Fast-Ion Transport and Acceleration Induced by Edge Localized Modes in MAST Upgrade and ASDEX Upgrade

Fast-Ion Transport and Acceleration Induced by Edge Localized Modes in MAST Upgrade and ASDEX Upgrade Author Juan Francisco Rivero Rodr´ıguez Supervisors Daniel Garc´ıa Vallejo Manuel Garc´ıa Mu˜noz A thesis presented for the degree of Doctor of Philosophy Departamento de Ingenier´ıa Mec´anica y Fabricaci´on Universidad de Sevilla Spain June 22, 2021 2 Declaration of Authorship I, Juan Francisco Rivero Rodr´ıguez, declare that this thesis titled, Fast- Ion Transport and Acceleration Induced by Edge Localized Modes in MAST Upgrade and ASDEX Upgrade and the work presented in it are my own. I confirm that: •This work was done wholly or mainly while in candidature for a research degree at this University. •Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. •Where I have consulted the published work of others, this is always clearly attributed. •Where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work. •I have acknowledged all main sources of help. •Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself. 3 4 A mi madre, que me ha dado la vida y la fuerza. 5 6 Abstract Fusion energy promises sustainable, low-emission, baseload power to the grid. The tokamak is the most advanced device to achieve fusion energy and it is approaching operation under power-plant conditions. Nowadays tokamaks aim to resolve the scientific and technical challenges of future power plants. In tokamaks, fast ions, – i.e., ions above the bulk plasma energy –, are employed to raise the plasma temperature up to fusion relevant energies. Fast ions may lose confinement due to the interaction with electromagnetic perturbations of different nature, posing a risk to the plasma performance and endangering the tokamak integrity. In the tokamak high confinement operation mode, known as H-mode, an explosive and repetitive magnetohydro dynamic (MHD) instability is driven in the plasma edge, known as Edge Localized Mode (ELM). ELMs release a great amount of energy and particles to the tokamak wall that are believed to be intolerable in future devices. Besides, recent experiments in the ASDEX Upgrade tokamak have observed an increase of fast-ion losses above their injection energy, correlated with the ELM activity. The observations suggest an interaction between the ELMs and the fast ions that results in the acceleration and loss of the latter. This work aims to study the fast-ion losses induced by ELMs. The thesis have covered the development of numerical tools and design of diagnostics to achieve this goal. Numerically, the implementation of time-evolving 3D electromagnetic perturbations in full-orbit codes, like ASCOT5, have made it possible to investigate the main transport and acceleration mechanisms suggested by the ELM-induced fast-ion losses. Experimentally, the first FILD for the MAST-U spherical tokamak was designed and installed in this thesis. The design includes a rotary and reciprocating mechanism that enables the adaptation of the FILD probe to different orientations [0◦, 90◦] and radial positions [1.40 m, 1.60 m]. The first MAST-U FILD data was expected during this thesis. However, due to several delays in the MAST-U experiments, the diagnostic has not been put in operation yet. For this reason, the experimental and the numerical plans have followed different paths. The experiments, carried out in the ASDEX Upgrade tokamak, have aimed to 7 8 expand the scope of the observations of ELM-induced fast-ion losses, performing new scans that would reveal the dominant parameters that affect the interaction between the fast ions and the ELMs. The modelling seeks to understand the transport and acceleration mechanism by first principle modelling and to reproduce the main experimental observations in ASDEX Upgrade. The modelling in MAST-U has also made it possible to reproduce the foreseen FILD signal and enabled comparisons between MAST-U and ASDEX Upgrade. Resumen La fusi´on nuclear se esboza como una fuente de energ´ıa sostenible, de bajas emisiones y que aportar´a potencia de base al mix el´ectrico. Los tokamaks son los dispositivos m´as avanzados para la obtenci´on de energ´ıa mediante fusi´on nuclear y operan en condiciones cercanas a las de las futuras centrales nucleares. Hoy en d´ıa, los tokamaks existentes tienen como objetivo resolver los desaf´ıos cient´ıficos y t´ecnicos que plantean las futuras plantas de fusi´on nuclear. En los tokamaks, los iones r´apidos, – aquellos iones cuya energ´ıa es superior a la del resto del volumen de plasma –, se emplean para aumentar la temperatura del plasma hasta energ´ıas donde la fusi´on se hace patente. Los iones r´apidos pueden perder el confinamiento debido a interacciones con perturbaciones electromagn´eticas de diversa naturaleza, siendo esto un riesgo para el rendimiento del plasma y para la integridad del reactor. En el modo de operaci´on de alto confinamiento, conocido por H-mode, aparecen de forma repetitiva inestabilidades magnetohidrodin´amicas (MHD) explosivas localizadas en el borde, conocidas como Edge Localized Modes (ELMs). Los ELMs liberan una gran cantidad de energ´ıa y momento hacia las paredes del plasma, cuyo efecto se estima intolerable en futuros tokamaks. Adem´as, investigaciones recientes en el tokamak ASDEX Upgrade han observado un aumento en las p´erdidas de iones r´apidos a energ´ıas por encima de su energ´ıa de inyecci´on, que aparecen relacionadas con la actividad de los ELMs. Estas observaciones sugieren que existe una interacci´on entre los ELMs y los iones r´apidos, que resulta en la aceleraci´on y p´erdida de estos ´ultimos. Este trabajo tiene como objetivo de estudio las p´erdidas de iones r´apidos inducidos por ELMs. Para conseguir este objetivo, la tesis ha abarcado el desarrollo de herramientas num´ericas y el dise˜no de diagn´osticos experimentales. En el aspecto num´erico, se han utilizado c´odigos de seguimiento de ´orbitas como ASCOT5 y se han implementado perturbaciones electromagn´eticas 3D que evolucionan en el tiempo. Esto ha permitido investigar los principales mecanismos de transporte y aceleraci´on planteados para esclarecer la p´erdida de iones r´apidos inducida por ELMs. En el ´ambito ex- 9 16 CHAPTER 1. INTRODUCTION 1.3 Tokamaks In magnetically confined fusion, the charged particles that make up the plasma are confined by means of the Lorentz force. Thus, the magnetic field lines must be shaped so that particles trace closed orbits along the field lines, following the gyromotion described in section 2.1. Depending on how the magnetic field lines are shaped, there are two main configurations of magnetically confined fusion reactors: stellarators and tokamaks. Stellarators [10] use complex shapes of the coils to produce a magnetic field were the orbit drifts are counterbalanced and therefore the plasma is confined. Tokamaks [7] use a simpler toroidal shape, as it is shown in figure 1.2, where the toroidal coils induce a toroidal magnetic field (blue) in the plasma, while an inner poloidal coil induce a current to the plasma, working as a transformer circuit (green). In turn, the plasma current produces a poloidal magnetic field, by means of the Ampere’s law. The combination of a toroidal and a poloidal magnetic field produces helical magnetic field lines that ensure the plasma confinement. Figure 1.2: Schematic of the tokamak plasma and magnetic coils [11]. The resulting helical magnetic lines lie on magnetic field surfaces around a toroidal magnetic axis, as it can be observed in figure 1.3. The magnetic field surfaces can be defined as the contours of the poloidal flux, ψ. ψ=ZS B·ndS (1.2) 1.3. TOKAMAKS 17 where Sis a circle centred in the tokamak centre whose perimeter crosses the poloidal position where ψis being evaluated. The helical shape of the magnetic field lines is crucial for plasma confinement, as we will see in chapter 2, and is described by the safety factor: q=m n=number of toroidal circulations number of poloidal circulations (1.3) When the safety factor is rational, it can be easily illustrated as the number of toroidal turns that a field line makes to close into itself. The plasma q value must always be greater than 1 to keep stability. Due to the high temperature of the plasma (≈10 keV), any material in contact with it would degrade, becoming a risk to the tokamak integrity. For this reason, poloidal coils (grey in figure 1.2) are employed to shape the poloidal cross section of the plasma and localize the contact with the tokamak wall to a reinforced region (limiters), as it can be observed in figure 1.3(a). However, material limiters transfer a large amount of impurities to the plasma, diluting the fuel and increasing radiation losses. Therefore, the use of magnetic divertors (figure 1.3(b)) is widespread, since they avoid the direct contact between the outer boundary of the plasma (separatrix) and the vessel wall, reducing the amount of impurities that enter into the plasma. Poloidal coils are also employed to shape the plasma separatrix, as it has a strong impact on the plasma performance [12]. The region between the plasma boundary and the wall is called scrape-off layer (SOL). Spherical Tokamaks (ST) [14, 15] are tokamaks with a more compact layout, by means of reducing the centre column to a minimum, retaining only the indispensable components. A very low aspect ratio is achieved, – A=R ain figure 1.4 –, being A∼1.5 in spherical tokamaks, whereas A∼2.5 in conventional tokamaks. The drastic aspect ratio reduction, in addition to the D-shape of the plasma poloidal section, give the plasma its characteristic spherical shape. Spherical tokamaks offer several advantages: •Higher relative pressures can be achieved. This is translated in a cost reduction, since, for similar plasma pressures, it requires lower magnetic fields and thus lower currents through the coils. β=p B2/2µ0 (1.4) •The maximum relative pressure at which severe plasma instabilities appear impeding the operation, βcrit, is higher in spherical tokamaks due to its significantly lower aspect ratio [16], which increases the magnetic field convexity. 