F(R, z, E, λ)dλdzdEdR, (4.18) but, owing to the fact that G ( z, λ|R )satisfies the normalization condition presented in equation (4.17), the latter is just the fast-ion distribution function weighted averaged 39 Synthetic INPA diagnostic λ (b) (c)(a) R =1.75 m σz[m] σλ 0.2 0.09 0.04 0.06 0.08 1.6 1.8 R [m] 2.0 1.6 1.8 R [m] 2.0 0.10 0.11 0.12 -0.5 0.0 z [m] 0.5 0.4 0.8 0.6 Fig. 4.8 INPA sensitivity in λ and z space. (a) Example at R = 1 . 75 m. (b) Width of the marginal distribution in z , as function of R . (c) Equivalent for the case of λ . Calculated for shot #40284, t= 1.17 s. on (z, λ)over the INPA field of view at a point R. Hence: Sij =Z˜ Wij(R, E)⟨F⟩INP A (E, R)dEdR. (4.19) An example of the numerically calculated G ( z, λ|R ), and the width of the marginal distributions, which are a function of R , can be seen in figure 4.8(a) and (b-c), respectively. Taking a discrete and uniform grid to represent the phase space, equation (4.19) can be written as a matrix problem: Sij =˜ Wαβ ij ⟨Fαβ⟩INP A ∆E∆R, (4.20) where Einstein’s summation criteria is assumed. As in equation (4.11), Greek letters span the phase space while Latin indexes run over the camera pixel space. The same equation can be written for the case that the camera signal is remapped using the strike map, as explained in the previous section. As the grid is finite, the two indexes can be collapsed into one: k=i+j#i,(4.21) γ=β+α#β,(4.22) where # i are the total number of rows of the signal matrix and # β the total number of rows in the fast-ion distribution matrix. Hence, equation (4.20) reads as: Sk=˜ Wγ k⟨Fγ⟩INPA ∆ζ≡˜ W′γ k⟨Fγ⟩INPA ,(4.23) where in the last step, the constant grid spacing was absorbed in the weight. It is possible now to further decompose ˜ W′ ij ( R, E )for its numerical calculation. Assuming the signal was remapped into the energy and radius space: Sγ=˜ W′ξ γ⟨Fξ⟩INPA =Tk γOp kΞξ p⟨Fξ⟩INPA ,(4.24) 40 4.4 The FILDSIM code (a)1.6 1.8 2.0 R [m] 1.6 1.8 2.0 R [m] (b) RS= 1.85m ES= 60keV 0.0 0.2 0.4 0.6 0.8 1.0 W [a.u.] 100 80 60 40 20 E [keV] Fig. 4.9 Example of weight function of the INPA diagnostic. (a) Gross weight of the INPA diagnostic. (b) Weight for a pixel centered at R = 1 . 85 m and E = 60 keV. Calculated for #40824 and t= 1.17 s. where the indexes have been collapsed in a similar manner to equation (4.22) and the different matrices are: •Tk γ (or Tij αβ with no collapse of indexes), the translation matrix, presented in the previous section. It relates camera pixel position with points of the phase space. It only depends on the diagnostic characteristics (foil energy loss and ionization rate, scintillator yield and detector geometry) and the magnetic field orientation at the head. •Op k (or Oab ij with no collapse of indexes), the optical model. It relates strike points in the scintillator with their position in the camera sensor, this matrix includes the finite focus and distortion of the optics and the overall transmission factor. • Ξ ξ p (or Ξ ητ ab with no collapse of indexes), the probability matrix. It contains the probability of a cell of the phase space given by ξ , to contribute at the emission in a point p of the scintillator. It depends on the plasma parameters and scales as presented in section 7.2.1. Tij αβ , can be fully determined as explained in the previous section. The distortion model included in the matrix Oab ij is adjusted to calibration frames as also described in the previous section and the absolute transmission factor is applied following the relation obtained in the first AUG INPA measurements during MHD quiescent phases, as shown in section 6.6. The last matrix is obtained from the synthetic signal where a delta fast-ion distribution function is used as input. This latter calculation is equivalent to the calculations done for other fast-ion diagnostics [57, 111]. In order to avoid numerical jitters coming from a finite number of MC markers in the FIDASIM simulation, the weight functions for each pixel of the phase space are fitted to bivariate Gaussian distributions. The gross weight of the diagnostic [ 110 ] (the sum of all ˜ Wij presented in equation (4.20)), calculated for the plasma profiles typical of a fast-ion experiment, can be seen in figure 4.9(a). Notice how the weight is heavily biased towards the larger radii, due to the NBI neutral density (see figure 2.9), 41 Synthetic INPA diagnostic 1.6 1.8 2.0 R [m] 20 30 40 50 60 70 80 E [keV] (a) Synthetic Signal [a.u.] 20 40 60 80 E [keV] 0.0 0.5 1.0 (b) Cut at 1.9 m [a.u.] Direct FIDASIM Matrix product 1.6 1.8 2.0 R [m] (c) Cut at 48 keV [a.u.] Direct FIDASIM Matrix product 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 4.10 Comparison of the synthetic signals calculated with a full Monte-Carlo simulation and using the weight function. (a) 2D synthetic signal. In color, the Monte-Carlo simulation, in gray contours, the signal calculated with the weight function. Dashed green lines indicate the performed cuts along the synthetic signal (shown in subplots (b) and (c)). (b) Energy distribution of the synthetic signal at R = 1 . 9m. (c) Radial distribution of the synthetic signal at E= 48 keV. and towards higher energies, due to the scintillator emission, transmission through the plasma and efficiency of the carbon foil. At large energies (above 80 keV), it starts to decrease because of the CX cross sections, as shown in figure 2.8. The weight for the scintillator position related to R = 1 . 85 m and E = 60 keV (with a width in radius of 1.5 cm and 2 keV in energy) can be seen in figure 4.9(b). As expected, only a small portion of the phase space contributes to that pixel. The width of this weight in energy and radius space determines the diagnostic resolution. The synthetic signals calculated with both methods are depicted in figure 4.10. Except for a minor deviation of the signal level near the core ( R∼ 1 . 75 m), due to the finite grid size, the agreement of both methods in terms of signal output is excellent. Both methods are equivalent also in computation time, as the calculation of the weight matrix requires a full FIDASIM simulation. Full details on the plasma profiles and used distribution function will be presented in chapter 7 when exploring the sensitivity of the INPA diagnostic against changes in plasma profiles. This weight function formalism of the synthetic diagnostic can be used for both the calculation of the neoclassical synthetic signal or the signal under the effect of a plasma perturbation. For the calculation of the neoclassical synthetic signal, the TRANSP distribution function can be used in equation (4.24). For the calculation of the synthetic signal in a situation where plasma perturbations are present, the neoclassical distribution function is first evolved in ASCOT under the effect of that perturbation and later introduced in equation (4.24). 42 Chapter 5 Design and simulations of the ASDEX Upgrade INPA diagnostic In this chapter the design and expected performance of the AUG INPA diagnostic are presented. The chapter begins with the selection of the best position to place the INPA diagnostic in order to match the phase space of the target fast-ion distribution functions. Afterwards, the mechanical design and resolution are presented. The chapter finishes with a detailed exploration of a synthetic INPA signal. The findings and developments presented in this chapter were partially published in [88, 109]. 5.1 Phase-space match The placement of the INPA diagnostic within the AUG tokamak involves a balance of four key considerations: the available space within the tokamak, the achievable phase-space resolution, the signal level, and the phase-space matching with the desired fast-ion distributions. The position of the INPA diagnostic in the AUG setup should be such that the pitch profile examined by the diagnostic matches the pitch profile of the slowing-down distributions generated by NBI#8 and #6. These specific NBIs were selected due to their capacity to produce on- and off-axis fast-ion distributions, respectively. Furthermore, they can induce toroidal Alfvén eigenmodes [ 112 , 113 ]. These TAE modes have a significant potential to interact with the fast-ion population, thereby causing notable redistribution and losses [ 68 , 114 ]. The fast-ion distributions of these NBIs are represented in figure 5.1. The off-axis distribution, figure 5.1(a), has its maximum in the LFS at R∼ 1 . 90 m and λ∼ 0 . 7. The on-axis distribution has a maximum in the core region, R∼ 1 . 70 m, at λ∼ 0 . 5. These are the target values set for the selection of the head position. A scan was carried out to select the position that best matches the required pitch profile. In a first stage, five locations along the toroidal direction are chosen, spanning the two closest sectors to the NBI diagnostic line, and five poloidal locations were 43 Design and simulations of the ASDEX Upgrade INPA diagnostic FWHM [cm] 0.0 0.2 0.4 0.6 0.8 1.0 Markers [a.u.] Fig. 5.10 Radial resolution of the INPA. (a) Comparison of the real radial position of the MC marker (y-axis) with the synthetic one as determined using the scintillator strike position (x-axis). The vertical dashed line represents an example of the cut done to estimate the radial resolution. The horizontal ones indicate the FWHM of the vertical cut. (b) FWHM of the cuts, along vertical lines, of the distribution in (a). The dashed line represents a quadratic fit to the data. There is no a priori reason for this behavior, it has been included just as a guideline for the eye. This is only valid for the active component of the signal. Calculated for shot #30585 and t= 1.25 s. Adapted from [88]. radius where they intercept the active source of neutrals, so the radial dispersion is minimized. The use of several NBI sources from the same NBI system can produce a degradation of the radial resolution. Nonetheless, the field of view of INPA is constrained enough so other beam sources have a small contribution, as depicted in figures 5.11 and 5.12. In figure 5.11, the contours of the NBI deposition are represented together with the birth position of CX neutrals, for the case of an on-axis fast-ion distribution function. Notice that even if the neutral distributions significantly differ, the actual birth position of the signal does not shift significantly nor does it become wider. This effect can be seen in more detail in figure 5.12, where the distribution of markers which fall into the strike map position R = 1 . 80 m is plotted for the case of the different NBI configurations. As can be observed, there are no significant changes, only a shift smaller than a centimeter when the NBI#4 is also active. Hence, the activation of other beams from box 1 does not hamper the INPA operation nor does it degrade its resolution. 5.4 Characteristics of INPA signals Three different components contribute to the INPA signal; each one coming from the interaction of the fast-ion population with a different source of neutral particles: NBI, halo and thermal neutrals. The contributions from NBI and halo are termed active, while the thermal is termed passive. These contributions are shown in figure 5.13, where an example of synthetic signal is presented. The simulation was performed for a core plasma electron density of ne (0) = 4 . 5 · 10 19 m −3 , in line with typical fast-ion 50 5.4 Characteristics of INPA signals 0 -40 -80 -120 y [cm] Neutral source: NBI#3 (b) (a) Neutral source: NBI#3 100 140 180 220 x [cm] Neutral source: NBI#3 & #2 (c) 100 140 180 220 x [cm] Neutral source: NBI#3 & #4 (e) 100 140 180 220 x [cm] Neutral source: NBI#3 & #2 (d) Neutral source: NBI#3 & #4 (f) 100 50 0 -50 -100 z[cm] 100 50 0 -50 -100 z[cm] 100 50 0 -50 -100 z[cm] 100 150 200 R [m] 1.0 0.8 0.6 0.4 0.2 0.0 100 150 200 R [m] 100 150 200 R [m] CXFlux [a.u.] Fig. 5.11 Birth position of the INPA signal. Colored contours represent the birth location of the neutrals measured at the INPA diagnostic. Grey scaled lines represent the density of NBI neutrals (the whitest line represents the contours of 75% of the maxima value and the darker one represents the 25%). Green dashed lines and crosses indicate the magnetic axis position and cyan lines the separatrix. (a-b) Simulation using only NBI#3, (c-d) the case of NBI#3 and NBI#2 while (e-f) the case of NBI#3 and #4. All cases were normalized to their respective maxima. Adapted from [87]. 0 1 2 3 1.75 1.80 1.85 R [m] Signal [a.u.] NBI#3 NBI#3 & #2 NBI#3 & #4 Fig. 5.12 Effect of multiple NBIs on the INPA signal. Distribution of birth radial position arriving to the strike map position related to R = 1 . 80 m. Dots represent the simulation data, while the solid line represents the performed fit to estimate the resolution. 51 Design and simulations of the ASDEX Upgrade INPA diagnostic TRANSP (a) Mag. Axis Separatrix Synthetic INPA (b) Mag. Axis Separatrix 60 40 20 E [keV] (c) (d) 1.0 0.5 0.0 Signal [a.u.] 1.6 1.8 2.0 R [m] 1.6 1.8 2.0 R [m] 1.6 1.8 2.0 R [m] 20 40 60 Es [keV] NBI Halo Passive Total NBI Halo Passive Total 1.0 0.5 0.0 Fig. 5.13 Example of synthetic signal for the AUG INPA diagnostic. (a) TRANSP fast-ion distribution function, integrated in λ and z coordinates. (b) Synthetic camera frame remapped in energy and radius. (c) Contributions to the signal per unit of radius, as reconstructed in the INPA scintillator. The legend indicates the population of neutrals with which the fast ion has interacted to produce the CX neutral. (d) Contributions to the signal per unit of energy, also as reconstructed in the scintillator. Es represents the energy of the particle impinging in the scintillator. Adapted from [87]. . experiments at the AUG tokamak [ 115 ] and a background neutral density with a maximum of 10 15 m−3 at the plasma edge, also in line with previous fast-ion studies [ 44 ]. The NBI fast-ion distribution function was calculated with TRANSP. It corresponds to NBI#3 (59 keV, on-axis source), as shown in figure 5.13(a). The complete synthetic signal is illustrated in subplot (b). As shown in subplot 5.13(c), the halo contribution turns out to dominate in the plasma core and is wider than the NBI contribution. The passive contribution is well below the active level and is apparently distributed along all possible radii (the relation between pitch and radial position can only be accurately established for CX neutrals originated along the NBI line [ 88 ]). For the case of the signal per unit of energy, depicted in subplot 5.13(d), the contribution is dominated by the active component at high energy, while the passive signal becomes significant at low energy, as neutrals coming from the edge suffer almost no attenuation due to re-ionization. 