18 CHAPTER 1. INTRODUCTION Limiter Last Closed Magnetic Surface Divertor Target Plates Last Closed Magnetic Magnetic axis Magnetic axis 1 1.5 2 2.5 -1 -0.5 0 0.5 1 X-Point 1 1.5 2 2.5 -1 -0.5 0 0.5 1 a) b) Surface ( )Separatrix Figure 1.3: Poloidal cross section of a material limited plasma (a) and a divertor limited plasma (b) [13]. •Smaller plasma inductance facilitates current drive and makes possible to investigate non-inductive current drive technologies, which will allow continuous operation in fusion reactors. However, the compactness of the spherical tokamak leads to minimum space in the centre column, complicating the power dissipation and preventing it from using superconductor magnets. Besides, due to the higher curvature in the spherical tokamaks, higher magnetic gradients are present, thus reducing the maximum operative magnetic field. Consequently, the total pressure in spherical tokamaks is lower than in conventional tokamaks. Tokamaks are the most extended fusion devices, having achieved D-T experiments in TFTR [17] and JET [18] and the highest fusion power gain in history [9]. There are many operating tokamaks worldwide, such as, EAST [19] (China), ASDEX Upgrade [20] (Germany) or DIII-D [21] (USA), among many others. It will also be the first configuration to aim for ignition, in ITER [22], where the international effort of a number of countries that comprise more than half the global population is put on building what is considered the ultimate experimental fusion reactor. After ITER, a first demonstration power plant, DEMO [23], is envisaged. With regards to spherical tokamaks, devices like NSTX-U [24] or MAST-U [25] explore this alternative with the aim to construct the first prototype of a ST fusion power plant in STEP [26]. 1.4. PLASMA HEATING AND FAST IONS 19 PLASMA Magnetic axis Figure 1.4: Poloidal cross section of a spherical tokamak [14]. 1.4 Plasma Heating and Fast Ions In addition to field line bending, current drive also contributes to plasma heating, following Ohm’s law [27]. But ohmic heating does not suffice to reach fusion relevant temperatures in the plasma. Therefore, external heating is required in tokamaks to reach higher temperatures. The main heating systems are: Neutral Beam Injection. The neutral beam injectors (NBI) [28] are highcurrent, high-power particle accelerators that inject a beam of neutral fuel (H, D or T) at an injection energy higher than the plasma temperature. Because they are neutrals, they are not affected by the magnetic fields and thus penetrate into the plasma, where they exchange charges with it. Once they are ionized, they get magnetically confined inside the plasma. The injected ions gradually transfer their energy to the plasma due to Coulomb collisions with other particles, thus increasing the plasma temperature. Radio frequency heating. Radio frequency (RF) heating uses the resonant interaction between the gyromotion frequency of charged particles in a magnetic field and externally applied high-power electromagnetic waves [29]. Thus, the RF heating accelerates the particle population that is resonant with the external waves. Two main heating systems 20 CHAPTER 1. INTRODUCTION are derived from this principle, depending on whether the external wave is resonant with the ions (Ion-Cyclotron Resonant Heating, ICRH) or the electrons (Electron-Cyclotron Resonant Heating, ECRH) in the plasma. Fusion reactions Fusion reactions release energy in the form of kinetic energy of the products. The charged particle products remain magnetically confined inside the plasma and therefore contribute to increasing the plasma temperature. These heating mechanisms produce a population of ions whose energy is significantly larger than the bulk thermal plasma, known as fast ions [30]. Due to their high energy, fast ions are characterized by a low collisionality, long mean free path, large gyroradius and large drift orbits [31]. The fast-ion distribution function is strongly correlated with the birth mechanism: •NBI produce fast ions at the injection energy and with an anisotropic spatial and angular distribution defined by the beam geometry and the plasma shape and density. Since the NBI ion source is not purely formed by detached ions but also molecules, (e.g., D+or D+ 3and D+ 3), fast ions with a half and a third the main injection energy are also found. •ICRH produce fast ions at the resonant layer (radially localized). The energy distribution is broad and their angular distribution strongly perpendicular to the magnetic field lines, as the acceleration occurs in the ion gyromotion direction. •Fast ions created in fusion reactions have a spatial distribution peaked at the magnetic axis (highest plasma density and temperature). Their angular distribution is nearly isotropic and their birth energy is defined by the fusion reactions. D+D→0.8 MeV 3He + 2.5 MeV n (1.5) D+D→1.0 MeV T +3.0 MeV p (1.6) D+3He →3.7 MeV α+14.6 MeV p (1.7) D+T→3.5 MeV α+ 14.1 MeV n (1.8) Fast ions are of paramount importance in tokamaks. They are responsible for achieving fusion relevant temperatures in the plasma. Besides, fusionproduct fast ions will be responsible for sustaining an ignited state in future power plants [32]. Fast ions not only contribute to plasma heating but also 1.5. MOTIVATION AND GOAL 21 to current drive and momentum. Conversely, due to their high momentum and energy, their losses can produce a great damage onto the tokamak wall [33]. For this reason, fast-ion confinement and their loss mechanisms are extensively studied in conventional and spherical tokamaks [34, 35, 36]. 1.5 Motivation and Goal The goal of this thesis is to develop the numerical tools and to design the diagnostic that would enable the investigation of fast-ion losses induced by edge perturbations in ASDEX Upgrade and MAST Upgrade. ASDEX Upgrade [20] is a medium size tokamak, renowned for discovering the H-mode regime [37]. It is one of the tokamaks in the world with the most active research nowadays and it is equipped with a large amount of state-of-the-art diagnostics [38]. MAST Upgrade is a spherical tokamak that stopped its operation in 2013 to undergo a mayor upgrade that would enable new physics, like the super-X divertor, which aims to improve the exhaust capabilities of tokamaks. During this upgrade it has been equipped with a wide variety of diagnostics, such as the FILD detector, designed in the scope of this thesis. Innumerable codes are available to model the many different aspects of fusion research. The fast-ion distributions are generally modelled as minority species inside the plasma, solving the Fokker-Planck equation [39] with a Monte-Carlo approach [40]. This is the case of ASCOT5 [41], LOCUST [42] or TRANSP [43]. The bulk plasma is frequently modelled as a fluid using the magneto-hydro dynamic (MHD) equations [44]. Two main families can be distinguished inside the MHD codes: the ideal and the non-ideal MHD codes. The ideal MHD models, like VMEC [45], do not account for magnetic reconnection events, whereas the non-ideal MHD codes, such as JOREK [46] or MARS-F [47], may solve the magnetic reconnection, including resistive effects. Ultimately, some hybrid codes, like MEGA [48], integrate the MHD equations and the fast-ion kinetic equation simultaneously, accounting for the wave-particle interaction. Some MHD codes like JOREK are including kinetic solvers [49] to account for the effect of the MHD perturbations on the minority species, like Tungsten impurity [50] or fast ions. The fast ions can be experimentally observed with several techniques that can be classified depending on whether they aim to measure the confined or the lost fast ions. Among other techniques, the confined fast ions are measured using the Fast-Ion Deuterium-α(FIDA) [51] or the Beam-Emission Spectroscopy (BES) [52]. They infer the fast-ion content and energy based on charge exchange spectroscopy, using the intensity and the Doppler shift of the light emitted when a fast ion exchange charges with a neutral from the 22 CHAPTER 1. INTRODUCTION NBI beam. The Neutral Particle Analyser (NPA) [53] is based on the same charge exchange reaction, although the NPA measures the neutralized fast ions that escape from the plasma confinement, colliding with the NPA probe. With regards to the lost fast ions, they can be directly measured with Faraday cups [54], which produce a current when the fast ions collide with them. The ubiquitous diagnostic to measure fast-ion losses is the scintillator-based Fast- Ion Loss Detector (FILD) [55], which infers the fast-ion loss velocity space working as a magnetic spectrometer. In this thesis, the first FILD for MAST-U has been designed as part of the main scope of the upgrade. The data of the MAST-U FILD was aimed to be analysed during this thesis to study the fast-ion losses induced by Edge Localized Modes (ELMs). However, due to several delays in the MAST- U restart, the diagnostic has not been put in operation yet. To overcome the lack of experimental data in MAST-U, different approaches have been followed on the experimental and the numerical areas of this thesis. The experiments have been carried out in ASDEX Upgrade, where ELM-induced fast-ion losses were previously reported [56, 57, 58]. Thus, the experiments of this thesis aim to augment the scope of the reported observations, seeking the plasma parameters that mostly affect the fast-ion interaction. With regards to the numerical results, first principle modelling of fast-ion transport and acceleration during ELMs is carried out with ASCOT5, to reproduce the reported observations in ASDEX Upgrade and characterize the transport and acceleration mechanism of the ELM-induced fast-ion losses. Additionally, the fast-ion modelling in MAST-U makes it possible to compare the expected FILD signals in ASDEX Upgrade and MAST Upgrade. 1.6 Content of this Thesis The content of this thesis is summarized here, with especial emphasis on the original work. The chapters are structured as follows: Chapter 1 reviews the context and state of the art of nuclear fusion research. Chapter 2 presents the charged particle equation of motion in an electromagnetic field, together with the resulting drifts and particle orbits in a tokamak. The Edge Localized Modes and the observation of ELM- induced accelerated fast-ion losses in AUG are introduced. Chapter 3 describes the numerical tools employed in this thesis. An orbit following code (FIOS) is developed to account for 3D electric perturbations, with the aim to study fast-ion acceleration. Besides, a 1.6. CONTENT OF THIS THESIS 23 time-evolving electromagnetic perturbation module is coded up in the Monte-Carlo full-orbit code ASCOT5. The multi-dimensional spline interpolation of the time-evolving module was also developed in the scope of this thesis. Chapter 4 describes the two tokamaks covered in this thesis. The first MAST-U FILD is designed, accounting for the probe design, optical system, signal resolution, mechanical design and structural analysis of the diagnostic. Chapter 5 shows the most recent experiments in ASDEX Upgrade dedicated to ELM-induced fast-ion losses, designed and analysed in this thesis. A dataset of AUG shots with similar plasma parameters has been constructed and the statistical analysis of the ELM-induced fastion losses is carried out. Chapter 6 presents the fast-ion modelling during an ELM with ASCOT5. The observed transport and acceleration mechanism is described and the ELM-induced fast-ion losses in AUG are estimated. The fast-ion modelling in MAST-U makes it possible to estimate the FILD signal in a MHD-quiescent scenario and during an ELM crash. Chapter 7 discusses and compares the presented experimental and numerical results. The future experiments and modelling plans are also described. 24 CHAPTER 1. INTRODUCTION Chapter 2 Theoretical Background 2.1 Charged Particle Motion in Electromagnetic Fields As it was mentioned in the previous chapter, tokamaks make use of magnetic fields to confine a plasma, – i.e., a ionized gas with quasi-neutral global behaviour –. Thus, it is of paramount importance to understand the chargedparticle motion in a magnetic field, which is the basis of magnetic confinement. Consequently, this section will describe the fast-ion orbits and their main attributes in a tokamak plasma. 2.1.1 Motion in an Homogeneous Magnetic Field The motion of charged particles in a magnetic field is governed by the Lorentz law [59]: mdv dt =q(v×B) (2.1) where m,qand vare the mass, charge and velocity of the particle, respectively, and Bis the magnetic field. This equation can be decomposed as follows: mdvk dt = 0 (2.2) mdv⊥ dt =q(v⊥×B) (2.3) where vkis the velocity component parallel to the magnetic field and v⊥ is the velocity component perpendicular to the magnetic field. In a uniform magnetic field, – constant in space and time –, the particle follows an uniform line motion along the direction of the magnetic field and a circular motion 25 32 CHAPTER 2. THEORETICAL BACKGROUND orbits due their characteristic shape in the poloidal plane (figure 2.4(a)). The bounce is a periodic event of frequency ωb, called bounce frequency. Since the qprofile normally grows at outer magnetic surfaces, the outer leg of the banana orbit covers a higher toroidal angle than the inner leg in a bounce period. This provokes a toroidal precession of the orbit, described by a bounce-averaged toroidal precession frequency, ωprec, that is illustrated in figure 2.4(b). Trapped orbits are also called banana orbits, for their characteristic poloidal projection. 2.2 The Edge Localized Modes A high confinement operating state, known as H-mode [37], spontaneously arises in divertor-limited plasmas above a certain power threshold. It is characterized by the formation of an edge transport barrier (ETB) that drastically reduces the energy and particle transport in a narrow region near the edge. Due to the profile stiffness [68], the transport in the core of the plasma is similar to the low confinement mode (L-mode), but the ETB enables steeper pressure gradients near the edge. This leads to high edge temperature and density, known as pedestal, that hoists the entire profile, thus increasing the stored energy by a factor greater than 2. The H-mode improves the fusion performance so it is the foreseen scenario for future fusion reactors [69]. In a H-mode regime, if the edge pressure gradient exceeds a certain limit, an explosive magneto-hydro dynamic (MHD) instability is driven, known as edge localized mode (ELM) [70, 71, 72, 73, 74, 75]. ELMs are fast (0.3 - 1 ms) filamentary plasma eruptions that expel particles and energy from the plasma leading to the collapse of the pedestal. ELMs decrease the plasma energy and particle content by a 5 - 10 %, leading to a high energy load on the divertor targets [76] and the plasma facing components (PFC) [77]. ELMs also lead to a contraction of the plasma volume. After the crash, the ETB and the pedestal slowly recover until the pressure gradient limit is exceeded again, triggering a new ELM [78]. Thus, ELMs are short repetitive MHD instabilities whose repetition frequency typically ranges between υELM = 10 −200 Hz. An ELM is generally detected by the spikes in the Dαlight emitted from the divertor target and the divertor shunt currents. The spikes are produced by the electrons ejected from the core plasma during the ELM. ELMs are also observed in the magnetic pick-up coils, depicting the high level of magnetic fluctuations during the ELM [79, 80]. Different types of ELMs can be distinguished based on a phenomenological classification. The two main type of ELM are distinguished by how the ELM repetition frequency 2.2. THE EDGE LOCALIZED MODES 33 varies with the energy flux through the separatrix, Psep1, which are described as follows: Type I ELMs The ELM repetition frequency increases with the energy flux: dυELM dPsep >0 (2.26) They are the largest and most common ELMs, since they are associated with high power H-mode regimes, desirable for high fusion performance. Type III ELMs The ELM repetition frequency decreases with the energy flux: dυELM dPsep <0 (2.27) They are usually observed close to the L-H transition power as very small ELMs with repetition frequencies above 1 kHz. Gradually, they become bigger and more easily distinguishable with increasing heating power. Eventually, the type-III ELMs give way to the larger type-I ELMs. Beside, other ELM types can be distinguished, generally smaller, associated to less common plasma conditions and without a clear correlation between ELM repetition frequency and the energy flux through the separatrix. These are the Type II ELMs [81], the grassy ELMs [82] or the type V ELMs [83], among others. The peeling-ballooning (PB) model [71, 84, 85, 86] is the most accepted theoretical framework that explains the onset of type I ELMs. The PB model sets an stability boundary which is a combination of the ballooning mode and the peeling mode. The ballooning mode is a pressure-gradient driven mode with medium to high toroidal mode number (n), whose perturbation is localized at the LFS of the plasma (referred to as ballooning structure). The kink-peeling mode is driven by the pedestal current, low nand extremely localized at the plasma edge. The coupled peeling-ballooning mode, with intermediate toroidal mode numbers (n≈10), is destabilized whenever the PB stability boundary is crossed by a combination of steep pressure gradients and large edge currents. When the PB mode is triggered, the edge pressure and current crashes. This returns the plasma to a stable condition, starting the ELM cycle again. The PB model may also provide an explanation for smaller ELMs, although more qualitatively. 1This is generally depicted as a variation in the heating power as Psep =Ptot−dW dt −Prad 34 CHAPTER 2. THEORETICAL BACKGROUND ELMs are beneficial to reduce the impurity content in the plasma [87]. However, it is predicted that the peak heat load on future fusion reactors will exceed the tolerable material limit of 10 MWm−2[88, 89, 90], causing a rapid erosion of the divertor target plates. Even though, the H-mode is the foremost regime to achieve a good fusion performance in future reactors. Therefore, ELM control techniques have been developed to mitigate or suppress ELMs in H-mode regimes, such as externally applied magnetic perturbations (RMP) [91], ELM pacing with pellets [92] or vertical kicks [93], among others. Additionally, alternative operational regimes that keep the high confinement and performance of the H-mode while supressing or mitigating the ELMs are also investigated [94], such as the QH-mode [95], the I-mode [96] and other small-ELM regimes [97]. 2.2.1 Observation of accelerated fast-ion losses induced by ELMs Recent experiments in ASDEX-Upgrade have reported a grow of the fast-ion losses correlated with the ELM bursts [98, 56, 57, 58]. The losses are measured with two scintillator-based Fast-Ion Loss Detector (FILD) located in different toroidal angles. Figure 2.5(a), shows the edge density, the magnetic loop voltage and the divertor current, typically employed to detect ELMs. Each peak in the signal corresponds to an ELM crash. An increase in the FILD signals (figure 2.5(b)) is seen to be correlated with the ELM activity. Figures 2.5(c) and 2.5(d) show a close-up of a single ELM crash, revealing that the peaks in the toroidally displaced FILDs are time-shifted. This suggests a 3D nature of the fast-ion losses during an ELM crash. Even though, Fast-Ion Deuterium-α(FIDA) measurements have shown that less than 0.3% of the total fast-ion population is lost in an ELM [99]. The intra-ELM velocity-space measurements of the fast-ion losses, which can be observed in figure 2.6(a), depict a population at energies tens of keV above the main NBI energy. A tomographic inversion of the measurement shows that this accelerated population is very localized, as it can be observed in figure 2.6(b). The accelerated population is observed during mitigated and non-mitigated ELMs but it is not seen in ELM-supressed regimes. This suggests a fast-ion acceleration induced by the ELM perturbation. Measurements of soft X-ray (SXR) and electron-cyclotron emission (ECE) depict bursts at the ELM onset, indicating electron acceleration, which further support this hypothesis [57, 100]. In the referenced works, the proposed acceleration mechanism is the parallel electric field arising during the ELM crash, when magnetic reconnection is believed to take place [101]. Observation of 2.2. THE EDGE LOCALIZED MODES 35 Div. Current (a.u.) FILD1 (a.u.) FILD2 (a.u.) AUG #28061 5 10 15 0 0 5 10 15 20 25 Time (s) 1.417 1.418 1.419 1.420 Time (s) 1.40 1.42 1.44 1.46 1.48 1.50 Density (10 m ) 19 -3 6 3.5 4.0 2 FILD2 (a.u.) FILD1 (a.u.) 10 Div. Current (a.u.) (a) (b) (d) 1.416 4.5 Loop Voltage (a.u.) B (a.u.) · r Figure 2.5: (a) Timetraces of the electron density (blue), the loop voltage (red) and the divertor current (black). (b) Timetraces of the two toroidally displaced FILDs. (c,d) Close-up of a single ELM for figure (a) and (b), respectively [56]. 2 4 6810 Gyroradius (cm) 0 4 3 2 1 FILD (a.u.) 5 FILD Experimental FILD Inversion FILDSIM NBI E0 NBI E /2 0High energy FILDSIM high energy FILDSIM E FILDSIM E /2 0 0 B =1.42 T FILD 20 40 60 80 2 4 6 8 10 Gyroradius (cm) Pitch angle (º) 0.0 0.7 1.4 NBI injection energy (~82 keV) High energy tails (140 keV) FILD1 (a.u.) AUG #33127 t=4.354s (a) ( )b Figure 2.6: (a) Intra-ELM velocity-space measurements of the fast-ion losses [56]. (b) Gyroradius profile of the intra-ELM measurements between Λ = [58◦,65◦]. The blue crosses are the experimental data, the red curves correspond to the tomographic inversion and the black curve is the reconstructed signal [56]. 