52 Chapter 6 Commissioning of the ASDEX Upgrade INPA diagnostic In this chapter, the first measurements of the AUG INPA diagnostic are presented, along with the characterization of the diagnostic and the benchmark of the synthetic diagnostic. The findings and developments presented in this chapter were partially published in [87]. 6.1 Resolution of the INPA diagnostic In section 5.3.1 the energy resolution of the INPA diagnostic was introduced. However, any spreading caused by differences in the energy loss due to thickness tolerance of the carbon foil or any optical non-ideality, such as limited camera pixel size, finite focusing or distortion, was not included in the calculation. As it is not possible to determine experimentally the precise thickness of the foil in each of its points, nor to account for possible deposition or oxidation during the campaign, the experimental signal must be analyzed to obtain the full energy resolution. A discharge with a MHD-quiescent phase, where NBI#3 and #8 were active, was taken as reference and the high-energy peak of the signal (93 keV) was fitted to a Gaussian. To avoid including the slowing-down tail, only the region above the injection energy was used, as illustrated in figure 6.1(a). The outcome of the fitting procedure yields σexp = 9 . 2 ± 0 . 3keV. This value of 9.2 keV contains both the intrinsic broadening of the fast-ion distribution, due to collisions with the thermal plasma [ 33 ], and the diagnostic resolution. The intrinsic broadening of the fast-ion distribution can be estimated performing the same fit on the neoclassical distribution calculated with TRANSP. This yields a value of σIntrinsic = 3 . 2keV. Provided that this is statistically independent from the INPA energy resolution: σE=qσ2 exp −σ2 Intrinsic = 8.5 keV (6.1) 53 Commissioning of the ASDEX Upgrade INPA diagnostic 20 40 60 80 100 Es [keV] 0.0 0.2 0.4 0.6 0.8 1.0 (a) INPA signal [a.u.] Excluded from fit Used to fit Fit 02 4 6 8 10 σo p tics [px] 6 8 10 12 14 σE[keV] with Distortion w/o Distortion (b) Fig. 6.1 INPA energy resolution. (a) Experimental INPA measurement and fit to estimate the INPA energy resolution. #40415, t= 6.86 s. NBI#3 and NBI#8 were active. (b) Effect of optical resolution on the INPA energy resolution. Blue dots represent the calculation performed with the measured level of distortion, red markers the case without any distortion. In both cases, the calculation was done for 93 keV ions and include the finite size of the camera pixels. The horizontal dashed line represents the experimental energy resolution, calculated as indicated in subplot (a). Adapted from [87]. This value surpasses the geometric energy resolution of 6.1 keV for 93 keV ions. As mentioned before, the tolerances in foil thickness, the scattering induced by the foil and the optical finite focus and distortion account for the difference. Ultimately, these effects can be approximated as a Gaussian broadening, hence, they can be consolidated into an effective optical resolution, σpx , i.e., the number of pixels over which a signal from a point-like source in the scintillator is distributed. In figure 6.1(b), the energy resolution of the system, calculated with FILDSIM, is depicted as a function of optical resolution. According to the empirical energy resolution, the value of the effective optical resolution is around 4.5 pixels. 6.2 Detector characterization 6.2.1 Background noise Two primary sources of background noise may impact the diagnostic signal. The first comes from the acquisition system: read-out noise and dark current in the case of the camera. The second arises from plasma light potentially entering through the pinhole and the openings in the optical head, which are necessary for out-gassing. Other noise sources, such as neutrons or electrons capable of inducing spurious scintillation, are neglected. This is deemed reasonable since electrons and neutrons make negligible contributions to noise in analogous diagnostics like the FILD diagnostic at AUG, which employ the same scintillator material. The primary source of camera noise is coming from the operation temperature of the camera and can be characterized using the initial frames recorded during each discharge. 54 6.2 Detector characterization ⟨ ⟩ 6000 5500 5000 4500 0.0 1.0 2.0 3.0 Time [s] 2 4 6 8 Signal NBI#3 Shutter open Shutter close Fig. 6.2 INPA noise contributions. Mean number of counts of the whole INPA signal during discharge #40259. Green shaded areas represent the periods where the INPA shutter is open. Vertical dashed lines represent the time points referenced in the text. The camera exposure time was 20 ms. Adapted from [87]. This noise can subsequently be subtracted from the remainder of the shot, as explained in [59]. To assess the noise due to plasma light entering through the out-gassing holes, the diagnostic shutter was opened and closed during the discharge #40259. In figure 6.2, the average count values recorded by the INPA camera during this discharge can be observed. At the early stage of the discharge (indicated with a red dashed line in figure 6.2), with no fast-ion source present, the count rate was 4275 counts per pixel. This corresponds to approximately 250 000 photons per second per pixel, considering the exposure time (20 ms), the camera quantum efficiency (0.35 electrons per photon), and the electron multiplication factor (2.45 counts per electron). The noise in a phase with plasma and where the shutter was closed (indicated with a magenta dashed line), t∼ 2 . 5s in figure 6.2, was of 4300 counts. This implies that only around 25 counts (equivalent to roughly 1400 photons per second per pixel) are coming due to light entering through the out-gassing holes, rendering this contribution negligible. When the shutter was open and no NBI#3 was active (indicated with a black dashed line), the measured count rate was 4400 counts. This suggests that the noise attributed to light entering through the pinhole amounted to approximately 100 counts (approximately 5800 photons per second per pixel). This last value includes passive contributions as well; for more details, please refer to section 6.4. In the context of this setup, a typical active INPA signal reaches approximately 35 000 counts at its peak. 6.2.2 Energy loss in the carbon foil The energy loss within the carbon foil depends on the energy of the incident particle. Figure 6.3 provides the calculated energy loss using the INPA signal from MHD- quiescent phases. For the calculation, the energy of the recorded peak at the scintillator is compared with the nominal energy from the NBI injection. The shaded area in 55 Commissioning of the ASDEX Upgrade INPA diagnostic 20 40 60 80 100 Energy [keV] 0 2 4 6 8 10 12 14 16 |ΔE| [keV] Fig. 6.3 Experimental energy loss in the carbon foil. Markers represent the measured energy loss while the dashed line the linear fit. The shaded area represents the uncertainty in the energy loss, assuming a 10% of uncertainty in foil thickness. Adapted from [87]. figure 6.3 represents the uncertainty, which is assumed to come from the uncertainties in the thickness of the graphite foil. The latter are assumed to be around 10%, as reported in [ 108 ], which use carbon foils from the same manufacturer. As it can be observed, the energy loss is well described with a linear dependency. This fitted linear dependency is included in the synthetic diagnostic. 6.3 INPA response to different fast-ion distributions To illustrate the ability of the INPA diagnostic to capture various fast-ion distributions, NBI and ICRH were used to tailor the fast-ion population. Figure 6.4 shows signals obtained during these experiments. Frames (a-c) correspond to a discharge with -2.5 T magnetic field on axis, while frame (d) corresponds to a -2.0 T scenario. All signals were normalized to their maximum peak values, with the noise estimated from the initial frames of the discharge and subtracted from the frames. In subplot (a), the signal for a case where only NBI#3 is activated is shown. NBI#3 is the primary source of neutrals for the diagnostic, hence, this is the baseline signal for the INPA diagnostic. Note that the signal is centered near the magnetic axis (located at R = 1 . 75 m in this discharge) and consists of two distinct energy components, representing the full and half energy components of NBI. The third energy component is less discernible due to several factors: its lower energy leads to increased re-ionization of the CX neutrals along the path to the detector, and both the ionization efficiency of the carbon foil and the scintillator yield are reduced at lower energies [ 14 , 59 , 108 ]. This radial localization aligns with expectations for NBI#3 since this beam generates on-axis fast-ion distributions [ 116 ]. At the experiment magnetic field, the NBI#3 full injection energy (59 keV) corresponds to 2.5 cm. Figure 6.4(b) corresponds to a scenario where the off-axis beam, NBI#6, is activated. The corresponding fast-ion population is detected around R= 1.95 m with a gyroradius around 3 cm. At the experimental magnetic field, 3 cm corresponds to 56 6.4 Passive signal characterization 200 300 400 Pixel 100 150 200 250 Pixel #40412 t = 2.88 s NBI#3 + NBI#6 (b) R [m] Gyroradius [cm] 1.5 2.2 1.00 3.75 NBI#3 NBI#6 Signal [counts / max(counts)] Signal [counts/max(counts)] Signal [counts/max(counts)] 200 300 400 Pixel #40412 t = 2.22 s NBI#3 (a) (e) (f) R [m] Gyroradius [cm] 1.5 2.2 1.00 3.75 NBI#3 100 150 200 250 Pixel #40415 t = 6.86 s NBI#3 + NBI#8 (d) R [m] Gyroradius [cm] 1.5 2.2 1.00 3.75 NBI#3 NBI#8 #40412 t = 3.86 s NBI#3 + ICRH (c) 1.00 R [m] Gyroradius [cm] 1.5 2.2 3.75 NBI#3 ICRH 250 200 150 100 Pixel 250 200 150 100 Pixel 200 300 400 Pixel 200 300 400 Pixel 1.0 0.8 0.6 0.4 0.2 0.0 R = 1.75 m R = 1.75 m R = 1.75 mR = 1.75 m 50 100 150 E [keV] 1.6 1.8 2.0 2.2 #40412, t=2.22s #40412, t=2.88s #40412, t=3.86s #40415, t=6.86s 0.0 0.5 1.0 0.0 0.5 1.0 R [m] Fig. 6.4 Example of INPA signals. All frames were normalized to their maxima. The spacing in the R direction of the map is uniform (0.05 m) as well as in the gyroradius [energy] direction (0.25 cm). (a-c) Are taken at -2.5 T on axis, (d) at -2.0 T. The approximate location of the magnetic axis is R = 1 . 75 m, this position is highlighted in cyan. (a) Baseline scenario, contribution of NBI#3, the diagnostic active source of neutrals. (b) Measurement with off-axis NBI#6 source. (c) Case where ICRH is activated. (d) Example of measurement with NBI#8 (AUG high energy on axis beam). (e) Integral of the signal along the radial direction, (f) integral of the signal along the energy direction. Adapted from [87]. the maximum injection energy of 93 keV. Subplot (c) illustrates the situation in which ICRH (36.5 MHz, corresponding to the cyclotron frequency of hydrogen at R = 1 . 72 m, on-axis) is activated together with NBI#3. The signal is similar to that in frame (a), but it exhibits a fast-ion tail extending to larger gyroradii (higher energies), as clearly seen in black in subplot (e). The final frame, subplot (d), illustrates a case in which NBI#8 was activated, this is an on-axis 93 keV beam source. The high-energy peak is prominently visible, while the second and third energy components are mixed with the full and half energy components of NBI#3, owing to the finite resolution of the system. 6.4 Passive signal characterization Thermal neutrals can make a contribution to the INPA signal, as they can penetrate the plasma and also undergo CX reactions with the fast ions. This particular signal is referred to as the passive contribution. For the AUG INPA diagnostic, the passive contribution tends to be minimal and typically falls below the noise threshold. This happens because the overlap between the measurable fast-ion distribution and the thermal neutral density is usually small, as observed in the case of the FIDA diagnostic [ 116 ]. The passive signal becomes noticeable only in scenarios with low-density discharges (line-integrated density below 3 · 10 19 m−2 ), where thermal neutrals have a greater penetration. Figure 6.5 illustrates an example of this passive signal in a low-density discharge. It is important 57 Commissioning of the ASDEX Upgrade INPA diagnostic 100 200 300 400 Pixel 100 150 200 250 300 Pixel #40414 t = 0.68 s NBI#6Passive 100 200 300 400 500 Counts Fig. 6.5 Example of INPA passive signal. Discharge #40414, t = 0 . 68 s, off-axis NBI#6 injection. Adapted from [87]. to note that, in contrast to figure 6.4, there is no discernible localization in the signal along the radial direction because there is no active source of neutrals, and therefore, no inherent relationship between the direction of neutrals and their birth positions. The passive signal typically reaches a maximum of 500 counts (approximately 25 000 photons per second per pixel) in the camera signal, while the active signal for the same shot in a later phase of the discharge reaches a maximum of 20 000 counts (equivalent to 1 160 000 photons per second per pixel). Consequently, the peak of the passive contribution constitutes less than 5% of the peak of the total signal. 6.5 Evolution of the INPA signal during a discharge Both density and temperature significantly influence the INPA signal. As density rises, the signal is expected to be reduced: the beam penetration is reduced and there is a higher probability of re-ionization of the CX neutrals on their way to the detector head. Both factors scale exponentially with density [ 24 ]. Furthermore, as density increases, collisionality also rises, leading to a decrease in the fast-ion slowing-down time, further diminishing the INPA signal. The temperature impact is two-fold: firstly, the CX reactivity changes with temperature [ 24 , 27 ]; secondly, higher temperatures result in a longer fast-ion slowing-down time, leading to a larger fast-ion content in the plasma and, consequently, a stronger signal. Among these effects, the latter dominates, as fast ions have velocities well above those of thermal ions and well below those of the thermal electrons. Therefore, variations in temperature do not cause substantial changes in reactivity. This general trend can be seen in figure 6.6, where the signal for a -2.5 T discharge is shown, together with the plasma density, temperature and the NBI time trace. Between 1.2 and 2.2 s, when the density is fairly constant, the INPA signal follows the trend of the plasma temperature, and it decreases in the phases when density is increasing, as expected. Notice that the active signal can be seen up to ne(0.2) ∼8·1019m−3. 58 6.6 Absolute calibration of the synthetic diagnostic 1 3 5 1 3 5 INPA counts [a.u.] NBI Power [MW] 2.0 #40284 t = 1.0 s (c) #40284 t = 1.5 s (d) #40284 t = 2.0 s (e) #40284 t = 2.5 s (f) 0.0 0.5 1.0 1.5 Counts[104] 300250200 300250200 300250200 300250200 Pixel Pixel Pixel Pixel Pixel nIDA e(ρ= 0.2)[1019m−3] TIDA e(ρ= 0.2)[keV] 225 175 125 2 6 8 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Time [s] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Time [s] 0.2 0.6 1.0 (a) (b) 1 Fig. 6.6 Evolution of the INPA signal during a typical -2.5T discharge. (a) Density (left scale) and temperature (right scale) taken at ρ = 0 . 2, shaded area represents the confidence interval. (b) INPA signal (left scale), integral of the camera frame over the scintillator, and NBI time-traces (right scale). (c-f) Regions or interest were the signal is registered in the camera. Adapted from [87]. To illustrate the impact of these factors on the INPA signal, a series of three consecutive shots were conducted while maintaining a similar plasma shape. During these shots, variations in density and temperature were induced by scanning the current and heating levels. The INPA signal was integrated within an energy range of 48 to 65 keV and a radial range of 1.70 to 1.85 meters. This integration was carried out within 20 ms time windows when NBI#3 was the only active on-axis source. This specific range was chosen to focus solely on the peak originating from the primary injection energy of NBI#3. The resulting integral is shown in figure 6.7 and is plotted against the electron collisionality at the pedestal top, calculated as in [117]: ν∗ e= 0.0012qR0Zeffne[1019m−3] ϵ3/2(Te[keV])2,(6.2) where ϵ is the aspect ratio, R0 the position of the magnetic axis and the rest of variables were introduced in previous sections. All profile quantities are evaluated near the core ( ρ∼ 0 . 25). As the collisionality increases the INPA signal decreases. The spreading in the figure can be attributed to the different profile shape at each time point as well as the presence of some MHD fluctuations which cause fast-ion redistribution. 6.6 Absolute calibration of the synthetic diagnostic As detailed in section 4.3, the FIDASIM code produces as output the absolute neutral flux reaching the INPA foil (in particles per second), therefore, a direct comparison can 59 Tomographic reconstruction of INPA measurements minimize can be written as1: C(F|α) = ||WF −S||2 2+α||F||2 2,(7.1) where || ||2 , indicates the L2 norm of the array and α is the regression hyper-parameter. For the case of the Elastic Net regularization, the cost function is: C(F|α, l1) = ||WF −S||2 2+αl1||F||2 1+1 2α(1 −l1)||F||2 2,(7.2) where || ||1 indicates the L1 norm of the vector, and l1 is a second hyper-parameter which weights the L1 and L2 norms of the distribution. Being the penalty in the 0 th Tikhonov regularization proportional to ||F||2 2 , sparse solutions of the inversion will be penalized. Therefore, 0 th Tikhonov inversions are expected to be smooth. For the Elastic Net regularization, the presence of the ||F||2 1 produces the opposite effect, it favors sparse solutions 2 . The actual performance of each inversion method is explored at section 7.3, which presents the inversions of synthetic signals. In both cases, the fast-ion distribution is obtained by minimizing the cost function: Fα= argminF(C(F)),(7.3) constrained to positive values of F, to avoid non-physical solutions. For the 0 th Tikhonov regularization, an analytical expression for the predicted signal (in the case of not imposing the restriction to positive solutions) can be easily found using the column vectors which diagonalize the weight matrix. As explained in [ 80 , 133 ], the weight function matrix, W, can be written as a product of 3 matrices: W=UΣVT,(7.4) where the columns of U and V are the eigenvectors of WWT and WTW respectively, and Σis a rectangular-diagonal matrix. The values of the diagonal elements of Σare the square roots of the non-zero eigenvalues of WWT . They are called singular values. With these matrices, the signal predicted by the 0 th Tikhonov regularization can be written as: STikhonov =WFTikhonov =N X i=1 U:i σ2 i σ2 i+αUT :iS,(7.5) 1 The notation for arrays was changed intentionally. ←→ a and a are reserved in this document to represent rotation tensors and 3D Cartesian vectors, respectively. In contrast, bold symbols are used to denote arrays in general. 2 The smoothness and sparseness of the solution obtained with both methods can be understood geometrically if the regression is seen graphically as the minimization of the ellipsoid of the cost function in the phase space. See [133] for full details. 66 7.2 INPA weight function where U:i are the columns of U and σi is the corresponding diagonal component of Σ. Hence, the best solution predicted by 0 th Tikhonov regularization is just the projection of the input signal in the column space of W , shrank by a factor σ2 i/ ( σ2 i + α ). Therefore, the regularization is simply a soft-thresholding [ 133 ]: the basis vectors of W for which σ2 i≪α are damped in the solution. For the case of the Elastic Net regularization, the presence of the ||F||2 1 makes the cost function to be not everywhere differentiable and an analytical predicted signal cannot be obtained; numerical calculations are needed. 7.2 INPA weight function This section introduces the INPA weight function, i.e., the sensitivity of the diagnostic to each portion of the phase space. Firstly, a simple analytical model will be derived, in order to visualize the main dependencies of the INPA sensitivity with plasma parameters. Afterwards, the singular values and eigenvectors of the INPA weight matrix will be presented, to determine the phase-space areas of larger INPA sensitivity. The section finishes with a quantification of the impact of the plasma profiles on the INPA weight function. 7.2.1 Zero order dependency The actual numerical procedure to calculate the weight function was already presented in section 4.4.7. The intention of this derivation is to illustrate how the weight function depends on plasma parameters. The number of fast CX neutrals per volume unit which are produced in a given point r is: ΦCX(r)≡ZZ σCX(vrel)vrelf2(vF,vn,r)d3vFd3vn,(7.6) where the subindex F refers to the fast ions while n to the neutral particles, σCX is the charge-exchange cross section, vrel the relative velocity between the fast ion and the neutral and f2(vF,vn,r) gives the number of fast ions and neutral particles with velocities vF and vn located at the point r , per volume unit. Considering the fast ions and the injected neutrals as independent species and neglecting the beam divergence this last term can be written as: f2(vF,vn,r) = F(vF,r)nn(r)δ(vn−v0),(7.7) where F ( vF,r )is the fast-ion distribution function, nn the neutral density and v0 the NBI injection velocity. Only one injection energy has been taken into account, the extension to the three energy components is trivial. If the halo is to be included, the only necessary change is to substitute the Dirac’s delta function by the appropriate Maxwellian distribution. With the Dirac’s delta function, the integration on the neutral 67 Tomographic reconstruction of INPA measurements Detector NBI Fig. 7.1 Representation of the INPA incoming flux. velocity is straightforward and yields: ΦCX(r) = ZσCX(vrel)vrelF(vF,r)nn(r)d3vF.(7.8) In equation (7.8), it is assumed that the relative velocity is calculated with the velocity of the neutral particle fixed by the injection geometry, v0 . Following the properties of the CX reactions, the outgoing CX neutral maintains the velocity of the initial fast ion. Therefore, only the region of the velocity space which points towards the pinhole must be considered. Converting the integral of equation (7.8) to spherical coordinates results in: ΦCX(r) = ZZ v2 FσCX(vrel)vrelF(vF,r)nn(r)dvFdΩvF.(7.9) As the NBI is located approximately one meter away from the pinhole, which has a radius of the order of few millimeters (1.5 mm for the AUG setup), the angular interval which is spanned by the integral is very small. Therefore, the integrand can be considered constant, evaluated at the central value, and taken out of the integration. This leaves the solid angle of the pinhole seen by the point r . This solid angle can be approximated as: ∆ΩvF=A 4π|r −rp|,(7.10) where A is the area of the pinhole and rp is its location. Introducing this in equation (7.9) and converting to energy-pitch coordinates results in: ΦCX(r) = ZσCX(vrel)vrelF(EF, λ0,r)nn(r)A 4π|r −rp|dEF.(7.11) Notice that F ( EF, λ0,r )is the fast-ion distribution function using energy and pitch as variables for the velocity space. The Jacobian of the transformation to the energy-pitch space was absorbed in this function. The notation λ0 indicates the pitch directed towards the detector pinhole. Figure 7.1, shows the relation of λ0 and vrel with the position: 68 7.2 INPA weight function λ0=−cos(β(r)),(7.12) vrel =|vF−v0|=s2 mE0+EF+ 2qE0EFcos(α(r)),(7.13) where the angles α and β are defined as shown in figure 7.1. Note, however, that this is not the flux reaching the detection region of the INPA diagnostic from a point r . A transmission factor, which accounts for re-ionization of CX neutrals along the path towards the pinhole, should be included, recovering equation (5.1): Φ(r)=ΦCX (r)e−|r−rp| d0(r).(7.14) Integrating this expression over all the volume probed by INPA (corresponding to the acceptance of the collimator) the total flux reaching the detection region, Φ T , can be calculated as: ΦT=ZV0 Φ(r)d3r = =ZV0 ZσCX(vrel)vrelF(EF, λ0,r)nn(r)A 4π|r −rp|e−|r−rp| d0(r)dEF!d3r. (7.15) Moving the origin to the pinhole and taking spherical coordinates, the expression simplifies to: ΦT=ZZZσCX(vrel)vrelF(EF, λ0,r +rp)nn(r +rp)A 4πe−|r| d0(r+rp)dΩdrdEF.(7.16) This Φ T is the CX flux entering the detection region. To translate this flux into photons reaching the camera, the ionization in the carbon foil, the scintillator efficiency and the absolute transmission of the optics must be taken into accounts. This joint dependency can be modeled as a quadratic term in energy, P2 ( E ), as explained in section 6.6. Hence, the photon flux reaching the camera would be: ΦTγ =ZZZσCX(vrel)vrelF(EF, λ0,r+rp)nn(r+rp)A 4πe−|r| d0(r+rp)P2(EF)dΩdrdEF.(7.17) From where the scaling of the INPA weight function read as: W∝σCX(vrel)vrelnn(r +rp)A 4πe−|r| d0(r+rp)P2(EF)(7.18) From this expression, it is clear that the overall response of the INPA diagnostic sensitivity to temperature is small. As changes in temperature only affect via changes in the re-ionization probability, contained in the mean free path d0 . Being the kinetic energy of fast CX neutrals well above the plasma temperature, moderate changes in Te implies only small changes in the mean free path of fast CX neutrals inside the 69 Tomographic reconstruction of INPA measurements (0) Σ = 1.00 (5) Σ = 0.47 (6) Σ = 0.46 (7) Σ = 0.40 100 80 60 20 Energy [keV] 100 80 60 20 Energy [keV] 100 80 60 20 Energy [keV] 40 40 40 1.5 1.7 1.9 2.1 R [m] 1.5 1.7 1.9 2.1 R [m] 1.5 1.7 1.9 2.1 R [m] 1.5 1.7 1.9 2.1 R [m] (1) Σ = 0.75 (2) Σ = 0.67 (3) Σ = 0.60 (4) Σ = 1.00 (8) Σ = 0.38 (9) Σ = 0.35 (10) Σ = 0.32 (11) Σ = 0.31 0.2 0.1 0.0 -0.1 -0.2 U:i [a.u.] Fig. 7.2 Eigenvectors of the INPA weight matrix. First 12 eigenvectors of the matrix U . The complete W was normalized to one by dividing by its maxima value. The written Σvalues correspond to σi/σ0. plasma. On the other hand, changes in electron density strongly affect the diagnostic signal. There is a double exponential dependency on density: the penetration of the beam neutrals into the plasma, nn , and the re-ionization of CX neutrals towards the diagnostic, e−|r| d0(r+rp) with d0 depending on density. Changes in main ion density will also affect exponentially to the weight function, as they affect the beam penetration and re-ionization probability. These changes can be considered via changes in Zeff , as for a fixed electron density, different Zeff implies different ion densities and different beam penetration and re-ionization probability. This is explored in detail in section 7.2.3. 7.2.2 Singular value decomposition of the weight function The eigenvectors corresponding to the first 12 singular values of the matrix WWT , presented in section 7.1, are depicted in figure 7.2. Notice that these first 12 eigenvectors are rather biased towards the high-energy, high-major radius part of the phase space, as expected seeing the overall weight shown in figure 4.9(a). The eigenvalues themselves are depicted in figure 7.3. 7.2.3 Dependence on plasma profiles In order to asses the effect of the plasma profiles on the INPA sensitivity a set of simulations where these profiles were varied was carried out. In each simulation, the 70 7.2 INPA weight function σ2 i 10-3 10-9 10-15 10-21 10-27 0 500 1000 Eigenvalue position Fig. 7.3 Square of the singular values of the INPA weight function. The eigenvalues are ordered in descending order when the diagonalization of the matrix is performed. 