36 CHAPTER 2. THEORETICAL BACKGROUND particle acceleration during magnetic reconnection events in MAST would support this hypothesis [102, 103]. However, this implies a pitch angle variation of the accelerated population, which is not observed in the experiments. For this reason, a recent work proposed a gyrotron acceleration induced by the vertical polarization of outwards-propagating ELM filaments with sizes smaller than the fast-ion gyroradius [104]. This mechanism would complement the parallel acceleration, thus keeping a constant pitch-angle structure. Chapter 3 Simulation Models This thesis employs state-of-the-art codes together with the development of new numerical tools that helps to understand the fast-ion transport and acceleration in MAST-U and AUG. The existing and developed codes used in this thesis are described in this chapter. 3.1 Numerical Tools 3.1.1 The ASCOT Code ASCOT [105, 106, 107, 40] is a Monte Carlo particle-following code that solves the distribution function of minority species in magnetically-confined fusion plasmas. It has been developed since the 1990s within the research groups of Aalto University and the VTT technological Research Center in Finland. ASCOT solves the distribution function of a minority species described by the Fokker-Planck equation [39]: δf δt +v· ∇f+a∇vf=C(f) (3.1) where f=f(r,v, t) is the distribution function in the particle phase space; a=q m(E+v×B) is the acceleration, given by the Lorentz law; and C is the collision operator. The test particles equation of motion is resolved using the Boris leap frog integration method [108]. ASCOT may also solve the Fokker-Planck equation in the guiding centre phase space. However, the guiding centre approximation is not used in this thesis, since the large fast-ion gyroradius and large magnetic field gradient in spherical tokamaks produce large variations along the gyromotion that are no accounted for by the guiding centre equations. 37 38 CHAPTER 3. SIMULATION MODELS To solve the particles orbits, some plasma parameters must be provided. ASCOT uses an axisymmetric magnetic field, where the toroidal magnetic field and the poloidal flux is given in 2D (R, z) matrices. The poloidal components of the magnetic fields are derived from the poloidal flux: BR=−∂ψ R∂z (3.2) Bz=∂ψ R∂R (3.3) In addition to this, a 3D magnetic perturbation in each component (δBR, δBϕ,δBz) can be included in 3D (R, ϕ, z) matrices. A one-dimensional electric field, radial to the plasma surface, is derived from the plasma potential. In ASCOT, the coupled effect of the fast ions on the electromagnetic perturbation, due to wave-particle interaction is not considered. The Monte Carlo algorithm resides in the collision operator, which aims to statistically model the Coulomb collisions between the test particles and the plasma. To evaluate the Coulomb collisions, the plasma kinetic profiles are needed. The standard set consists of one-dimensional temperature and density profiles as a function of the normalized poloidal flux: ρpol =sψ−ψa ψs−ψa (3.4) where ψaand ψsare the poloidal flux values at the magnetic axis and the separatrix, respectively. 2D (R, z) profiles may also be given, but they are not generally necessary. Optionally, the plasma rotation can be taken into account in the evaluation of the collisions. This is essential for the impurity studies, but it is not important for fast particles studies and it is generally neglected. Due to the short orbital times and the low collisionality of the fast ions, the collision operator is neglected in this thesis, thus avoiding the stochastic term in the orbit following simulations. One of the crucial features of the ASCOT code is the detection of the orbit intersection with the tokamak wall, that enables to estimate the fast-ion losses. ASCOT works with both 2D (R, z) and 3D (R, ϕ, z) wall geometries. The code uses a modified computergraphics ray-tracing implementation [109] to find an intersection of a line (orbit) and a planar element (wall). The 3D description of the wall makes it possible to study the wall loading in a realistic geometry with sharp elements. Additionally, this algorithm enables the development of synthetic diagnostics based on the estimation of the fast-ion loss velocity-space. ASCOT includes external tools to model the initial distribution of markers, such as BBNBI [110], a beamlet-based neutral beam ionization model that takes into account 3.1. NUMERICAL TOOLS 39 the fine structure of the injector and the plasma kinetic profiles to produce a birth distribution of a given number of markers. BBNBI includes the beam geometries of the most relevant tokamaks, like AUG, ITER, JET or MAST- U. The inputs in ASCOT follows the COCOS3 sign convention [111]. The data is numerically interpolated using splines [112] in cylindrical coordinates (R, ϕ, z). The ASCOT latest version, ASCOT5 [41], has undergone a full rewrite in C, using OpenMP + MPI to leverage the capabilities of Single-Instruction Multiple-Data Central Processing Units (SIMD-enabled CPUs). In comparison to its older version, ASCOT5 enables simulations with nearly an order of magnitude more markers with the same computational effort. Moreover, due to the large number of followed test particles, the parallelization of particles tracking becomes natural and thus a good scalability is foreseen. An ideal scalability would follow the curve tn=t1 n, where t1is the time consumed by a simulation run in a single node and nis the number of employed nodes. The scalability of the ASCOT5 code is tested in the Marconi High Performance Computer, based on Intel Xeon 8160 (Skylake) nodes. A test with 400000 markers, followed in ITER for 60 ms were performed for different numbers of CPUs. Figure 3.1 shows that ASCOT5 scales near-ideally with the number of cores. The inputs in ASCOT5 are implemented via interfaces, thus allowing to add new features to the code without affecting the codebase. Different implementations for the inputs will be added as the need arises, such as analytical models of the kinetic profiles or Fourier decomposed magnetic perturbation. In turn, the interfaces improve the long-term maintenance of the code. ASCOT has enabled a wide variety of studies in fast-particle physics. The power load on the plasma facing components provoked by fast-ion losses have been estimated in tokamaks [113] and stellarators [114]. Besides, ASCOT has been employed to investigate the fast-ion transport due to externally applied resonant magnetic perturbations (RMP) [115, 65] and MHD modes [116, 117, 118], and it even has made it possible to create a loss map that connect losses to the underlying transport mechanisms. The code has modelled fast ions from different sources, such as NBI [119], fusion products [120] and ICRH [121]. ASCOT has also been used in runaway electrons [122] and impurity studies [123]. 3.1.2 The FILDSIM Code The FILDSIM code [124, 58] resolves the instrument function of a scintillatorbased fast-ion loss detector (FILD). The FILD diagnostic [55] provides direct measurements of the fast-ion losses and gives information of their velocity 40 CHAPTER 3. SIMULATION MODELS 0 200 400 600 800 1000 1200 1400 #cpus 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 time [s] 10 4 ASCOT5 ideal Figure 3.1: ASCOT5 scalability diagram in MARCONI SkyLake. space. It works as a magnetic spectrometer, as it is illustrated in figure 3.2: Using the tokamak magnetic field, it collimates the escaping fast ions from the plasma that reaches the probe head following their gyromotion, until they strike a scintillator plate. The fast-ion velocity space, – i.e., gyroradius and pitch angle –, is inferred by the strike point on the scintillator plate, using a strike map that translates the plate coordinates into velocity space coordinates. FILDSIM uses the 3D geometry of the FILD main components and performs orbit tracing from the FILD pinhole to the scintillator plate to estimate the strike map and the signal intensity, coverage and resolution. The FILD main components are: Protective shield Its distinctive D-shape enhances the pitch-angle detection range, avoiding self-obstruction of the measurements. Collimator Its inner geometry is responsible for the diagnostic resolution and collimator factor (number of ions reaching the scintillator plate per ions on the collimator aperture). Scintillator plate It is the active component of the detector. It consists of a plate coated with an ionoluminescent material, whose properties (scintillator yield, decay time, etc.) determine the detector capabilities [67]. 3.1. NUMERICAL TOOLS 41 Figure 3.2: (a) FILD probe head and escaping fast-ion orbit reaching the aperture. (b) Scintillator plate where the collimated ions are dispersed using the tokamak magnetic field. FILDSIM simulates ion trajectories started at the detector pinhole with a range of fixed gyroradii and pitch angles and a random distribution of gyrophase and position in the pinhole. The code assumes that the local magnetic field in the volume of the probe head is constant, and therefore the orbits are traced as geometrical helices. FILDSIM provides the location where each particle hits the scintillator plate, showing that orbits with equal gyroradius and pitch angle form a Gaussian distribution on the plate, due to the finite resolution of the detector. The strike map is computed using the centroid of these distributions. The size and shape of the distribution also provides numbers of the collimator factor and the signal resolution. Other codes like EfipDesign [125] have previously followed this same method to construct the strike map of the W7-AS [125] and the JET [126] detectors. Besides, FILDSIM carries out backward tracing to estimate the velocityspace range that would be self-obstructed by the probe external geometry. The finite detector resolution will provoke the FILD signal to be a distortion of the fast-ion loss velocity space reaching the detector pinhole. FILDSIM uses the orbit tracing to develop a weight function that relates the fastion loss distribution at the pinhole and the distorted distribution at the scintillator plate. With this, FILDSIM applies tomographic inversion techniques [127] to the FILD signal and infers the undistorted fast-ion loss distribution, similarly to that obtained in other fast-ion diagnostics [128, 129, 130, 131, 132]. In particular, the 0th order Tikhonov regularization is implemented in FILDSIM. The weight function can be used oppositely to generate a syn- 48 CHAPTER 3. SIMULATION MODELS Now, each knot is comprised of 4 coefficients. Namely, the given data, fij; the second derivative along the xcoordinate, Mx ij; the second derivative along the ycoordinate, My ij; and the cross fourth derivative, Mxy ij =d4(fij (xi,yj)) dx2dy2. The existence and uniqueness of a solution of the derivatives can be demonstrated as an extension of the 1D spline [112]. Since the interpolation is continuously twice differentiable in the xand ycoordinates, it is clear that fij(x, y) is continuously twice differentiable in its entire domain. Equations (3.13), combined with (3.14) and (3.15), can be expressed as follows: fij(x, y) = 4 X ιy=1 φy(ιy) 4 X ιx=1 (φx(ιx)C(ιx, ιy))!(3.18) where φχ=hqχ, pχ,hχ 6sχ,hχ 6rχifor χ∈ {x, y}and Cis the coefficient matrix: C=    fij fij+1 My ij My ij+1 fi+1jfi+1j+1 My i+1jMy i+1j+1 Mx ij Mx ij+1 Mxy ij Mxy ij+1 Mx i+1jMx i+1j+1 Mxy i+1jMxy i+1j+1     (3.19) Equation (3.18) minimizes the computational effort of the multi-dimensional splines evaluation. Note that this construction is commutative and therefore, the order in which the coordinates are interpolated is indifferent. The same extension may be carried out for a third dimension, z, and a fourth dimension, t, resulting in the following 4D spline evaluation: fijkm(x, y, z, t) = 4 X ιt=1  φt(ιt) 4 X ιz=1  φz(ιz) 4 X ιy=1 φy(ιy) 4 X ιx=1 (φx(ιx)C(ιx, ιy, ιz, ιt))!   (3.20) where, similarly to the 2D splines, φχ=hqχ, pχ,hχ 6sχ,hχ 6rχifor χ∈ {x, y, z, t} and Cis a 4D coefficient matrix that follows the same construction criteria as (3.19). In the same way, the existence and uniqueness of a solution of the derivatives can be demonstrated and the resulting piecewise polynomials are continuously twice differentiable on the entire domain. In this case, each knot stores 16 different coefficients, corresponding to the given value and the cross derivatives of the different coordinates. This results in a total data storage of 16 ×Nx×Ny×Nz×Nt. As an example, a typical grid of 3.2. CODE DEVELOPMENT 49 NR= 100, Nϕ= 320, Nz= 100, Nt= 50 would consume 25 GB of memory for each component of the perturbation. The 4D spline evaluation is implemented inside the Single Instruction Multiple Data (SIMD) vectorized loop in ASCOT. Thus, it can be technically considered that it will be executed in serial. For this reason, the minimized computational effort of equation (3.20), an operation that is repeated in ASCOT more than 3 times every integration step, is of paramount importance. In fact, it has been observed that the 4D splines evaluation using equation (3.20) is faster (3.3 ms/(marker ·µs)) than the conventional 3D spline evaluation (3.7 ms/(marker ·µs)). Besides, in contrast to Fourier series, a grid refinement would not affect the evaluation speed, even though it would have a drastic effect in memory consumption. The boundary conditions employed in ASCOT are natural in the R,zand tcoordinates and periodic in the ϕ coordinate. This new ASCOT5 interface enables the study of fast ions in the presence of fast MHD events in which the perturbed electromagnetic field changes on the fast-ion orbital time-scales. The module has been tested against a perturbation described with Fourier series that rotates along the field lines. The two methods were compared, showing equivalent results of the given values. Besides, the divergence, which must be zero according to Maxwell’s equation [140], showed undetectable magnification due to numerical errors. The new time-dependent interface has been crucial for the investigation of fast-ion transport and acceleration during ELMs, presented in chapter 6. 50 CHAPTER 3. SIMULATION MODELS Chapter 4 The Fast-Ion Loss Detector FILD is a unique diagnostic in the detection of MHD-induced fast-ion losses [146, 147, 148, 149, 56, 57, 58] and it is crucial in the detection of fusionproduct α-particle losses in D-T experiments [150]. It has been used in most large tokamaks, such as TFTR [55], JET [126], ASDEX Upgrade [151, 152, 153, 154], or DIII-D [155, 156]; spherical tokamaks like NSTX [157]; and stellarators like W7-SA [125]. The goal of this thesis, among others, is the development and analysis of a fast-ion loss detector in MAST Upgrade [158], at the Culham Centre for Fusion Energy, United Kingdom. However, due to several delays in the MAST Upgrade restart, the experiments of this thesis are carried out in ASDEX Upgrade, at the Max Planck Institute for Plasma Physics in Garching, Germany. A description of the two devices is presented here, together with a description of the main diagnostic employed in this thesis, the Fast-Ion Loss Detector. 4.1 The ASDEX Upgrade Tokamak The Axially Symmetric Divertor Experiment Upgrade tokamak (ASDEX Upgrade or AUG) [20, 159, 160, 38] is a conventional medium size tokamak with an approximate aspect ratio of A=R a≈3.0. It is renowned for discovering the H-mode regime in tokamak plasmas [37]. AUG is mainly conditioned for divertor-limited, lower single-null, D-shape plasmas but allows for a wide variety of alternative configurations. An upcoming upgrade of its upper divertor will enable the investigation of alternative magnetic configurations in divertor physics [161]. The maximum toroidal field is 3.1 T near the magnetic axis (R= 1.65 m), and the maximum plasma current is 1.6 MA. The typical plasma electron density and temperature in AUG are ∼1.0×1019 m−3and ∼7 keV, respectively. A total external heating of 30 MW is provided using 51 52 CHAPTER 4. THE FAST-ION LOSS DETECTOR the following systems: •20 MW of NBI heating, provided by 8 beams [162]. The beams are organized into two boxes with a different operating voltage, 60 kV (Q1-Q4) and 93 kV (Q5-Q8). The injection geometry of each beam can be observed in figure 4.1. •4 MW of ICRH power with 4 ICRH antennas toroidally distributed around the low field side wall [163]. •6 MW of ECRH power, fed through 8 gyrotrons [164]. The ECRH systems are fitted with movable mirrors that enable localized heating and current drive (ECCD). AUG holds 16 RMP coils distributed in two rows (8 up and 8 down). The plasma facing components are made of Tungsten (W) with frequent boronization coating to reduce the tungsten impurity released into the plasma [165]. AUG experiments use deuterium and hydrogen as fuel. The main AUG parameters are summarised in table 4.1. Parameter AUG Major radius (m) 1.65 Minor radius (m) 0.5 Plasma current (MA) 1.6 Magnetic field at R= 0.85 m (T) 3.1 NBI power (MW) 20.0 ICRH power (MW) 4.0 ECRH power (MW) 6.0 Pulse length (s) 10 Table 4.1: AUG key parameters. AUG is equipped with an extensive range of diagnostics, including Charge Exchange Recombination Spectroscopy (CXRS) [63, 166], Fast-Ion Deuteriumα(FIDA) [51], Neutral Particle Analiser (NPA) [53], Visible and IR cameras. The most relevant in the scope of this thesis are the following: Magnetic diagnostics An extensive range of magnetic coils are placed around the vacuum vessel. The current induced by magnetic fluctuations across these coils is used to infer many plasma parameters, such as the plasma current and position, halo currents, diamagnetic flows, high-frequency magnetic fluctuations, etc. 4.2. THE MAST UPGRADE TOKAMAK 53 14 16123 4 5 6 7 8 9 10 11 12 13 NBI box 1 b) a) FILD1 FILD4 FILD5 FILD2 FILD3 FILD3 FILD1&2 FILD4 FILD5 Separatrix (Q -Q )1 4 NBI box 2 (Q -Q )5 8 Figure 4.1: Top (a) and poloidal (b) view of the ASDEX Upgrade vacuum vessel. Density and temperature profiles AUG employs Thomson Scattering (TS) [167], lithium beam emission spectroscopy (LIS) [168], deuterium cyanide laser interferometry (DCN) [169] and electron cyclotron emission (ECE) [170]. These heterogeneous diagnostics are combined to coherently estimate the electron temperature and density profiles through integrated data analysis (IDA) using Bayesian probability theory [171]. Divertor current The use of shunts in the AUG divertor makes possible to measure the current flowing through the divertor tiles [172]. FILD poloidal array AUG is fitted with 5 FILD diagnostics [151, 152, 153, 154] distributed around the AUG wall, as it can be seen in figure 4.1. Their different poloidal and toroidal positions make it possible to resolve the poloidal distribution of the fast-ion losses [173]. A fast reciprocating system in FILD4 makes it possible to adapt the radial position of the detector during the discharge, allowing for radial resolution of the fast-ion losses [174]. All the detectors are equipped with high resolution and high sampling frequency sensors. 4.2 The MAST Upgrade Tokamak The Mega Amp Spherical Tokamak (MAST) [175, 176] was built in 1999, motivated by the favourable properties of low aspect ratio tokamaks (A=R a≤ 54 CHAPTER 4. THE FAST-ION LOSS DETECTOR Parameter MAST MAST-U Major radius (m) 0.85 0.85 Minor radius (m) 0.65 0.65 Plasma current (MA) 1.3 2.0 Magnetic field at R= 0.85 m (T) 0.52 0.75 On-axis NBI power (MW) 3.8 2.5 Off-axis NBI power (MW) 0.0 2.5 Total NBI power (MW) 3.8 5.0 Pulse length (s) 0.6 5 Table 4.2: MAST and MAST-U key parameters. 1.5) observed in its predecessor, START [177], where record breaking high-β plasmas (βT≥30%) were achieved. In addition, MAST provides a larger plasma cross section, similar to other conventional medium size tokamaks, like ASDEX Upgrade or DIII-D, thus enabling cross-machine aspect ratio scaling of tokamak phenomena. In 2013, MAST stopped its operation to undergo a major upgrade (MAST-U) [178, 179] that would improve its overall performance, enable new exhaust physics and equip it with state-of-the-art diagnostics. The key parameters of MAST and MAST-U can be observed in table 4.2. The MAST-U vacuum vessel consists of an up-down symmetric cylinder with an aspect ratio A≈1.3, as it can be observed in figure 4.2. It is equipped with 21 poloidal coils and 12 RMP coils distributed in two rows (4 up and 8 down). It contains two symmetric closed divertor chambers fitted with cryopumps. The plasma facing component and divertor tiles are protected by fine grain graphite. The toroidal field coils will provide toroidal magnetic fields up to 0.75 T near the magnetic axis (R= 0.85 m). Its upgraded solenoid makes it possible to sustain pulse durations up to 5 s and plasma currents up to 2 MA, although not concurrently. The solenoid will provide a maximum ohmic heating of 1 MW. Additionally, 2.5 MW on-axis and 2.5 MW off-axis NBI heating will be provided, also contributing to non-inductive current drive. The core plasma electron density and temperature will be ∼1.5×1020 m−3and ∼3 keV, respectively. MAST-U will use deuterium and hydrogen as fuel. MAST-U is equipped with a wide variety of diagnostics, especially dedicated to plasma control and divertor physics, such as ∼600 magnetic coils, Thomson Scattering (TS) near the midplane and in the divertor chamber [180], Charge Exchange Recombination Spectroscopy (CXRS) with unique resolution at the plasma edge [181], Visible and Infra-red (IR) cameras, etc. 4.2. THE MAST UPGRADE TOKAMAK 55 Toroidal field coils Solenoid Main chamber Divertor chamber Poloidal field coils RMP coils Graphite shield Figure 4.2: Cross section of MAST-U. The confined fast-ion profile is measured with the Fast-Ion Deuterium α (FIDA) spectrometer [182] and the solid-state Neutral Particle Analyser (ss- NPA) [183]. The first MAST-U FILD is designed in the scope of this thesis and described in the following sections. The MAST-U research plan [25, 184] will aim to explore the viability of spherical tokamaks as future fusion devices. Foremost, it will be focused on the investigation of alternative divertor configurations, which are illustrated in figure 4.3; such as the super-X configuration [185], which aims to extend the divertor legs, thus increasing the magnetic field lines connection length; or the snowflake configuration [186], which by means of achieving a secondorder X-point, will spread the heat loads onto a larger wall surface. On the fast-ion physics, the MAST-U research plan will focus on optimizing fastion confinement in spherical tokamaks and investigate non-inductive current drive for non-pulsed tokamak operation. The MAST Upgrade project endeavours further enhancements that will be combined with the upcoming experimental campaigns. These enhancements cover the installation of two additional NBIs, achieving a total neutral beam power of 10 MW; new diagnostic upgrades and the installation of a cryoplant to service the divertor cryopump and improve the solenoid cooling. RF heating based on Electron Bernstein Waves [187] is also envisaged. 56 CHAPTER 4. THE FAST-ION LOSS DETECTOR Figure 4.3: All the alternative divertor configurations in MAST-U (single and double-null). 4.3 The MAST-U FILD Design The MAST-U FILD is designed to measure the NBI born fast-ion population in a wide variety of plasma scenarios, with the distinctive spatial and temporal resolution of the scintillator-based detectors. As it is mentioned in section 3.1.2, FILD works as a magnetic spectrometer, collimating the escaping ions from the plasma that reaches its probe head. Thus, the FILD probe is generally installed in regions where the fast-ion orbit drifts benefit the detection of escaping fast ions from the plasma, usually the scrape-off layer at the low field side, near the plasma edge. The MAST-U FILD is allocated in one of the main diagnostic ports, 109 mm above the midplane. Its design enables the adaptation of its radial position [1.40 m,1.60 m] and orientation with respect to the horizontal plane [0◦,90◦]. 4.3.1 Probe Design As it was mentioned in section 3.1.2, the main components of the FILD probe are the protective shield, the collimator and the scintillator plate. The geometry and relative position of these parts strongly affect the FILD measurement range and resolution, so they require a careful design and signal 4.3. THE MAST-U FILD DESIGN 57 Species Energy (keV) Gyroradius (mm) D (1st energy) 75 101 D (2nd energy) 37.5 72 D (3rd energy) 25 58 H (1st energy) 75 72 H (2nd energy) 37.5 51 H (3rd energy) 25 41 Table 4.3: Energy and gyroradius (B= 0.55 T) of the fast-ion population in MAST-U. mock-up with FILDSIM. The measurement range must be so that it covers the fast-ions gyroradii while it blocks the thermal ions from the bulk plasma. The fast ions in MAST-U are produced by the two neutral beams. The maximum beam voltage is 75 kV and thus, they inject hydrogen or deuterium at a main energy of 75 keV and half and third energies of 37.5 keV and 25 keV, respectively. Considering that the local magnetic field on the FILD probe will be approximately 0.55 T, the different fast-ion energies and gyroradii in MAST-U are summarised in table 4.3. The thermal ions gyroradii are considered to be below 20 mm (<3 keV). Thus, the MAST-U FILD measurement range is aimed to be above 40 mm gyroradius. Forward modelling with FILDSIM makes it possible to calculate the strike map and the synthetic signal for different collimator geometries and relative positions of the scintillator plate, with the aim to find a compromise between signal resolution, intensity and range. As an example, figure 4.4 illustrates the strike maps and the synthetic signals for three different collimator heights. The modelled fast-ion distribution is an artificial distribution that covers the three NBI energies in the entire pitch angle range. The results show that a larger height restricts the ion entrance, reducing the signal intensity and measurement range, but it improves the resolution. A height of 12 mm is chosen, because it allows to measure gyroradii above 35 mm and provides enough resolution to distinguish the three NBI energies, as it can be observed in figure 4.4(b). The same logic can be followed with the remaining geometrical parameters, whose final design is shown in figure 4.5(a). Figure 4.5(b) shows the detector resolution for different gyroradii. It can be observed that the resolution lessens at larger gyroradius values, which is geometrically caused by the collimator finite dimensions and is difficult to avoid. On the contrary, the resolution in pitch angle remains almost constant for the entire velocity-space range, Λ = ±2◦. Gyroradii above the maximum shown in the strike map of figure 4.4(b) may be measured, only limited by the plate size. 64 CHAPTER 4. THE FAST-ION LOSS DETECTOR FILD mechanism APD camera CCD camera camera support CCD 1 234 5 78 APD 60 22.5 20 40 70 3 5.2 20 31.3 a) b) Figure 4.9: (a) Light acquisition and support system. (b) Optical system layout (mm) and ray tracing of the two branches. The numeration corresponds to table 4.4. nostic PCs via an Ethernet connection. The APD camera is an APDCAM 10G [192] developed by Fusion InstrumentsR for high-frequency low-level light measurements. It consists of an 8 ×8 array of Hamamatsu avalanche photodiodes arranged in a 20 mm ×40 mm sensor. Its maximum sampling frequency is 2 MHz, its typical quantum efficiency is 85% at 650 nm, higher than photomultiplier tubes, and unlike normal photodiodes, it has internal gain. It has a 14 bit resolution and a noise-to-signal ratio of 0.2 %. Due to its high sampling frequency, the data transmission of the APDCAM can reach up to 2.5 GB during a single shot. For this reason, a 10 Gigabit Ethernet connection with the control PC is used. The two optical branches produced by the beam splitter are designed with the Qioptic WinlensR software. WinlensR is a ray tracing code that allows to dimension the optical system and calculate the resulting image magnification and Seidel aberrations with a large database of standard lenses. In the design, the scintillator plate is defined as a 100 mm object at 1.10 m from the first lens. Due to the different size of each sensor, the magnification of each optical branch is different: m= 0.048 for the CCD camera and m= 0.25 for the APD camera. The resulting optical system is shown in figure 4.9(b). Firstly, the APD branch has been produced based on the Tessar, Heliar or the Super-Dynarex lens designs [193], using an achromat, a plano-convex 4.3. THE MAST-U FILD DESIGN 65 Element Type Focal length (mm) (mm) 1 Achromat 120 50 2 Plano-convex 300 31.5 3 Symmetric-concave -50 22.4 4 Beam splitter 5 Symmetric-convex 50 31.5 6 Aspheric 15 18 7 Plano-concave -30 12.7 8 Plano-convex 12.5 8 Table 4.4: Lens description, according to the numeration in figure 4.9(b). Figure 4.10: Seidel aberrations of each optical component. The numeration corresponds to table 4.4. and a symmetric-concave lens. The beam splitter is allocated between the third lens and the APD sensor, introducing the CCD branch. Therefore, the first three lenses are common to the two branches. Further, in the CCD branch, a symmetric-convex lens works as a field lens and 3 additional lenses achieve the required magnification. The different lenses of the optical system are described in table 4.4. The two branches produce a very low Seidel aberration, as it can be observed in figure 4.10. The scintillator plate consists of a stainless steel plate coated with a layer of ionoluminescent material powder (<0.1 mm). The employed scintillator material is yttrium aluminum garnet activated by cerium (Y3Al5O12 :Ce3+, YAG:Ce or P46). The P46 is characterized for a rather low yield (≈103 photons/ions) but fast decay time (≈70 ns) [194], which makes it possible 66 CHAPTER 4. THE FAST-ION LOSS DETECTOR Figure 4.11: MAST-U FILD data acquisition and control system. to detect fast-ion loss fluctuations up to the 1 MHz bandwidth of the APD camera. The P46 emits in a broad wavelength distribution within the visible spectrum, peaked at 550 ns, making it possible to use sensors in the visible range. The material has shown a very low degradation to ion fluence [195], thus providing a similar performance during long experimental periods. An halogen lamp has been added to the probe head to light up the scintillator plate allowing to set up the cameras and calibrate the optical system. The two PCs that control and save the FILD data acquisition are placed in an electrically isolated area from the tokamak. One PC controls the APD camera and the other controls the CCD camera, the two stepper motors that drive the FILD mechanism and other auxiliary systems like the thermocouple or the halogen lamp. Electric/optical data converters are used at both ends of the isolation barrier. The FILD data acquisition layout can be seen in figure 4.11. The absolute calibration of the MAST-U FILD is expected to be carried out in the next MAST-U opening. It is constructed using the scintillator material yield [195] and the collimator factor. It also requires to calibrate the optical system and sensors using an Ulbricht sphere [196, 197, 198]. With this, absolute measurements of the fast-ion flux, in terms of ions/s, can be provided. Chapter 5 Experimental Results This chapter aims to further investigate the experimental observations of ELM-induced fast-ion losses in ASDEX Upgrade, described in section 2.2.1. The aim of this chapter is to broaden the understanding of the interaction between ELMs and fast ions and provide some conclusions of the parameters that mostly affect this interaction. The investigated shots consist of lower single-null plasmas with Ip= 800 kA, Baxis t= 1.8−2.5 T, low density (naxis e∼6·1019 m−3,Taxis e∼5 keV) and low collisionality (0.4< ν∗ e<1.4) plasmas. The original shots presented in [56, 57, 58] used beams NBI7 and NBI8 to feed the FILD1 probe with passing and trapped particles, respectively. However, the most recent shots aim to populate a wider fast-ion velocity space, covering the measurable range of the entire FILD poloidal array. For this reason, a heating sequence of 0.2 s blips with NBI5, NBI6 and NBI8 is repeated during the entire shot. Additionally, 0.4 s of ICRH are added, completing a repetitive sequence of 1 s. The NBI3 is active during the entire discharge to enable FIDA/BES measurements. FIDA/BES measure the content of confined fast-ions based on charge exchange spectroscopy, using the intensity of the light emitted when a fast ion exchange charges with a neutral from NBI3. The heating scheme, which is shown in figure 5.1(a), is used in the four most recent shots dedicated to investigate ELM-induced fast-ion losses, providing a constant heating power and a varied fast-ion velocity-space distribution. The four dedicated shots, described in the following sections, aimed to investigate the resonant interaction between the ELM perturbation and the fast-ion orbits (#37700 and #37701) and to investigate the effect of varying the ELM stability with edge electron heating (#38020 and #38022). 