0 2 4 6 8 n [1019m-3] ne ni nimp × 10 0.0 0.2 0.4 0.6 0.8 1.0 ρt 0.0 0.2 (a) (b) 0.4 0.6 0.8 1.0 ρt 0 1 2 3 4Te [keV] Ti [keV] Zeff #40284 t = 1.17 s Fig. 7.4 Initial plasma profiles for the sensitivity scan. (a) Density profiles for electrons, ions and impurities (assumed to be carbon). Temperature and Zeff profiles. Discharge #40284, t= 1.17 s. magnitude of the electron density, temperature and Zeff profiles were independently varied while ∆ ni was extracted from quasineutrality. For the ion temperature the following relation was imposed: ∆ Te/Te = ∆ Ti/Ti . The original plasma profiles are shown in figure 7.4. The results from the scan can be seen in figure 7.5. Subplots (a-c) show the overall trend in the instrument response for different values of Zeff while subplots (d-i) show the projection of the gross instrument function when just one of the variables is varied. As shown in sections 6.2 and 7.2.1, changes in plasma temperature have a small effect on the diagnostic response (less than 5% even for a change of 25% in temperature). On the contrary, changes in the plasma density strongly affect the INPA response. Changes in Zeff have a moderate impact on the overall response of the INPA. This is because the changes in main ion density are related to the changes in Zeff via: ∆ni ni =−∆Zeff ZImp −Zeff ,(7.19) where ZImp represents the charge of the main impurity, assumed to be carbon, ZImp = 6. This implies that for moderate values of the effective charge, Zeff ∼ 2the relative changes in the main ion density would be: ∆ni ni∼∆Zeff 4 . Hence, a large relative variation in Zeff of 25% translates into a moderate change in the main ion density of about 10%. 71 Tomographic reconstruction of INPA measurements (d) (e) (f) (g) (h) (i) ΔTe[%] -0.25 -0.12 0.00 0.12 0.25 1e−4 ΔTe[%] -0.25 -0.12 0.00 0.12 0.25 1e−6 Δne[%] -0.25 -0.12 0.00 0.12 0.25 1e−4 Δne[%] -0.25 -0.12 0.00 0.12 0.25 1e−7 ΔZeff [%] -0.25 -0.12 0.00 0.12 0.25 1e−4 ΔZeff [%] -0.25 -0.12 0.00 0.12 0.25 (a) 25 25 0 0 -25 -25 ΔT [%] Δn [%] ΔZeff = -25.0% ΔZeff = 0.0%(b) 250-25 Δn [%] (c) 0 1 2 3 250-25 Δn [%] ∫W dΨ ∫W0 dΨ ΔZeff = 25.0% W [(#m2)/(#FIkeVs)]W [(#m2)/(#FIkeVs)]W [(#m2)/(#FIkeVs)] 0 2 4 6 1e−7 0 2 4 6 8 0.0 0.4 0.8 1.2 W [(#m)/(#FIs)]W [(#m)/(#FIs)] 0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2 0.0 1.0 2.0 W [(#m)/(#FIs)] 20 40 60 80 100 Energy [keV] 1.6 1.8 2.0 R [m] Fig. 7.5 Changes in the weight function due to plasma profile changes. (a-c) relative changes in the integrated INPA sensitivity as a function of the changes in temperature and density for each change in Zeff . (d-i) Projection of the instrument response in energy and radius axes. (d-e) are calculated for ∆ ne = ∆ Zeff = 0, (f-g) for ∆ Te = ∆ ne = 0, and (h-i) for ∆Te= ∆Zeff = 0. 72 7.3 Inversion of synthetic signals 7.3 Inversion of synthetic signals To demonstrate the capabilities of the INPA tomography, a series of inversions in increasing order of complexity are presented: 1. An inversion with a synthetic signal generated as S = WF and the same grid used for the signal and the reconstructed distribution. This case corresponds to the so-called inversion crime. The use of the weight matrix to calculate the signal and the use of the same grid for the forward and inverse problems makes the algorithm to perform artificially better [ 134 ]. This inversion is an example of the maximum possible performance of the INPA tomography via the 0 th Tikhonov and Elastic Net regularization techniques. 2. An inversion with a signal generated as S = WF but with a finer grid in the signal than in the inverted distribution, i.e., an over-constrained problem. This overconstrained grid will be the case for the experiments. Notice that the generation of the signal as S = WF will still imply that the inversion will be artificially better than the experimental inversions. This case shows the upper limit of the inversion performance in the case of over-constrained problems. 3. An inversion with the signal generated by an independent Monte-Carlo simulation compared to the simulation used for the weight matrix. It features different grids for the signal and reconstructed fast-ion distributions. This is the closest situation to an experimental case. It will reveal the actual performance of the inversion methods. Four fast-ion distribution functions in increasing order of complexity were used: (a) a distribution with a single peak in energy and ρp , (b) a distribution with a single peak in energy and a double peak in ρp , (c) a distribution with a double peak in energy and a single peak in ρp , and (d) a TRANSP slowing-down distribution, which is the closest case to the experiment condition. An overview of these distributions can be seen in figure 7.6. For the sake of brevity, only the distribution (b), which has the largest radial complexity, will be used for the first two inversion levels. The inversions of the rest of distributions will only be shown for the case of the independent Monte-Carlo simulations, the most realistic case. In all inversion examples, noise is added to replicate the experimental conditions. To mimic the noise the following model will be used: Snoise =Sexact(1 + kN),(7.20) being N a normally distributed random number with mean zero and variance one. With this definition, kis directly the inverse of the signal-to-noise ratio (SNR). 73 Tomographic reconstruction of INPA measurements FI [a.u.] FaFb Fc TRANSP #40284 (a) Fa Fb ρp Fa Fb Fc TRANSP #40284 (b) Fc TRANSP #40284 (c) (d) (e) (f) Magnetic axis Magnetic axis Magnetic axis Magnetic axis 1.0 0.8 0.6 0.4 0.2 0.0 20 40 60 80 1000 0.2 0.4 0.6 0.8 1.0 80 60 40 20 E [keV] 80 60 40 20 E [keV] 1.2 1.4 1.6 1.8 2.0 2.2 R [m] 1.2 1.4 1.6 1.8 2.0 2.2 R [m] 0.0 0.5 1.0 E [keV] Fig. 7.6 Test fast-ions distributions functions. (a) Energy profile of the used distributions. (b) Radial profiles as function of ρp . (d-f) Weighted average of the distributions in z and λ over the field-of-view of the INPA diagnostic. 7.3.1 Level 1 inversion As explained above, the first inversion level corresponds to the inversion of a synthetic signal generated with the weight matrix using the same grid for the forward calculation and the inversion. This will establish an upper limit for the inversion performance. The synthetic signal and the best inversions for each regression method can be seen in figure 7.7, subplots (b) to (k). For low noise, k = 0 . 00 or k = 0 . 01, the ground truth (the test fast-ion distribution function) is perfectly reconstructed for both inversion methods. For larger noise, k = 0 . 10, the HFS and central peaks are relatively well reconstructed, but the inversion in the LFS fails to yield a smooth solution. This is due to the large variance of the inversion in that region (a quantitative estimate is presented in section 7.3.4). Nevertheless, notice that three distinct peaks in the radial direction can be identified in the reconstructed distribution, which cannot be observed in the signal. Even in this noisy situation, the tomographic inversion is still revealing valuable information. The integrals of the reconstructed fast-ion distributions are compared 74 7.3 Inversion of synthetic signals to the ground truth in figures 7.8(a-b), for the intermediate case of 1% of noise, the expected level in experiments. Notice the excellent agreement in energy in all the energy range and in radius up to the magnetic axis. For major radius below 1.6 m, close to the limit of the INPA FoV, both inversion methods yield slightly different results, being the Elastic Net regularization the one with the worst results as it predicts a too thin distribution. The 0th Tikhonov method is essentially still matching the ground truth. 7.3.2 Level 2 inversion The second step is to invert a synthetic signal generated with the weight matrix, but using a different grid to generate the signal and to reconstruct the distribution. In this case, an over-constrained inversion (2.25 times more data points than unknowns) has been performed. This is the situation which will take place in the experiments, due to the high INPA resolution and the amount of pixels in the camera. The results are shown in figures 7.7(l-p) and 7.8(c-d). The case of zero noise is not shown, as it is essentially equivalent to the case of k = 0 . 01. Similar results of those of the level 1 inversion were found, demonstrating that the use of different grids does not imply an intrinsic problem for the INPA tomography. 7.3.3 Level 3 inversion The third and more realistic level is the inversion of synthetic signals generated with an independent Monte-Carlo simulation from the weight function calculation. In this case, also an over-constrained inversion was performed, the same grid as in the level two inversion was used. Notice that the synthetic signal coming from the MC simulations will have its intrinsic noise, on the order of a few percent, due to the finite number of markers in the simulations. The results for the case of 1% of added noise are shown in figures 7.9 and 7.10. In figure 7.9 each row shows the synthetic signal, the ground truth and the best inversion results for each test distribution. For the single energy on-axis fast-ion distribution (subplots a-d), both algorithms reconstruct perfectly the amplitude center and width of the distribution, as can be seen in the fits shown in subplots (a) and (e) of figure 7.10. For the second distribution function (subplots e-h of figure 7.9), the combination of on- and off-axis distributions, both methods agree perfectly in energy with the ground truth and match the central and outer peaks, but they disagree in the inner off-axis peak, as shown in figure 7.10(f). Both of them predict an off axis peak in the HFS of the right amplitude but centered ∼ 1 cm outwards compared to the ground truth. Also, they predict smaller values for the distribution function near R = 1 . 7m. For the case of the third test distribution, the radial location and width of the ground truth is well captured, as well as the energy location and width of the peaks. As depicted in subplot (c) of figure 7.10, the inversion underestimates the height of the low-energy peaks by 10 %. For the last test distribution, both inversion 75 Tomographic reconstruction of INPA measurements 1.6 1.8 2.0 R [m] 30 40 50 60 70 σ(F)/F k = 0.01 (a) (a) E [keV] 0.00 0.02 0.04 0.06 0.08 0.10 1.6 1.8 2.0 R [m] 30 40 50 60 70 E [keV] 0.0 0.1 0.2 0.3 0.4 0.5 σ(F)/F k = 0.10 (b) Fig. 7.13 Relative error in the inversion due to variance of the input synthetic signal. Calculated for a set of 100 inversions. Gray contour represents the average inversion of the ensemble. Only cells of the phase space with values larger than 5% of the maximum value were considered. for the case of the 10% variance input signal. The larger variance of the reconstruction in the LFS relates to the larger values of the INPA weight matrix in this region. 7.4 Inversion of experimental signals To show an example for the inversion of the experimental signal, discharge #40284 at t = 1 . 17 s was chosen. This discharge, corresponding to a BT = − 2 . 5T and Ip = 800 kA scenario, was selected because it is a representative case of moderate densities ( ne (0) ∼ 8 · 10 19m−3 ), quiescent MHD activity and on-axis NBI injection (both sources 3 and 8 were on at 2.5 MW). The signal analyzed here coincides with the one shown in figure 6.6. The plasma profiles were already shown in figure 7.4. The inversion results can be seen in figure 7.14. An overall agreement with the neoclassical distribution predicted by TRANSP can be seen in subplots (c-d), where the marginal distributions from TRANSP and the inversions are compared. The inferred energy profile agrees well with TRANSP, apart from the low-energy region (below 20 keV) and close to the NBI#3 injection energy (59 keV). The first can be explained due to uncertainties in the background neutral density profile introduced in TRANSP to calculate the neoclassical distribution. This background profile affects the CX losses of fast ions, which can reduce significantly the number of fast ions at low energies [ 135 ]. The second deviation (near 59 keV) is in line with the observations presented in section 6.6 when comparing the neoclassical synthetic signal with the INPA measurements. For the radial profile, the TRANSP prediction is more extended radially and has a smaller maximum value compared to the inversion. This is also in line with the observations discussed in section 6.6. 82 7.4 Inversion of experimental signals 80 20 60 40 E [keV] 80 20 60 40 E [keV] ⟨FI⟩ [cm-3] 0.0 0.4 0.8 1.2 Inversion TRANSP (a) 1.6 1.8 2.0 R [m] 0.0 0.5 1.0 20 40 60 80 100 E [keV] ⟨FI⟩ [cm-3keV-1] 0 8 6 4 2 (b) (c) (d) 1.5 1.7 1.9 2.1 R [m] 1.5 1.7 1.9 2.1 R [m] Fig. 7.14 Inversion of experimental data. (a) Neoclassical distribution predicted by TRANSP, for discharge #40284 at time 1.17 s. (b) Best inversion obtained with 0 th Tikhonov regularization. (c) Energy profile of the distribution. (d) Radial profile of the distribution. The shaded areas represent the uncertainties in the reconstruction. This uncertainties are estimated using different weight matrices for the inversions, calculated with different plasma density profiles inside the IDA uncertainty. 83 Chapter 8 Fast-ion transport in phase space In this chapter, two redistributions of fast ions in phase space measured with the INPA diagnostic are discussed. First, the diffusion of fast ions due to ICRH heating is presented. For the first time ICRH accelerated fast-ions are measured with an INPA diagnostic. This measurement is compared to modeling based on TRANSP/TORIC calculations. Afterwards, the fast-ion transport caused by toroidal Alfvén eigenmodes is analyzed and compared to the predictions based on the LIGKA and ASCOT codes. 8.1 ICRH induced energy diffusion One of the foreseen heating schemer for the ITER ICRH system is the 2 nd harmonic deuterium heating [ 136 ]. 