67 68 CHAPTER 5. EXPERIMENTAL RESULTS 4.0 2.0 0.0 Heating [MW] -4.8 -5.2 -5.6 -6.0 1.0 2.0 3.0 4.0 5.0 6.0 Time [s] q95 NBI3 NBI8 NBI5 NBI6 ICRH #37700 #37701 6.0 a) b) Figure 5.1: (a) Heating scheme used in discharge #37700, #37701, #38020 and #38022. (b) q95 scan covered in shots #37700 and #37701. 5.0.4 Scan in the Plasma Helicity In the earliest reported shots, the accelerated distribution during the ELM perturbation in the FILD1 signal depicted a pitch-angle splitting, which was correlated with the plasma helicity [56, 57, 58]. In those shots (#34614 and #34615), a scan in the plasma q951depicted changes in the dominant pitch angle of the accelerated population. The affected fast ions were born in NBI7 and followed passing orbits, so it was suggested that the pitch-angle splitting was caused by a resonant interaction between the ELM perturbation and the fast-ion orbits. When the plasma q95 is modified, the geometrical resonances of passing particles change, as they depend on the plasma helicity [141, 142]. Consequently, the signal depicts the pitch angle of the resonant orbits for each q95. To deepen the understanding of the interaction between the fast-ion orbits and the ELM perturbation, shots #37700 and #37701 were prepared to investigate the pitch angle structure of the accelerated fast-ion losses with trapped orbits born in NBI8. An extended scan in the plasma q95 is performed by ramping the toroidal magnetic field from - 2.5 T to -2.25 T and from -2.5 T to -2.75 T, respectively, while keeping a constant toroidal current of 800 kA. Thus, the total q95 scan ranges from -4.8 to -6, as it can be observed in figure 5.1(b). Figure 5.2(a) shows the divertor shunt current, used to monitor the ELMs, 1Safety factor, q, evaluated in the flux surface ρpol = 0.95. 69 Figure 5.2: (a) Time trace of the inner and outer divertor current, used to monitor ELMs. The dashed grey lines mark the ELM onset and the shaded area mark the ELM duration. (b) Time trace of the FILD1 signal. The dashed grey lines mark the ELM onset. (c) Velocity-space of the FILD1 signal. and figure 5.2(b) shows the gyroradius of the fast-ion losses resolved with the 1 kHz CMOS camera installed in FILD1. The vertical dashed lines mark the ELM onset, defined by the rise in the divertor current, while the shaded areas mark the ELM duration. The gyroradii above the injection energy can be observed to be well correlated with the ELM onset in figure 5.2(b), evidencing an ELM-induced accelerated population. Although the distribution is distorted due to the collimator finite resolution, the accelerated population is localized 1.5 cm (∼60 keV) above the injection energy, as it can be observed in figure 5.2(c). No pitch-angle splitting was observed on the trapped particles signal during the q95 scans. Conversely, the accelerated population depicted a pitch angle centred with respect to the NBI prompt losses during the entire experiments. It must be noted that trapped orbits have geometrical resonances which are broader in phase space and do not depend on the plasma q. Thus, the absence of a pitch angle splitting on trapped orbits supports the hypothesis of a resonant interaction between the fast ions and the ELM perturbation. 5.0.5 Scan in Edge Electron Heating Recent experiments have used ECRH applied on the edge to modify the pedestal pressure and current density profile, with the aim to test the peelingbalooning model [199, 200, 201]. In those experiments, the ECRH power was gradually directed from the plasma core up to ρpol = 0.9. It was observed, 70 CHAPTER 5. EXPERIMENTAL RESULTS 1.0 1.5 2.0 2.5 R [m] -1.0 -0.5 0.0 0.5 1.0 z [m] Gyr: 1 Gyr: 2 Gyr: 3 Gyr: 4 Gyr: 7 Gyr: 8 0.0 0.2 0.4 0.6 0.8 1.0 Gyr: 1 Gyr: 2 Gyr: 3 Gyr: 4 Gyr: 7 Gyr: 8 0.0 0.2 0.4 0.6 0.8 1.0 -0.15 -0.10 -0.05 0.00 0.05 Current [MA/m ] 2 Gyr: 5 Gyr: 7 Gyr: 8 Gyr: 1 Gyr: 2 Gyr: 3 Gyr: 4 0 2 4 6 8 10 Power [MW/m3] Gyr: 5 Gyr: 7 Gyr: 8 Gyr: 1 Gyr: 2 Gyr: 3 Gyr: 4 Gyr: 5 Gyr: 7 Gyr: 8 Gyr: 1 Gyr: 2 Gyr: 3 Gyr: 4 #38022 5.10@ s #3802 100@4. s #3802 100@4. s #3802 102@5. s ρpol ρpol #3802 102@5. s -0.15 -0.10 -0.05 0.00 0.05 Current [MA/m ] 2 0 2 4 6 8 10 Power [MW/m3] a) b) c) d) e) resonant layer Total Total Total Total Figure 5.3: (a) Poloidal view of the EC waves from the different ECRH launchers. (b) Profile of the ECRH power deposition in shot #38020. (c) Current drive profile in shot #38020. (d) Profile of the ECRH power deposition in shot #38022. (e) Current drive profile in shot #38022. that when the ECRH heating was directed towards the edge, the ELM repetition frequency increased above the values foreseen by the peeling-ballooning model. The ECRH waves can be oriented making use of their movable mirrors, whose toroidal and poloidal angles can be resolved with the TORBEAM code [202]. TORBEAM is a beam tracing code for electron-cyclotron (EC) waves that resolves the propagation and absorption of the wave in the plasma. It calculates the position and profiles of deposited power and current driven given a resonant layer, whose position depends on the plasma magnetic field, as the resonant condition occurs when the wave frequency matches the electron cyclotron frequency ωe=eB me. Motivated by the aforementioned results, two dedicated shots are designed to investigate the effect of ELM stability on the ELM-induced fastion losses. In these shots, a scan in the edge ECRH power is performed by sequentially starting the 8 ECRH launchers in steps of 1 s, matching the repetitive heating sequence in figure 5.1(a). Gyrotrons 1 and 4 were pointing toward the plasma core and were active during the entire shot to avoid impurity accumulation that would destabilize the plasma. The orientation of the remaining gyrotrons was optimized with TORBEAM to concentrate 71 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] 0 2 Heating [MW] 1 f [Hz] ELM 100 200 300 1.0 0.5 0.0 Counts [a.u.] Gyroradius [cm] Gyroradius [cm] 6 4 2 6 4 2 1 2 3 4 5 6 Time [s] a) b) c) d) e) NBI3 NBI8 NBI5 NBI6 ICRH ECRH FILD1 FILD4 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] 1.0 0.8 0.6 0.4 0.0 0.2 Counts [a.u.] Heating [MW] f [Hz] ELM Counts [a.u.] Gyroradius [cm] Gyroradius [cm] 2 3 4 5 6 Time [s] 1 f) g) h) i) j) NBI3 NBI8 NBI5 NBI6 ICRH ECRH FILD1 FILD4 FILD1 FILD4 0 4 2 100 200 1.0 0.5 0.0 6 4 2 6 4 2 #38020 #38022 FILD1 FILD4 Figure 5.4: (a, f) External heating, (b, g) ELM repetition frequency, (c, h) time traces of FILD1 and FILD4, (d, i) histogram of FILD1 and (e, j) histogram of FILD4 in shots #38020 and #38022, respectively. the power deposition within ρpol = [0.8,0.9]. Besides, the toroidal angles of the edge gyrotrons were set to minimize the current driven in shot #38020 and maximize it in shot #38022. Yet, a fault in the mirror actuators slightly mismatched the desired configuration. The actual power and current deposition is simulated with TORBEAM and shown in figure 5.3, where it can be observed that the ECRH heating was deposited in a broader edge region (ρpol = [0.65,0.9]) and the edge current drive in shot #38022 was drastically reduced. Moreover, gyrotron 6 was not available and gyrotron 5 failed during shot #38020. This reduced the maximum achieved ECRH power, reaching 2 MW in shot #38020 and 4 MW in shot #38022, as it can be observed in figures 5.4(a) and 5.4(f). The first edge gyrotron is started at t= 2 s. This caused a drastic increase in the ELM repetition frequency from 100 Hz to 200 Hz, as it can be observed in figure 5.4(b) and 5.4(g). Afterwards, the ELM repetition frequency remains stable and it even decreases at the latest stage of the discharge, when the edge ECRH power is maximum. This suggests that a small amount of edge ECRH power is sufficient to observe the reported changes in the ELM frequency while higher ECRH power does not cause any further effect. The FILD signals, shown in figures 5.4(c) and 5.4(h), follow a gradual growth during the shots which drastically increases during the ELMs, especially in FILD4, suggesting that the ELM-induced fast-ion losses scale up 72 CHAPTER 5. EXPERIMENTAL RESULTS Figure 5.5: (a, c) Density profiles and (b, d) FIDA emission in shot #38020 and #38022, respectively. 1.00 -10 0 20 Counts [a.u.] Current [kA] inner divertor outer divertor FILD4 #38022 10 0.66 0.33 0.00 5.86 5.87 5.88 5.89 5.91 5.93 5.94 Time [s] Gyroradius [cm] FILD4 5.925.90 6 4 2 Counts [a.u.] 1.0 0.8 0.6 0.4 0.2 0.0 a) b) c) Figure 5.6: (a) Time trace of the inner and outer divertor current, (b) time trace of FILD4 and (c) histogram of FILD4 during an ICRH phase in shot #38022. The dashed lines mark the ELM onset and the shaded area mark the ELM duration. The red circles in (b) mark the detected peaks of fast-ion losses correlated with the ELM activity. 