1 This is because no strong resonance (energy absorption) can occur in a single species plasma at the ion cyclotron frequency. At this frequency, the cyclotron motion of ions short-circuits the circularly polarized wave which can efficiently transfer energy to the ions [ 137 ]. Thus, higher-order resonances are needed to heat the main ion species via ion cyclotron resonance heating. In the case of deuterium, the frequency of the second harmonic resonance coincides with the first harmonic frequency of hydrogen (see equation (2.5)), which is always present in a deuterium plasma a as residual species. Therefore, in a deuterium plasma, when using the 2 nd harmonic ICRH scheme, there is a direct competition in the ICRH power being deposited in the hydrogen population (via 1 st harmonic) and the deuterium population (via 2 nd harmonic). The absorption at the first harmonic resonance of a minority species is highly efficient because it does not suffer the same short-circuit mechanics as the main ion population. An approximate expression for the partitioning of power between deuterium and hydrogen, PD/PH, can be written as [33]: PD PH=βDnD nH,(8.1) 1 2 nd harmonic means that instead of using a wave at the ion cyclotron frequency to heat the plasma, a wave at twice the ion cyclotron frequency is used. 85 Fast-ion transport in phase space (a) (c) 0.8 Counts [a.u.] #40412 t = 3.5 - 3.9 s 1.0 0.6 0.2 0.0 0.4 60 100 140 Es [keV] Time to ICRH [s] 0.05 0.15 0.25 0.35 (b) For E = 120 keV 3 5 7 Time to ICRH [s] Counts to NBI peak 0.0 0.2 0.4 τ [ms] 60 50 40 30 80 100 120 Es [keV] Fig. 8.1 Evolution of the ICRH population measured with the INPA diagnostic. (a) Temporal evolution of the measured energy profile, normalized to the signal at the NBI injection energy. (b) Temporal evolution of the ratio of the signal at 120 keV with respect to the signal at the NBI injection energy. (c) Fitted constant τ for each energy, in blue the results for the first ICRH phase, between 3.5 and 3.9 s, in red the results for the second ICRH phase, between 5.6 and 6.0 s. A parabolic fit was added to guide the eye. Discharge #40412. Adapted from [ 87 ]. where βD is the ratio of the kinetic to magnetic pressure of the majority deuterium species, and nD and nH the deuterium and hydrogen densities, respectively. Hence, small residual hydrogen concentrations of few percents can absorb almost all the ICRH power. The absorption of power in the second harmonic scales with the Larmor radius [ 33 , 36 ]; hence, the larger the energy of the deuterium ions is, the more energy from the ICRH wave they will absorb. Thus, the energy absorption by deuterium will be more pronounced in the NBI FI population than in the thermal component. Experimental characterization of the acceleration of fast deuterium has been carried out in the last decades in different machines and with different diagnostics (see for example [ 33 , 98 ] and the references therein). Nonetheless, an agreement between measurements and simulations of the energy distribution measured by an NPA during NBI only and NBI + ICRH phases has not been achieved simultaneously at the AUG tokamak [ 98 ]. Thus, new measurements are needed for the code validations and their application to future reactor scenarios. The spatially localized measurements with good energy resolution of the INPA diagnostic makes this detector an attractive tool to further extend the measurements of this phenomena and perform the validation of the simulation tools. An example of the measurements of accelerated deuterium NBI ions was already shown in figure 6.4(b) when introducing the main fast-ion populations measurable by the INPA diagnostic. The temporal evolution of this measurement is shown in detail in figure 8.1(a). As the time evolves, the population of ions with energies larger than the NBI#3 injection energy (59 keV) grows. This growth is shown, for ions with an energy of 120 keV (twice the injection energy), in figure 8.1(b), where the temporal variation of the signal at this energy, normalized to the signal at the NBI injection energy, is depicted. The temporal evolution of this ratio, denoted here as η , (for all 86 8.1 ICRH induced energy diffusion Fig. 8.2 Comparison of the number of neutrons between experiment and simulation. The solid black line is calculated as the moving average of the experimental data from the neutron diagnostic, represented as a shaded gray collection of points. All data from the TRANSP simulations was scaled by 3.4. The percentage of the legend indicates the hydrogen concentration in the simulation. energies above the injection one) can be fitted with the following relation: η=A1−exp−t−tICRH τ,(8.2) where t−tICRH represents the time from the onset of the ICRH and A and τ are the two parameters of the fit. Both A and τ are energy dependent. The constant τ measures the timescale on which the ICRH ion population stabilizes via diffusion. This is the time scale in which the kick given by the wave compensates the drag by the bulk plasma. The corresponding values of τ for each energy are depicted in figure 8.1(c) for two different ICRH phases, characterized by varying plasma density and temperature and, consequently, collisionality. In order to compare the observed evolution with the model (TRANSP/TORIC) the precise amount of hydrogen must be known. There is no diagnostic at the AUG tokamak which can yield values of the hydrogen concentration with precision below 1%. Hence, a scan in the hydrogen concentration was performed in the simulation to find the best agreement. The first variable which was used to narrow down the level of hydrogen was the neutron emission. The comparison of the neutron emission obtained by TRANSP and the neutron emission measured by the AUG neutron diagnostic is shown in figure 8.2. A scaling factor was added in the simulated neutron time trace in order to match the measured flux. This is the typical procedure done in AUG studies [ 98 ]. The factor is calculated to best fit the neutron time trace in the early stage of the discharge where there is no ICRH nor strong MHD activity. As the fusion cross section from hydrogen are much smaller than the deuterium ones [ 3 ], the less power goes into deuterium (the more hydrogen concentration) the less neutrons are produced. From the neutrons, the level of hydrogen present in the plasma is between 0.5 and 1% (purple lines in figure 8.2). Note that this is the concentration input in TRANSP, which is defined as hydrogen 87 Fast-ion transport in phase space 60 80 100 120 140 10−4 10−3 10−2 10−1 100 5 % 1 % 4 % 0.5 % 3 % 0.25 % 2 % E xp Energy [keV] INPA signal [a.u.] Fig. 8.3 Synthetic INPA signals under the effect of ICRH. All signals were normalized to the NBI maximum peak. The black line shows the experimental INPA signal. Calculated for shot #40412, t= 3.88 s. 0.0 0.1 0.2 0.3 0.4 For E = 120 keV (b) Ratio to NBI peak [%] (a) 1 % 0.5 % 0.25 % 80 90 100 110 120 130 8 6 4 2 0 τ [ms] 90 70 50 30 Time to ICRH [s] Energy [keV] Fig. 8.4 Evolution of the TRANSP calculated ICRH tail. (a) Temporal evolution of the ratio of the signal at 120 keV with respect to the signal at the NBI injection energy. The percentage of the legend indicates the hydrogen concentration. (c) Fitted constant τ for each energy. Experimental data, from the first temporal interval in figure 8.1(c) was plotted as a reference in black. Color code is equivalent to subplot (a), the shaded areas represent uncertainties in the fit. density divided by the electron density and not as the ratio between deuterium and hydrogen densities. This hydrogen concentration of about 0.5-1.0% is also in line with the measured INPA energy profile. The comparison can be seen in figure 8.3. For values of the hydrogen concentration above 1%, the tail of the INPA signal at energies larger than the NBI injection clearly deviates from the INPA signal, yielding values up to two orders of magnitude below the measured ratio. Simulations performed with a hydrogen concentration equal or below 1% match well the tail at these energies. Notice that all simulations strongly decay above 140 keV due the finite number of markers used in the TRANSP/FIDASIM simulations. The temporal evolution of the accelerated deuterium population can be seen in figure 8.4 for the three TRANSP simulations with a hydrogen concentration below or equal to 1%. Figure 8.4(a) represents the ratio between the fast-ion distribution at 120 keV and 60 keV, averaged in the INPA FoV in ( λ, z )and integrated along radius. As 88 8.2 TAE induced fast-ion transport can be seen, this ratio follows a clear exponential growth, as the INPA signal depicted in figure 8.1(b). Although this corresponds to the evolution of the simulated fast-ion distribution itself and not the INPA signal, the time scale of its temporal evolution should match the measured signal. This temporal evolution was also fitted to the model shown in equation (8.2). The results from this fit are depicted in figure 8.4(b) where the experimentally determined τ from the INPA measurements is also shown. Within error bars, both simulations for 0.5 and 0.25% of hydrogen concentration agree with the measured τ and the case with 1% of hydrogen only differs around 10 ms from the measured time constant. This confirms the TRANSP/TORIC capabilities to reproduce the temporal behavior of the evolution of the fast-ion distribution within the ICRH phase. 8.2 TAE induced fast-ion transport Previous studies in the ASDEX Upgrade tokamak (AUG) made extensive use of the fast-ion loss detector (FILD) [ 9 , 17 , 18 , 50 , 68 , 69 , 113 – 115 , 138 ] and fast-ion D-alpha (FIDA) and neutron diagnostics [ 116 , 139 ] to study transport and losses caused by Alfvén modes. However, the FILD diagnostic is limited to the lost portion of the phase space, while FIDA integrates over large regions of it, so the local interactions in phase space of fast ions with the magnetic fluctuations cannot be fully explored. This investigation is essential to validate predictive modeling, which is key for the development of future actuators and control techniques. In this regard, the INPA capabilities of measuring simultaneously energy and radial position of fast ions in a narrow pitch range are a major advance. The visualization of the flows induced by reversed-shear Alfvén eigenmodes (RSAE) was performed for the first time at the DIII-D tokamak [ 140 , 141 ]. These flows were also successfully compared against numerical simulations [ 142 ]. Here the first visualization of the fast-ion flows under the effect of toroidal Alfvén eigenmodes at the AUG tokamak is discussed. 8.2.1 Experimental characterization of the TAE induced flow The studied scenario corresponds to an H-mode plasma 2 with toroidal magnetic field on-axis of BT = − 2 . 5T, a plasma current of Ip = 600 kA, a core electron collisionality of ν∗ e (0 . 25) ∼ 0 . 06, and a core safety factor of q (0) > 3; following well-established fast-ion scenarios in the AUG tokamak [ 115 , 145 ]. The temporal evolution of this discharge in the time range of interest can be found in figure 8.5. 2.5 MW of steady off-axis neutral beam injection (NBI) and 2.5 MW of ion cyclotron resonance heating (ICRH) are applied to generate a large ratio of energetic particle beta, βEP ≡ 2 µ0pEP /B2 , to the 2 The H-mode (high confinement mode) is a plasma regime with an edge transport barrier and a steep gradient in density and temperature profiles [ 143 ]. This confinement mode is also characterized by the existence of edge localized modes [144]. 89 Fast-ion transport in phase space 0 2 4 6ne(0.10) ne(0.85) 2 4 6NBI ICRH Prad 3.9 Te [keV] 4.0 4.1 4.2 4.3 Time [s] 1 2 3 Te(0.10) Te(0.85) 0.1 0.3 0.5 Energy [MJ] WMHD WFI (b) (a) 90 110 130 F [kHz] 3.9 4.0 4.1 4.2 4.3 Time [s] 90 110 130 F [kHz] (c) Magnetics (d) ECE (ρ ~ 0.5) f (n=3) TAE f (n=2) TAE Power [MW] ne [1019m-3] 1.0 0.6 0.4 0.2 0.8 Signal [a.u.] Signal [a.u.] 1.0 1.5 0.5 Fig. 8.5 Evolution of plasma parameters for #41090 near t = 4 . 0s. (a) Electron density (left axis) and temperature (right axis). (b) Injected and radiated power (left axis) and stored energy (right axis). (c) Spectrogram from a magnetic coil, B31-14, at the LFS. The vertical lines represent the time points selected for the comparison of INPA signals. (d) Spectrogram from an ECE channel near ρp∼ 0 . 5; the analytic frequencies (Doppler shifted) for the center of the TAE gap are plotted in dashed lines. thermal plasma one, βTh ≡ 2 µ0p/B2 , of the order of 1. This is depicted in figure 8.5(b), where the fast-ion stored energy is shown together with the thermal stored energy. The used ICRH frequency is 36.5 MHz, which at the experiment magnetic field corresponds to the cyclotron frequency of H at R = 1 . 72 m. On top of this, 40 ms on-axis NBI blips (NBI#3) are added for diagnostic purposes. Despite this large input power, the electron temperature remains low (below 2 keV) due to the large radiated power, also shown in figure 8.5(b). The off-axis NBI source, NBI#6, was tilted to the maximal off-axis injection angle of 6.95 o , maximizing the spatial gradient of the FI distribution, and hence achieving a stronger drive for the modes [ 146 , 147 ]. Up to t = 4 . 2s, during each beam blip, the plasma density peaks up to the same values (near 4 · 10 19m−3 in the core), while the temperature is gradually changing from blip to blip but less than 10% in the time range. This small increase should not have any impact on the INPA sensitivity as seen in chapter 7. Only a moderate increment in the INPA signal at the plasma core is expected, as predicted from the increase in the slowing-down time due to the electron temperature raise. Figure 8.5(c) shows the spectrogram from a coil located in the LFS while figure 8.5(d) shows the spectrogram associated with temperature fluctuations near mid-plasma, as measured by the electron cyclotron emission (ECE) diagnostic. On both diagnostics, fluctuations in the frequency range of toroidal Alfvén eigenmodes can be observed. Under the effect of these modes, neoclassical predictions do not hold, as fast ions interact with the TAEs and are redistributed [ 68 , 115 ]. To visualize this particle transport 90 8.2 TAE induced fast-ion transport ρp t=4.10 t=4.18 2.0 1.0 1.5 0.5 0.0 0.0 |ξ| [cm] 0.2 0.4 0.6 0.8 1.0 Fig. 8.6 Field line radial displacement caused by the TAE. Dashed line corresponds to n = 2 modes while solid line corresponds to the n= 3 component. caused by the TAE, two frames measured by the INPA diagnostic are compared: one without significant TAE activity (highlighted in magenta in figure 8.5(c)) and a second one with a large amplitude of the TAE, in a subsequent blip. The differences in the signal will correspond to the differences in the fast-ion distribution function, as the density profile is similar among them and hence, the INPA sensitivity will be equivalent. Two groups of TAEs can be observed, the ones destabilized by NBI#3 ( f < 110 kHz), which appear only when NBI#3 is activated; and the ones driven by NBI#6 and ICRH ( f∼ 120 kHz), which are present even in the absence of NBI#3. The focus will be put in the modes at around 120 kHz, which are present throughout the whole time range. The magnetic field line displacement, ξ , of the TAE mode can be extracted from the measurements of the perturbed electron temperature, δTe , observed by an electroncyclotron-emission radiometer thanks to the relation [148]: δTe Te =−(γ−1)∇ξ−ξ∇Te Te ,(8.3) where Te stands for the electron temperature and γ is the ratio of specific heats [ 149 ]. Neglecting the compressional term, ( γ− 1) ∇ξ , due to the shear nature of the TAE mode, the radial displacement can be obtained as: ξ≃ − δTe ∇Te .(8.4) This radial displacement is shown in figure 8.6. Two groups of peaks can be observed, one near the plasma core, centered at about ρp = 0 . 15, and another one off-axis, near ρp∼ 0 . 60 for the n = 3 mode and near 0.40 for the n = 2 mode. The toroidal mode number n was obtained from the phase correlations of the set of toroidally distributed magnetic coils in AUG, as done in [9, 150]. The INPA signals at the time points represented in figure 8.5(c) are plotted in figure 8.7(a-c). Subplots (d) and (g) show the electron density and temperature profiles, respectively. The differences in the signal (FI population) between the frames at 91 Summary and outlook This thesis contributes to the study of the fast-ion dynamics in tokamak plasmas. The confinement of fast ions is key for future fusion reactors, as they are an important source of energy (temperature) and momentum (current drive) for the system. A comprehensive characterization of their dynamics is crucial for the development of control techniques which will ensure the confinement of the fast-ion population in future reactors. The study presented in this thesis is based on the imaging neutral particle analyzer, which is able to probe the energy and position of the confined fast-ion population. This diagnostic complements the strong set of fast-ion diagnostics already present on the ASDEX Upgrade tokamak. The manufacturing, installation, characterization and physics exploitation of the AUG INPA diagnostic was carried out during this thesis. All the tools necessary to model the diagnostic response and analyze its signals in a systematic way were developed and benchmarked against experimental data. Based on the TRANSP and FIDASIM codes, a synthetic diagnostic has been constructed to predict the INPA signal during MHD-quiescent discharges. This synthetic diagnostic has been validated against experimental data during the first INPA experiments carried out at the ASDEX Upgrade tokamak. The synthetic diagnostic allows not only the understanding of the measured experimental signals but also the design of future INPA diagnostics, both for the ASDEX Upgrade tokamak and other magnetic fusion devices. The implementation of the matrix formalism in the synthetic diagnostic addresses the inverse problem, i.e., the extraction of the fast-ion distribution function from the measurements, and also allows us to deeply understand the variation of the diagnostic sensitivity with changes in plasma parameters such as density or temperature. A systematic study was carried out to explore this sensitivity, showing that the plasma density strongly affects the diagnostic response while plasma temperature merely makes an impact; this information is vital for the design of experiments to explore the fast-ion redistributions due to MDH fluctuations. Several fast-ion distribution functions with different complexity in radial and energy profiles were used to test the capabilities of the INPA inversions. These inversions, limited to the ( R, E )-space, proved to be robust 99 Summary and outlook even in the presence of large levels of noise (10%). Hence, the INPA diagnostic can yield significant information about the fast-ion distribution function even in cases of low signal-to-noise ratio, making it the ideal diagnostic to study fast-ion redistributions in phase space. When applied to experimental data, tomographic inversions show good agreement with neoclassical calculations during MHD-quiescent phases. The acceleration of deuterium NBI ions due to second harmonic ICRH has been studied for the first time with an INPA diagnostic. A good agreement of the energy distribution of the measured CX flux with the simulated one using the TRANSP/TORIC code is found. This serves as a further validation for the use of these codes for future scenario design. For the first time at the ASDEX Upgrade tokamak, the flow of fast ions in phase space has been directly measured. Fast ions are redistributed outwards while losing energy, as predicted by the conservation of E′ . Linear simulations based on the LIGKA and ASCOT codes suffice to reproduce the location and magnitude of the observed flow. The demonstrated capabilities open new avenues for experiments in the ASDEX Upgrade tokamak, to be carried out in the following campaigns after the long experimental break due to the installation of new shaping coils. The transport caused by a broad range of instabilities, not only by TAE as presented in this thesis, but by other Alfvénic fluctuations, could be systematically studied. The inclusion of a fast channel with PMTs will allow for the identification of the coherent fluctuations of the fast-ion population with the mode. This will add further benchmark capabilities for different codes and will help for extrapolation to future machines. The inclusion of a second INPA view, with a different pitch profile, could allow the diagnosis of a broader portion of the phase space, and hence obtain more details on the fast-ion dynamics. Along the same line, the development of new tomographic techniques, combining the FIDA and INPA diagnostics, can be used to reconstruct the fast-ion distribution with unprecedented precision. 100 Appendix A List of publications and contributions A.1 Publications as main author 1. J. Rueda-Rueda, M. Garcia-Munoz, E. Viezzer, P. A. Schneider, J. Garcia- Dominguez, J. Ayllon-Guerola, J. Galdon-Quiroga, A. Herrmann, X. Du, M. A. Van Zeeland, P. Oyola, M. Rodriguez-Ramos, and the ASDEX Upgrade team. Design and simulation of an imaging neutral particle analyzer for the ASDEX Upgrade tokamak. Review of Scientific instruments, 2021 2. J. Rueda-Rueda, M. Garcia-Munoz, E. Viezzer, P. A. Schneider, J. Garcia- Dominguez, P. Oyola, J.Galdon-Quiroga, J. Hidalgo-Salaverri, X. Du, M. A. Van Zeeland, M. Videla-Trevin, A. Jansen van Vuuren, and the ASDEX Upgrade team. Commissioning of the Imaging Neutral Particle Analyzer for the ASDEX Upgrade tokamak Plasma Physics and Controlled Fusion, 2024 3. J. Rueda-Rueda, M. Garcia-Munoz, E. Viezzer, P. A. Schneider, P. Oyola, J. Galdon-Quiroga, M. Salewski, B. Simmendefeldt-Schmidt, J. Garcia-Dominguez, and the ASDEX Upgrade team. Tomographic reconstructions of the fast-ion phase space using Imaging Neutral Particle Analyser measurements. Plasma Physics and Controlled Fusion. In preparation 4. J. Rueda-Rueda, M. Garcia-Munoz, E. Viezzer, P. A. Schneider, J. Galdon- Quiroga, J. Garcia-Dominguez, P. Oyola, L. Velarde, J. Hidalgo-Salaverri, X. Du, M. A. Van Zeeland, M. Videla-Trevin, J. Gonzalez-Martin, Ph. Lauber, T. Lunt, A. Herrmann, J. Ayllon-Guerola, and the ASDEX Upgrade team. Phase-space measurements of MHD-induced fast-ion transport in the ASDEX Upgrade tokamak. Nuclear Fusion, Submitted 101 List of publications and contributions A.2 Publications as coauthor 1. J. Galdon-Quiroga, G. Birkenmeier, P. Oyola, H. Lindl, A. Rodriguez-Gonzalez, G. Anda, M. Garcia-Munoz, A. Herrmann, J. Kalis, K. Kaunert, T. Lunt, D. Refy, V. Rohde, J. Rueda-Rueda, M. Sochor, B. Tal, M. Teschke, M. Videla, E. Viezzer, S. Zoletnik, and ASDEX Upgrade Team. First measurements of an imaging heavy ion beam probe at the ASDEX Upgrade tokamak. Review of Scientific Instruments, 2024 2. J. Hidalgo-Salaverri, J. Rueda-Rueda, J. Galdon-Quiroga, P. Cano-Megias, E. Viezzer, J. Garcia-Dominguez, D. Silvagni, M. Videla-Trevin, P. Oyola, J. Ayllon- Guerola, M. Garcia-Munoz, T. Happel and the ASDEX Upgrade Team. First measurements of a scintillator-based fast-ion loss detector in reversed Ip/Bt at the ASDEX Upgrade tokamak. Fusion Engineering and Design, 2023 3. J. Poley-Sanjuán, J. Galdón-Quiroga, E. Viezzer, J. Rueda-Rueda, P. Cano- Megias, P. Oyola, D. J. Cruz-Zabala, ASDEX Upgrade Team. Conceptual study for velocity space resolved thermal ion loss detection in tokamaks. Review of Scientific instruments, 2022 4. J. Garcia-Dominguez, J. Rueda-Rueda, E. Viezzer, P. Schneider, J. Ayllon-Guerola, M. Garcia-Munoz, J. Hidalgo-Salaverri, A. Mancini, M. Videla, A. Herrmann, and the ASDEXC Upgrade Team. Imaging Neutral Particle Analyzer Engineering Design and Installation for the ASDEX Upgrade Tokamak. IEEE Transactions on Plasma Science, 2022 5. A. Jansen van Vuuren, S. A. Lazerson, A. LeViness, M. Garcia-Munoz, D. Gates, J. Galdon-Quiroga, J. Hidalgo-Salaverri, J. Rueda-Rueda, J. Garcia-Dominguez, J. Ayllon-Guerola. Conceptual Design of a Scintillator-Based Fast-Ion Loss Detector for the Wendelstein 7-X Stellarator. IEEE Transactions on Plasma Science, 2022 6. J. Hidalgo-Salaverri, J. Gonzalez-Martin, J. Ayllon-Guerola, M. Garcia-Munoz, B. Sieglin, J. Galdon-Quiroga, D. Silvagni, E. Viezzer, J. Rueda-Rueda, T. Lunt, A. Herrmann and the ASDEX Upgrade team. Thermo-mechanical limits of a magnetically driven fast-ion loss detector in the ASDEX Upgrade tokamak. Journal of Instrumentation, 2022 7. P. Oyola, J Gonzalez-Martin, M Garcia-Munoz, J Galdon-Quiroga, G. Birkenmeier, E. Viezzer, J. Dominguez-Palacios, J. Rueda-Rueda, J.F. Rivero-Rodriguez, Y. Todo and the ASDEX Upgrade Team. Implementation of synthetic fast-ion loss detector and imaging heavy ion beam probe diagnostics in the 3D hybrid kinetic-MHD code MEGA. Review of Scientific Instruments, 2021 102 A.3 Contributions to conferences and seminars as main author 8. Bo S. Schmidt, J. Poley-Sanjuán, J. Rueda-Rueda, J. Galdon-Quiroga, M. Baquero- Ruiz, H. Järleblad, B. C. G. Reman, M. Rud, A. Valentini, M. García-Muñoz, M. Salewski. A new FILDSIM model for improved velocity-space sensitivity modelling and reconstructions. Plasma Physics and Controlled Fusion, submitted 9. Bo S Schmidt, J. Galdon-Quiroga, J. Rueda-Rueda, J. Poley-Sanjuán, M. García- Muñoz. H. Järleblad, B. C. G. Reman, M. Rud, A. Valentini, M. Salewski, the ASDEX Upgrade Team and the EUROfusion Tokamak Exploitation Team. Determination of the optimal regularization parameter and iterative reconstruction methods for fast-ion loss detector data analysis. Plasma Physics and Controlled Fusion, in preparation 10. R Bilato, C Angioni, A Banon Navarro, V Bobkov, T Bosman, A Di Siena, E Fable, R Fischer, J Galdon-Quiroga, T Görler, J Hidalgo-Salaverri, F Jenko, T Johnson, A Kappatou, Ye O Kazakov, O Kudlacek, J Manyer, M Mantsinen, R Ochoukov, E Poli, J Rueda-Rueda, PA Schneider, S Sipilä, M Weiland, ASDEX Upgrade Team, EUROfusion MST1 Team. Progress in understanding the effects of ICRF/NBI fast ions on core turbulence and Alfvenic activity in ASDEX Upgrade. Nuclear Fusion, submitted 11. J. Dominguez-Palacios, S. Futatani, M. Garcia-Munoz, A. Jansen van Vuuren, E. Viezzer , J. Gonzalez-Martin, M. Toscano-Jimenez, P. Oyola , Y. Todo, Y. Suzuki, L. Sanchis, H. Chen, J. Galdon-Quiroga, J. Hidalgo-Salaverri, J. F. Rivero- Rodriguez, J. Rueda-Rueda, L. Velarde-Gallardo. Simulations predict key role of energetic ion kinetic effects on Edge Localized Modes in tokamak plasmas. Nature Physics, submitted 12. P. Oyola, M. García-Muñoz, M. Vallar, E. Viezzer, J. Rueda-Rueda, J. Dominguez- Palacios, J. Gonzalez-Martin, Y. Todo, D.J. Cruz-Zabala, A. Fasoli, B. Duval, M. Dreval, A. N. Karpushov, S. Coda, O. Sauter, K. Särkimäki, A. Jansen van Vuuren, the TCV team and the EUROfusion Tokamak Exploitation team. Mitigation of Alfvén Eigenmodes in Negative Triangularity plasmas at the TCV tokamak. Nature Physics, in preparation A.3 Contributions to conferences and seminars as main author 1. Status and plans for the INPA diagnostic. ASDEX Upgrade Program Seminar, 2020 (Oral). 2. Development of the Imaging Neutral Particle Analyzer diagnostic for the ASDEX Upgrade Tokamak. 22nd High Temperature Plasma Diagnostics, 2020 (Poster). 103 List of publications and contributions 3. Fast-ion losses during ramp-up phase and disruptions of the ASDEX Upgrade tokamak. IAEA 17 th Technical Meeting on Energetic Particles and Theory of Plasma Instabilities in Magnetic Confinement Fusion, 2021 (Oral). 4. Imaging Neutral Particle Analyzer for the ASDEX Upgrade Tokamak. Fusenet PhD event, 2022 (Oral, best slides prize). 5. First Measurements with an Imaging Neutral Particle Analyzer in the ASDEX Upgrade tokamak. 23 rd European Conference on Plasma Diagnostics, 2023 (Oral) 6. First Measurements with an Imaging Neutral Particle Analyzer in the ASDEX Upgrade tokamak. 12 th ITER International School, 2023 (Poster, best poster prize) 7. Phase-space measurements of MHD-induced fast-ion transport in the ASDEX Upgrade tokamak. 29th IAEA Fusion for Energy Conference, 2023 (Poster) 104 Appendix B Parameters of the ASDEX Upgrade tokamak B.1 General parameters The main parameters of the ASDEX Upgrade tokamak can be seen in table B.1 while a schematic representation of the machine can be seen in figure B.1. Total height of the experiment 9 m Total radius over all 5 m Weight of the experiment 800 T Number of toroidal field coils 16 Number of poloidal field coils 12 Maximum magnetic field 3.1 T Plasma current 0.4 - 1.2 MA Pulse duration < 11 s Time between pulses 15 - 20 min Plasma species D, H, He Plasma volume 14 m3 Plasma mass 3 mg Electron density 1020m−3 Plasma temperature Up to 10 keV Plasma heating: Up to 31 MW Ohmic heating 1 MW NBI heating 20 MW ICRH 6 MW (30 MHz - 120 MHz) ECRH 4 MW (120 GHz) Table B.1 Main parameters of the AUG tokamak. Data extracted from [155]. 105 Parameters of the ASDEX Upgrade tokamak Fig. B.1 Schematic overview of the AUG tokamak. Extracted from [155]. Injector 1 Injector 2 14 15 16 1 3456 7 8 9 10 11 12 13 NBI#2-3 NBI#1-4 NBI#6-7 NBI#5-8 Sector 15 7 Number of sources 4 4 Extraction voltage 60 kV 93 kV Sources and geometry 1 Radial 5 Tangential 2 Tangential 6 Off-axis 3 Tangential 7 Off-axis 4 Radial 8 Tangential Table B.2 Basic parameters of the AUG NBIs. Data extracted from [155]. The figure on the left shows the toroidal view of the NBI geometry. B.2 NBI parameters 8 NBI sources, with 2.5 MW each, thus providing up to 20 MW of heating power are installed at AUG. The eight sources are installed in two NBI boxes located in opposite sectors of the machine. Their main parameters can be found in table B.2, and a toroidal representation of their geometry in the figure inside it. 106 Acknowledgments I would like to begin by thanking my supervisors, Prof. Manuel García-Muñoz and Prof. Eleonora Viezzer, for giving me the opportunity to work in this project and giving me all the tools and resources needed to complete the tasks; and also, for their infinite patience and orientation during our physics discussions. I also want to acknowledge the efforts of all the people from IPP who contributed to the immense task of designing, fabricating, and operating the INPA diagnostic. I would like to express special thanks to Wolfgang Popken, Wolfgang Zeidner, and Michael Ebner for their assistance with the assembly and installation process. Additionally, I greatly appreciate Dr. Philip Schneider’s continuous involvement and contributions to the project since its early stages, in late 2018, including his help with programming and analyzing the conducted experiments. I extend my gratitude to many others who have contributed in various ways to this work: Dr. Athina Kappatou and Dr. Philipp Lauber for their assistance in preparing discharge programs and scenarios for the experiments, and Dr. Tilmann Lunt, Moritz Gruber, and Michael Rolffs for their support with cameras and optical fibers. I would also like to acknowledge the efforts of the entire AUG team, who consistently provided valuable support and kindness, ensuring the success of this project. I extend my thanks to Prof. Mirko Sawlesky and Dr. Bo Schmidt, from the DTU team, for their readiness to discuss everything necessary for the development of INPA tomography and weight functions. Their consistent willingness to participate in exciting projects related to new data analysis techniques is highly appreciated. I want to express my immense gratitude to all my office mates. Those who came before us, paving the road for our journey: Mauri, Joaquín, Juanfran, Javi, Lucía, Jesus, Pilar, Diego, and Alessio. From them, we learned how to navigate this complex field. However, my special thanks go to those who embarked on this thesis journey with us, turning it into a wonderful experience: Javi García, the mastermind behind the mechanical design of INPA, who transformed the concept into a reality ready for installation. Jorge, always ready to lend a hand with any problem. Kiera, to whom I owe the readability and well-structured nature of this document. And Lina, the heir to the FILDSIM throne. I couldn’t think of better hands to entrust with the tools I 107 References [1] Qazi, A. et al., “Towards Sustainable Energy: A Systematic Review of Renewable Energy Sources, Technologies, and Public Opinions,” IEEE Access, vol. 7, pp. 63837–63851, 2019. [2] Sources of Greenhouse Gas Emissions, Greenhouse Gas (GHG) Emissions, US EPA | https://www.epa.gov/ghgemissions/sources-greenhouse-gas-emissions. [3] Kikuchi, M. et al., Fusion physics. International Atomic Energy Agency (IAEA), 2012. [4] ITER - the way to new energy | https://www.iter.org/. [5] Lawson, J. D., “Some Criteria for a Power Producing Thermonuclear Reactor,” Proceedings of the Physical Society. Section B, vol. 70, no. 1, pp. 6–10, 1957. [6] Wesson, J. A., Tokamaks. Oxford University Press, 4ºed., 2011. [7] Abu-Shawareb, H. et al., “Lawson Criterion for Ignition Exceeded in an Inertial Fusion Experiment.,” Physical review letters, vol. 129, no. 7, p. 075001, 2022. [8] Joffrin, E. et al., “Overview of the JET preparation for deuterium–tritium operation with the ITER like-wall,” Nuclear Fusion, vol. 59, no. 11, p. 112021, 2019. [9] Galdon-Quiroga, J. et al., “Velocity space resolved absolute measurement of fast ion losses induced by a tearing mode in the ASDEX Upgrade tokamak,” Nuclear Fusion, vol. 58, no. 3, p. 036005, 2018. [10] Weiland, M. et al., “Enhancement of the FIDA diagnostic at ASDEX Upgrade for velocity space tomography,” Plasma Physics and Controlled Fusion, vol. 58, no. 2, p. 25012, 2016. [11] Medley, S. S. et al., “Invited review article: Contemporary instrumentation and application of charge exchange neutral particle diagnostics in magnetic fusion energy experiments,” Review of Scientific Instruments, vol. 79, no. 1, pp. 1–16, 2008. [12] Nocente, M., Neutron and gamma-ray emission spectroscopy as fast ion diagnostics in fusion plasmas. PhD thesis, Universita di Milano-Bicocca, 2012. [13] Garcia-Munoz, M. et al., “Scintillator based detector for fast-ion losses induced by magnetohydrodynamic instabilities in the ASDEX upgrade tokamak,” Review of Scientific Instruments, vol. 80, no. 5, 2009. [14] Du, X. D. et al., “Development and verification of a novel scintillator-based, imaging neutral particle analyzer in DIII-D tokamak,” Nuclear Fusion, vol. 58, no. 8, p. 082006, 2018. [15] Chen, F. F., Plasma physics and controlled fusion. Springer, 2ºed., 2006. 115 References [16] Galdon-Quiroga, J., Velocity-space resolved measurements of fast-ion losses due to magnetohydrodynamic instabilities in the ASDEX Upgrade tokamak. PhD thesis, University of Seville, 2018. [17] Gonzalez-Martin, J. et al., “First measurements of a magnetically driven fast-ion loss detector on ASDEX Upgrade,” Journal of Instrumentation, vol. 14, no. 11, 2019. [18] Gonzalez-Martin, J. et al., “First measurements of a scintillator based fast-ion loss detector near the ASDEX Upgrade divertor,” Review of Scientific Instruments, vol. 89, no. 10, p. 10I106, 2018. [19] Oyola, P., “Characterization of edge toroidal Alfvèn Eigenmodes,” Master thesis, University of Seville, 2020. [20] Tokamak, Max-Planck-Institut für Plasmaphysik | https://www.ipp.mpg.de/ 14869/tokamak. [21] Zohm, H., Magnetohydrodynamic Stability of Tokamaks. Wiley, 2015. [22] Viezzer, E. et al., “High-accuracy characterization of the edge radial electric field at ASDEX Upgrade,” Nuclear Fusion, vol. 53, no. 5, 2013. [23] Shinohara, K. et al., “Solid state neutral particle analyzer array on national spherical torus experiment,” in Review of Scientific Instruments, vol. 75, pp. 3640– 3642, American Institute of Physics, 2004. [24] Hutchinson, I. H., Principles of plasma diagnostics. Cambridge University Press, 2ºed., 2002. [25] Heidbrink, W. W. et al., “A code that simulates fast-ion D α and neutral particle measurements,” Communications in Computational Physics, vol. 10, no. 3, pp. 716– 741, 2011. [26] FIDASIM | Luke Stagner. https://d3denergetic.github.io/FIDASIM/. [27] Geiger, B., Fast-ion transport studies using FIDA spectroscopy at the ASDEX Upgrade tokamak. PhD thesis, Ludwig-Maximilians-Universität München, 2012. [28] Staebler, A. et al., “Design of the neutral beam injection system for ASDEX- Upgrade,” in Fusion Technology 1988, pp. 620–624, Elsevier, 1989. [29] Staebler, A. et al., “Perfomance of the first ASDEX Upgrade neutral beam injector,” Fusion Technology, p. 593, 1994. [30] Fantz, U. et al., “Physical performance analysis and progress of the development of the negative ion RF source for the ITER NBI system,” Nuclear Fusion, vol. 49, no. 12, p. 125007, 2009. [31] Franzen, P. et al., “Spectroscopic Determination of the Species Distribution and the Divergence of the ASDEX Upgrade Neutral Beam Injection System,” in 26th EPS Conference on Controlled Fusion and Plasma Physics, European Physical Society, 1999. [32] Geiger, B. et al., “Progress in modelling fast-ion D-alpha spectra and neutral particle analyzer fluxes using FIDASIM,” Plasma Physics and Controlled Fusion, vol. 62, no. 10, p. 105008, 2020. [33] Heidbrink, W. W. et al., “The behaviour of fast ions in tokamak experiments,” Nuclear Fusion, vol. 34, no. 4, pp. 535–615, 1994. 116 References [34] Garcia-Munoz, M. et al., “MHD induced fast-ion losses on ASDEX Upgrade,” Nuclear Fusion, vol. 49, no. 8, p. 85014, 2009. [35] Garcia-Munoz, M., Fast Response Scintillator Based Detector for MHD Induced Energetic Ion Losses in ASDEX Upgrade. PhD thesis, Ludwig-Maximilians- Universität of Munich, 2006. [36] Stix, T., “Fast-wave heating of a two-component plasma,” Nuclear Fusion, vol. 15, no. 5, pp. 737–754, 1975. [37] Kerbel, G. D. et al., “Kinetic theory and simulation of multispecies plasmas in tokamaks excited with electromagnetic waves in the ion-cyclotron range of frequencies,” The Physics of Fluids, vol. 28, no. 12, pp. 3629–3653, 1985. [38] JET, T. et al., “ICRF results in D-T plasmas in JET and TFTR and implications for ITER,” Plasma Physics and Controlled Fusion, vol. 40, no. 8A, pp. A87–A103, 1998. [39] Noether, E., “Invariant variation problems,” Transport Theory and Statistical Physics, vol. 1, no. 3, pp. 186–207, 1971. [40] Heidbrink, W. W. et al., “Mechanisms of energetic-particle transport in magnetically confined plasmas,” Physics of Plasmas, vol. 27, no. 3, 2020. [41] Todo, Y., “Introduction to the interaction between energetic particles and Alfven eigenmodes in toroidal plasmas,” Reviews of Modern Plasma Physics, vol. 3, no. 1, p. 1, 2019. [42] Podesta, M. et al., “A reduced fast ion transport model for the tokamak transport code TRANSP,” Plasma Physics and Controlled Fusion, vol. 56, no. 5, 2014. [43] Klages, R., “From Deterministic Chaos to Anomalous Diffusion,” Reviews of Nonlinear Dynamics and Complexity, vol. 3, pp. 169–227, 2010. [44] van Vuuren, A. J., Edge fast-ion transport study using passive FIDA spectroscopy at the ASDEX Upgrade tokamak. PhD thesis, Ludwig–Maximilians–Universität, 2020. [45] Heidbrink, W. W. et al., “The diffusion of fast ions in Ohmic TFTR discharges,” Physics of Fluids B: Plasma Physics, vol. 3, no. 11, pp. 3167–3170, 1991. [46] Zonca, F. et al., “Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas,” New Journal of Physics, vol. 17, no. 1, p. 013052, 2015. [47] Boozer, A., “Establishment of magnetic coordinates for a given magnetic field,” tech. rep., Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States), 1981. [48] Van Zeeland, M. A. et al., “Measurements, modelling and electron cyclotron heating modification of Alfvén eigenmode activity in DIII-D,” Nuclear Fusion, vol. 49, no. 6, p. 065003, 2009. [49] Sanchis, L., Fast-ion transport induced by externally applied Resonant Magnetic Perturbations in the ASDEX Upgrade tokamak. PhD thesis, University of Seville, 2019. [50] Garcia-Munoz, M. et al., “Fast-Ion Losses due to High-Frequency MHD Perturbations in the ASDEX Upgrade Tokamak,” Physical Review Letters, vol. 100, no. 5, p. 055005, 2008. 117 References [51] Viezzer, E., Radial electric field studies in the plasma edge of ASDEX Upgrade. PhD thesis, Ludwig-Maximilians-Universität München, 2012. [52] Schneider, P. A., Characterization and scaling of the tokamak edge transport barrier. PhD thesis, Ludwig–Maximilians–Universität München, 2012. [53] Wolle, B., “Tokamak plasma diagnostics based on measured neutron signals,” Physics Reports, vol. 312, no. 1-2, pp. 1–86, 1999. [54] Koleva, M. et al., “Calibration of neutron detectors at ASDEX Upgrade, measurement and model,” Fusion Engineering and Design, vol. 170, p. 112702, 2021. [55] Tardini, G. et al., “First neutron spectrometry measurements in the ASDEX Upgrade tokamak,” Journal of Instrumentation, vol. 7, no. 03, pp. C03004–C03004, 2012. [56] Moseev, D. et al., “Recent progress in fast-ion diagnostics for magnetically confined plasmas,” Reviews of Modern Plasma Physics, vol. 2, no. 1, p. 7, 2018. [57] Galdon-Quiroga, J. et al., “Velocity-space sensitivity and tomography of scintillator-based fast-ion loss detectors,” Plasma Physics and Controlled Fusion, vol. 60, no. 10, 2018. [58] Garcia-Munoz, M. et al., “Characterization of scintillator screens for suprathermal ion detection in fusion devices,” Journal of Instrumentation, vol. 6, no. 4, 2011. [59] Rodríguez Ramos, M., Calibración absoluta y aplicación de los detectores de pérdidas de iones rápidos basados en materiales centelleadores para dispositivos de fusión nuclear. PhD thesis, University of Seville, 2017. [60] Darrow, D. S. et al., “Measurement of loss of DT fusion products using scintillator detectors in TFTR (invited),” Review of Scientific Instruments, vol. 66, no. 1, pp. 476–482, 1995. [61] Fisher, R. K. et al., “Scintillator-based diagnostic for fast ion loss measurements on DIII-D,” Review of Scientific Instruments, vol. 81, no. 10, 2010. [62] Rivero-Rodriguez, J. F. et al., “A rotary and reciprocating scintillator based fast-ion loss detector for the MAST-U tokamak,” Review of Scientific Instruments, vol. 89, no. 10, 2018. [63] Reimerdes, H. et al., “Overview of the TCV tokamak experimental programme,” Nuclear Fusion, vol. 62, no. 4, p. 042018, 2022. [64] Kim, J. et al., “Initial measurements of fast ion loss in KSTAR,” Review of Scientific Instruments, vol. 83, no. 10, 2012. [65] Baeumel, S. et al., “Scintillator probe for lost alpha measurements in JET,” Review of Scientific Instruments, vol. 75, no. 10, pp. 3563–3565, 2004. [66] Ayllon-Guerola, J. et al., “Thermo-mechanical assessment of the JT-60SA fast-ion loss detector,” Fusion Engineering and Design, vol. 167, p. 112304, 2021. [67] Garcia-Munoz, M. et al., “Conceptual design of the ITER fast-ion loss detector,” Review of Scientific Instruments, vol. 87, no. 11, 2016. [68] Garcia-Munoz, M. et al., “Fast-ion losses induced by ACs and TAEs in the ASDEX Upgrade tokamak,” Nuclear Fusion, vol. 50, no. 8, p. 084004, 2010. 118 References [69] Garcia-Munoz, M. et al., “NTM induced fast ion losses in ASDEX Upgrade,” Nuclear Fusion, vol. 47, no. 7, pp. L10–L15, 2007. [70] McGuire, K. et al., “Study of High-Beta Magnetohydrodynamic Modes and Fast-Ion Losses in PDX,” Physical Review Letters, vol. 50, no. 12, pp. 891–895, 1983. [71] Galdon-Quiroga, J. et al., “Beam-Ion Acceleration during Edge Localized Modes in the ASDEX Upgrade Tokamak,” Phys. Rev. Lett., vol. 121, no. 2, p. 25002, 2018. [72] Van Zeeland, M. A. et al., “Modulation of prompt fast-ion loss by applied n = 2 fields in the DIII-D tokamak,” Plasma Physics and Controlled Fusion, vol. 56, no. 1, p. 015009, 2014. [73] Karpushov, A. N. et al., “Neutral particle analyzer diagnostics on the TCV tokamak,” Review of Scientific Instruments, vol. 77, no. 3, p. 33504, 2006. [74] Kusama, Y. et al., “Fast-ion radial diffusivity evaluated from vertical neutral particle measurements following short pulse beam injection into a TFTR ohmic plasma,” tech. rep., Princeton Plasma Physics Laboratory (PPPL), Princeton, NJ (United States), 1992. [75] Schneider, P. A. et al., “A new compact solid-state neutral particle analyser at ASDEX Upgrade: Setup and physics modeling,” Review of Scientific Instruments, vol. 86, no. 7, p. 73508, 2015. [76] Bartiromo, R. et al., “Design and calibration of the JET neutral particle analyzer,” Review of Scientific Instruments, vol. 58, no. 5, pp. 788–795, 1987. [77] Hamamatsu |https://www.photonis.com/products/ channeltron-electron-multipliers. [78] Heidbrink, W. W., “Fast-ion Da measurements of the fast-ion distribution (invited)a),” Review of Scientific Instruments, vol. 81, no. 10, 2010. [79] Stagner, L. et al., “Orbit tomography of energetic particle distribution functions,” Nuclear Fusion, vol. 62, no. 2, p. 026033, 2022. [80] Jacobsen, A. S. et al., “Inversion methods for fast-ion velocity-space tomography in fusion plasmas,” Plasma Physics and Controlled Fusion, vol. 58, no. 4, p. 045016, 2016. [81] Jacobsen, A. S. et al., “Benchmark and combined velocity-space tomography of fast-ion D-alpha spectroscopy and collective Thomson scattering measurements,” Plasma Physics and Controlled Fusion, vol. 58, no. 4, p. 42002, 2016. [82] Salewski, M. et al., “On velocity-space sensitivity of fast-ion D-alpha spectroscopy,” Plasma Physics and Controlled Fusion, vol. 56, no. 10, 2014. [83] Podestà, M. et al., “Experimental studies on fast-ion transport by Alfvén wave avalanches on the National Spherical Torus Experiment,” Physics of Plasmas, vol. 16, no. 5, 2009. [84] Muscatello, C. M. et al., “Extended fast-ion D-alpha diagnostic on DIII-D,” Review of Scientific Instruments, vol. 81, no. 10, 2010. [85] Heidbrink, W. W. et al., “Measurements of fast-ion acceleration at cyclotron harmonics using Balmer-alpha spectroscopy,” Plasma Physics and Controlled Fusion, vol. 49, no. 9, pp. 1457–1475, 2007. 119 References [86] Murakami, M. et al., “Progress toward fully noninductive, high beta conditions in DIII-D,” Physics of Plasmas, vol. 13, no. 5, 2006. [87] Rueda-Rueda, J. et al., “Commissioning of the imaging neutral particle analyzer for the ASDEX Upgrade tokamak,” Plasma Physics and Controlled Fusion, Submitted. [88] Rueda-Rueda, J. et al., “Design and simulation of an imaging neutral particle analyzer for the ASDEX Upgrade tokamak,” Review of Scientific Instruments, vol. 92, no. 4, p. 043554, 2021. [89] Hawryluk, R. J., “An Empirical Approach To Tokamak Transport,” Phys. Plasmas Close to Therm. Cond., vol. 1, pp. 19–46, 1981. [90] TRANSP Code. https://transp.pppl.gov DOI: 10.11578/dc.20180627.4. [91] Ongena, J. et al., “Numerical transport codes,” in Fusion Science and Technology, vol. 61, pp. 180–189, American Nuclear Society, 2012. [92] TRANSP at JET TOKAMAK. https://users.euro-fusion.org/expert/transp/ Webpages/index.html. [93] Mukhovatov, V. et al., “Plasma equilibrium in a Tokamak,” Nuclear Fusion, vol. 11, no. 6, pp. 605–633, 1971. [94] LoDestro, L. L. et al., “On the Grad-Shafranov equation as an eigenvalue problem, with implications for q solvers,” Physics of Plasmas, vol. 1, no. 1, pp. 90–95, 1994. [95] Pankin, A. et al., “The tokamak Monte Carlo fast ion module NUBEAM in the National Transport Code Collaboration library,” Computer Physics Communications, vol. 159, no. 3, pp. 157–184, 2004. [96] Brambilla, M., “Numerical simulation of ion cyclotron waves in tokamak plasmas,” Plasma Physics and Controlled Fusion, vol. 41, 1999. [97] Tamor, S., “ANTIC: A code for calculation of neutral transport in cylindrical plasmas,” Journal of Computational Physics, vol. 40, no. 1, pp. 104–119, 1981. [98] Weiland, M. et al., “Phase-space resolved measurement of 2nd harmonic ion cyclotron heating using FIDA tomography at the ASDEX Upgrade tokamak,” Nuclear Fusion, vol. 57, no. 11, p. 116058, 2017. [99] Bertelli, N. et al., “Full-wave simulations of ICRF heating regimes in toroidal plasma with non-Maxwellian distribution functions,” Nuclear Fusion, vol. 57, no. 5, p. 056035, 2017. [100] Asunta, O., Fast ions in fusion plasmas - Towards Numerical Tokamaks. PhD thesis, Aalto University. [101] Hirvijoki, E. et al., “ASCOT: Solving the kinetic equation of minority particle species in tokamak plasmas,” Computer Physics Communications, vol. 185, no. 4, pp. 1310–1321, 2014. [102] Salmi, A., Fast ions and momentum transport in the JET tokamak plasmas. PhD thesis, Aalto University, 2012. [103] Laboratory, N. R., ed., no. 3–67, (Washington D. C), Naval Research Laboratory, Naval Research Laboratory, 1970. 120 References [104] ASCOT Code. https://www.aalto.fi/en/department-of-applied-physics/ particle-orbit-simulations-ascot. [105] Stagner, L., Inference of the Fast-ion Distribution Function. PhD thesis, University California Irvine, 18. [106] Oyola, P. et al., “Implementation of synthetic fast-ion loss detector and imaging heavy ion beam probe diagnostics in the 3D hybrid kinetic-MHD code MEGA,” Review of Scientific Instruments, vol. 92, no. 4, 2021. [107] iHIBPsim Code. https://poyo.pages.mpcdf.de/ihibpsim/. [108] Du, X. D. et al., “Resolving the fast ion distribution from imaging neutral particle analyzer measurements,” Nuclear Fusion, vol. 60, no. 11, 2020. [109] Garcia-Dominguez, J. et al., “Imaging Neutral Particle Analyzer Engineering Design and Installation for the ASDEX Upgrade Tokamak,” IEEE Transactions on Plasma Science, vol. 50, no. 11, pp. 4138–4143, 2022. [110] Schmidt, B. S., “A new fildsim model for improved velocity-space sensitivity modelling and reconstructions,” [111] Salewski, M. et al., “Tomography of fast-ion velocity-space distributions from synthetic CTS and FIDA measurements,” Nuclear Fusion, vol. 52, no. 10, p. 103008, 2012. [112] Pinches, S. D., Nonlinear Interaction of Fast Particles with Alfvén Waves in Tokamaks. PhD thesis, University of Nottingham, 1996. [113] Garcia-Munoz, M. et al., “Fast-ion transport induced by Alfvén eigenmodes in the ASDEX Upgrade tokamak,” Nuclear Fusion, vol. 51, no. 10, p. 103013, 2011. [114] Garcia-Munoz, M. et al., “Convective and diffusive energetic particle losses induced by shear alfvén waves in the ASDEX upgrade tokamak,” Physical Review Letters, vol. 104, no. 18, 2010. [115] Gonzalez-Martin, J. et al., “Active Control of Alfvén Eigenmodes by Externally Applied 3D Magnetic Perturbations,” Physical Review Letters, vol. 130, no. 3, p. 035101, 2023. [116] Geiger, B. et al., “Fast-ion D-alpha measurements at ASDEX Upgrade,” Plasma Physics and Controlled Fusion, vol. 53, no. 6, 2011. [117] Sauter, O. et al., “Neoclassical conductivity and bootstrap current formulas for general axisymmetric equilibria and arbitrary collisionality regime,” Physics of Plasmas, vol. 6, no. 7, pp. 2834–2839, 1999. [118] Fischer, R. et al., “Integrated data analysis of profile diagnostics at asdex upgrade,” Fusion Science and Technology, vol. 58, no. 2, pp. 675–684, 2010. [119] Geiger, B. et al., “Quantification of the impact of large and small-scale instabilities on the fast-ion confinement in ASDEX Upgrade,” Plasma Physics and Controlled Fusion, vol. 57, no. 1, p. 014018, 2015. [120] McDermott, R. M. et al., “Extensions to the charge exchange recombination spectroscopy diagnostic suite at ASDEX Upgrade,” Review of Scientific Instruments, vol. 88, no. 7, 2017. [121] Dux, R. et al., “A forward model for beam emission spectroscopy at asdex upgrade,” in 42nd EPS Conference on Plasma Physics, European Physical Society, 2015. 121 References [122] Geiger, B. et al., “Fast-ion transport and neutral beam current drive in ASDEX upgrade,” Nuclear Fusion, vol. 55, no. 8, p. 083001, 2015. [123] Egedal, J. et al., “Reconstruction of gyrotropic phase-space distributions from one-dimensional projections,” Physics of Plasmas, vol. 11, no. 5, pp. 2191–2198, 2004. [124] Salewski, M. et al., “On velocity space interrogation regions of fast-ion collective Thomson scattering at ITER,” Nuclear Fusion, vol. 51, no. 8, p. 83014, 2011. [125] Salewski, M. et al., “Combination of fast-ion diagnostics in velocity-space tomographies,” Nuclear Fusion, vol. 53, no. 6, 2013. [126] Salewski, M. et al., “High-definition velocity-space tomography of fast-ion dynamics,” Nuclear Fusion, vol. 56, no. 10, p. 106024, 2016. [127] Salewski, M. et al., “MeV-range velocity-space tomography from gamma-ray and neutron emission spectrometry measurements at JET,” Nuclear Fusion, vol. 57, no. 5, p. 56001, 2017. [128] Madsen, B. et al., “Velocity-space tomography using prior information at MAST,” Review of Scientific Instruments, vol. 89, no. 10, p. 10D125, 2018. [129] Madsen, B. et al., “Fast-ion velocity-space tomography using slowing-down regularization in EAST plasmas with co- and counter-current neutral beam injection,” Plasma Physics and Controlled Fusion, vol. 62, no. 11, p. 115019, 2020. [130] Madsen, B. et al., “Tomography of the positive-pitch fast-ion velocity distribution in DIII-D plasmas with Alfvén eigenmodes and neoclassical tearing modes,” Nuclear Fusion, vol. 60, no. 6, p. 66024, 2020. [131] Su, J. et al., “Reconstructions of velocity distributions from fast-ion D-alpha (FIDA) measurements on EAST,” Plasma Science and Technology, vol. 23, no. 9, p. 95103, 2021. [132] Salewski, M. et al., “Alpha-particle velocity-space diagnostic in ITER,” Nuclear Fusion, vol. 58, no. 9, p. 96019, 2018. [133] Mehta, P. et al., “A high-bias, low-variance introduction to Machine Learning for physicists,” Physics Reports, vol. 810, pp. 1–124, 2019. [134] Hansen, C., Discrete Inverse Problems: Insight and Algorithms. Society for Industrial and Applied Mathematics, 2010. [135] Lin, D. J. et al., “Validation of the imaging neutral particle analyzer in nearly MHD quiescent plasmas using injected beam ions on DIII-D,” Nuclear Fusion, vol. 60, no. 11, 2020. [136] Swain, D. et al., “ITER ion cyclotron system: Overview and plans,” Fusion Engineering and Design, vol. 82, no. 5-14, pp. 603–609, 2007. [137] Bilato, R. et al., IPP Summer University for Plasma Physics. 2018. [138] Günter, S. et al., “Interaction of energetic particles with large and small scale instabilities,” Nuclear Fusion, vol. 47, no. 8, p. 025, 2007. [139] Geiger, B. et al., “Fast-ion transport in the presence of magnetic reconnection induced by sawtooth oscillations in ASDEX Upgrade,” Nuclear Fusion, vol. 54, no. 2, 2014. 122 References [140] Gonzalez-Martin, J. et al., “Hybrid kinetic-MHD modelling of observed fast ion migration induced by Alfvénic instabilities,” APS, vol. 2021, p. CP11.079, 2021. [141] Du, X. D. et al., “Visualization of fast ion phase-space flow in plasmas well-below, near and well-above Alfvén eigenmode stability threshold in tokamak,” Nuclear Fusion, vol. 63, no. 4, p. 046020, 2023. [142] Gonzalez-Martin, J. et al., “Modelling the Alfvén eigenmode induced fast-ion flow measured by an imaging neutral particle analyzer,” Nuclear Fusion, vol. 62, no. 11, p. 112003, 2022. [143] Wagner, F. et al., “Regime of Improved Confinement and High Beta in Neutral- Beam-Heated Divertor Discharges of the ASDEX Tokamak,” Physical Review Letters, vol. 49, no. 19, pp. 1408–1412, 1982. [144] Zohm, H., “Edge localized modes (ELMs),” Plasma Physics and Controlled Fusion, vol. 38, no. 2, pp. 105–128, 1996. [145] Lauber, P. et al., “Strongly non-linear energetic particle dynamics in ASDEX Upgrade scenarios with core impurity accumulation,” 27th IAEA Fusion Energy Conference (FEC), pp. 3–10, 2018. [146] Ph. Lauber et al, “Strongly non-linear energetic particle dynamics in asdex upgrade scenarios with core impurity accumulation,” EX/1-1 Proc. 27th IAEA FEC, 22-27 October 2018, Gandhinagar (Ahmedabad) Gujarat, India. [147] Lauber, P., “Off-axis nb heated discharges at asdex upgrade: Egams,tae bursts, and rsaes,” IAEA TM on Energetic Particles Vienna, Austria (2015), 2015. [148] Van Zeeland, M. A. et al., “Radial Structure of Alfvén Eigenmodes in the DIII-D Tokamak through Electron-Cyclotron-Emission Measurements,” Physical Review Letters, vol. 97, no. 13, p. 135001, 2006. [149] Cheng, C., “Kinetic extensions of magnetohydrodynamics for axisymmetric toroidal plasmas,” Physics Reports, vol. 211, no. 1, pp. 1–51, 1992. [150] Lauber, P. et al., “Kinetic Alfvén eigenmodes at ASDEX Upgrade,” Plasma Physics and Controlled Fusion, vol. 51, no. 12, p. 124009, 2009. [151] Lauber, P. et al., “LIGKA: A linear gyrokinetic code for the description of background kinetic and fast particle effects on the MHD stability in tokamaks,” Journal of Computational Physics, vol. 226, no. 1, pp. 447–465, 2007. [152] Lauber, P., “Super-thermal particles in hot plasmas—Kinetic models, numerical solution strategies, and comparison to tokamak experiments,” Physics Reports, vol. 533, no. 2, pp. 33–68, 2013. [153] Lauber, P. et al., “Analytical finite-Lamor-radius and finite-orbit-width model for the LIGKA code and its application to KGAM and shear Alfvén physics,” Journal of Physics: Conference Series, vol. 1125, p. 012015, 2018. [154] Bierwage, A. et al., “Orbit-based analysis of resonant excitations of Alfvén waves in tokamaks,” Physics of Plasmas, vol. 21, no. 11, 2014. [155] ASDEX Upgrade. https://www.aug.ipp.mpg.de/wwwaug/. 123