73 with the edge ECRH power. The enhancement of the fast-ion losses is caused by a gradual increase of the density profile during the shots, as it is shown in figures 5.5(a) and 5.5(c). The increased density alters the fast-ion deposition profile, reducing the fast-ion content on the core, as it can be observed in the FIDA emission (figures 5.5(b) and 5.5(d)), which can be correlated with the fast-ion content at the given poloidal coordinate. Consequently, a larger fast-ion population is expected to concentrate in the edge, available to interact with the ELM stability. As a matter of fact, the FIDA emission does not reveal a higher fast-ion content in the plasma edge, what suggests that the fast ions are being expelled. This is in line with the enhancement in the FILD signal. Therefore, an increase in the plasma density leads to more fast ions in the edge which ultimately results in larger ELM-induced fast-ion losses. The results motivate further investigations of the interaction between fast ions and ELMs using ELM pacing techniques that, in turn, enhance the plasma density, such as gas puffing [200] or pellet injection [203]. The enhancement of the ELM-induced fast-ion losses makes it possible to observe peaks correlated with the ELM activity from all the employed NBI sources, thus suggesting that the ELM enhances fast-ion losses in a wide range of velocity-space distributions. Besides, an enhancement of the fastion losses clearly correlated with the ELM onset can be observed during the ICRH phase in FILD4, as it is shown in figure 5.6(b). This suggests that the ELM activity also enhance the losses of ICRH fast ions. Nonetheless, the velocity-space of these losses cannot be distinguished, as it is illustrated in figure 5.6(c), because the 60 Hz CCD camera installed in FILD4 does not offer enough temporal resolution to resolve the intra-ELM velocity space of the fast-ion losses. Nonetheless the 64 px sensor of the APD camera reveals that the ELM-induced losses are localized in phase-space. 5.0.6 Statistical Analysis Since the ELMs are quasi-periodic fluctuations with typical repetitive frequencies of ∼100 Hz, the number of ELMs during a 6 s discharge is enough to perform a statistical analysis of the FILD signal induced by ELMs. This makes it possible to achieve a more general understanding of the ELM- induced fast-ion losses and find the parameters that most strongly affect them. A dataset of AUG H-mode discharges with FILD data has been constructed, covering the original shots in reference [56, 57, 58], some shots dedicated to ELM mitigation with RMPs and the latest shots dedicated to the investigation of ELM-induced fast-ion losses presented in the previous sections. An overview of the included shots is given in table 5.1. An algorithm has been developed to detect the several peaks appearing in 80 CHAPTER 6. NUMERICAL RESULTS 2.28 1.71 1.14 0.57 0.00 -0.57 -1.14 -1.71 -2.28 -2.85 4.5 3.6 2.7 1.8 0.9 0.0 -0.9 -1.8 -2.7 -3.6 -0.2 -0.4 -0.6 -0.8 -1.0 -0.2 -0.4 -0.6 -0.8 -1.0 x10-21 [keV] [kg·m2/s] separatrix Pitch Pitch b) TΔ a) PΔ φ 3.5 3.0 2.3 2.0 1.6 1.2 0.230 0.380 trapped passing 3.5 3.0 2.3 2.0 1.6 1.2 0.230 0.380 trapped passing 1.90 1.95 2.00 2.102.05 2.15 2.20 R [m] 1.90 1.95 2.00 2.102.05 2.15 2.20 R [m] separatrix 2200 1760 1320 880 440 0 -440 -880 -1320 -1760 -0.2 -0.4 -0.6 -0.8 -1.0 [J/T] Pitch c) μΔ 3.5 3.0 2.3 2.0 1.6 1.2 0.230 0.380 trapped passing 1.90 1.95 2.00 2.102.05 2.15 2.20 R [m] separatrix Figure 6.3: (a) Variation of the toroidal canonical momentum and ωpol/ωtor contour lines. (b) Variation of the kinetic energy and ωpol/ωtor contour lines. (c) Variation of the magnetic moment and ωpol/ωtor contour lines. 6.1. FAST-ION TRANSPORT AND ACCELERATION IN AUG 81 trix, where the electromagnetic perturbation is stronger. Besides, it can be seen that the transport and acceleration does not reveal the same patterns in the phase-space. This indicates that the transport and the acceleration are decoupled, being δBthe dominant responsible for the particle transport and δEthe dominant responsible for the acceleration, independently. As a consequence, particles with both transport and acceleration will occur in the narrow phase-space regions where the two effects overlap. To evaluate a resonant interaction, the orbits resonance condition is expressed in terms of the fraction ωpol/ωtor, having different expressions depending on the orbit nature [141, 142]. As an example, for an electromagnetic perturbation with a toroidal mode number n, the linear resonance condition of a trapped orbit is: ωpol ωtor =n p(6.3) where ωpol is the orbit poloidal frequency, ωtor is the orbit precession frequency and pis the bounce harmonic, an arbitrary integer. A clear correlation between the high |∆Pϕ|lines and the ωpol/ωtor contour lines can be observed in figure 6.3(a). This clearly suggests a resonant interaction between the fast-ion orbits and the magnetic perturbation, which produces the fast-ion transport. The patterns in figures 6.3(b) and 6.3(c) do not follow the ωpol/ωtor contour lines. Nonetheless, they are analogous to each other, depicting that the energy variation is associated with a variation of the perpendicular velocity component in the cyclotron time scale. It can be demonstrated that the energy variation is mainly produced by the v×Bterm of the electric field, while the resistive term has a negligible effect. Thus, the fast-ion acceleration is caused by the enhanced transport during the ELM, which produces high electric fields in the plasma edge, as it is shown in figure 6.4(a). Due to the high electric field gradients in the scale of the fast-ion gyroradii, the conservation of the magnetic moment is lost, thus producing a net energy variation. Figure 6.4(b) shows the energy variation of a set of markers distributed with z= 0, ϕ= 0, Λ = −0.3 and ranged over the initial radial position, R= [1.90,2.20] m, and the initial energy, T= [30,150] keV. Lines of high energy variation can be observed, highlighting the phase-space where this effect occurs resonantly. The lines extend toward energies above 60 keV, which suggests that particles can resonantly increase their energy for a few tens of keV, while they cannot lose energy below 60 keV. In addition to the variation in the constants of motion, a poloidal Poincar´e plot has been made. The Poincar´e plot depicts the intersections of the magnetic field lines with a poloidal angle (ϕ= 0). Thus, the Poincar´e plots can reveal structures such as magnetic islands or chaotic field lines [211]. Figure 6.5 shows the resulting Poincar´e plot of the ELM perturbation, where it can be observed 82 CHAPTER 6. NUMERICAL RESULTS 1.90 1.95 2.00 2.10 2.202.15 R [m] 140 Energy [keV] 120 100 80 60 40 2.05 4.2 b) ΔT 3.4 [keV] 2.5 1.7 0.8 0.0 -0.8 -1.7 -2.5 -3.4 a) δE 0.10 separatrix -0.10 0.00 -0.05 0.05 z [m] 2.05 2.20 R [m] 2.152.10 1.0 Δp [a.u.] 0.5 0.0 -0.5 -1.0 -1.5 60 keV Figure 6.4: (a) Poloidal δEvector field. (b) Variation of the kinetic energy in a R−Tgrid. that magnetic field lines become chaotic beyond ρpol >0.81. Thus, besides the discussed transport and acceleration mechanisms, particle transport is also affected by the chaotic field lines beyond ρpol >0.81. The chaotic effect is mostly visible in the passing region of figure 6.3(a), breaking the lines of high |∆Pϕ|. The results above foresee a maximum fast-ion acceleration of 4.5 keV per 10 poloidal turns. Considering that the orbital time scale ranges between 3 - 8 µs, the necessary time to accelerate fast ions over 50 keV is in the order of the ms. However, the duration of the MEGA simulation, marked by the unshaded region in figure 6.1, is only 0.1 ms. No wonder, the growth rate in MEGA is an order of magnitude higher than the observed experimentally. Consequently, the duration of the MEGA perturbation has been artificially extended up to 1 ms with the aim to observe relevant fast-ion acceleration. This effectively reduces the growth rate to γ= 104s−1. In turn, since the growth rate is proportional to the plasma resistivity (γ∝η1/3) [212], it must be noted that the resistive effects will be overestimated in the results. Nonetheless, it has been discussed already that its effects are negligible in the fast-ion transport and acceleration. The transport and acceleration of the NBI-born fast-ion distribution in AUG are assessed by modelling the 8 beams with BBNBI, using the updated data from the beam calibration in 2017. BBNBI produces a distribution of 6 million markers for each beam, that will be treated in ASCOT independently. The markers are first tracked in an unperturbed field to filter out the NBI prompt losses. Then, the orbits of the confined fast ions are tracked during the ELM perturbation. Table 6.1 shows the fraction of the 6.1. FAST-ION TRANSPORT AND ACCELERATION IN AUG 83 0.75 0.80 0.85 0.90 0.95 1.00 0 50 150 300 350 θ [ ] o ρpol 100 200 250 Figure 6.5: Poloidal Poincar´e map of the perturbed magnetic field lines during the ELM. NBI# 1 2 3 4 5 6 7 8 % ELM losses 0.24 0.27 0.31 0.24 0.36 0.63 0.59 0.36 Λmain -0.2 -0.4 -0.45 -0.25 -0.35 -0.55 -0.7 -0.45 Table 6.1: Fraction of lost fast ions during an ELM and main pitch angle of the NBI distribution of each beam. fast-ion distribution lost during the ELM perturbation. In total, a 0.37% of the fast ions are lost during the ELM. The patterns of the ELM-induced losses on the AUG wall are shown in figure 6.6(a). It can be observed that the ELM-induced losses are concentrated on the limiters at the LFS and the FILD probes. Simulations carried out with an axisymmetric wall, thus removing the protruding elements, show that the ELM-induced losses, follow the high-n, field-aligned patterns of the ELM perturbation, as it can be observed in figure 6.6(b). This illustrates the strong impact of the ballooning perturbation on the fast-ion confinement. Table 6.1 also shows the main pitch angle of the beams, – i.e., the pitch angle corresponding to the maximum of the fast-ion distribution –. The beams with pitch angles near the high ∆Tregions in figure 6.3(b), – i.e., NBI1 and NBI4 –, depict a fast-ion energy distribution that spreads over high energies during the ELM, forming a local maximum at 30 keV above the injection energy, as it can be observed in figure 6.7. As a result, an accelerated population can be distinguished. The velocity space of the fast ions colliding with the FILD probes can be 84 CHAPTER 6. NUMERICAL RESULTS 60 30 0 -30 -60 -90 -120 60 30 0 -30 -60 -90 -120 0 60 120 180 240 300 360 φ [ ] o θ [ ] oθ [ ] o 1.0 0.5 0.0 Particles [a.u.] 1.0 0.5 0.0 Particles [a.u.] a) b) FILD1FILD2 FILD5 FILD4 Figure 6.6: ELM-induced fast-ion losses on the AUG 3D wall (a) and 2D wall (b) projected in the toroidal and poloidal angles. used as an estimate of the FILD signal. Figure 6.8 shows the velocity space of the NBI1 and NBI4 distributions colliding with FILD1 and FILD2. A high energy component, 40 keV above the injection energy, that converges to a narrow pitch angle range, can be observed in the velocity space of the fast ions impinging on FILD1 and FILD2. This high-energy feature is only observed with the distributions from NBI1 and NBI4, since the accelerated population is larger in these distributions, as it was discussed before. The high energy feature is not observed in FILD3, FILD4 or FILD5. This might be explained by the strong effect of the ballooning perturbation, as it was mentioned before, expelling fast ions near the midplane, where only FILD1 and FILD2 are located. 6.2 FILD Signal for the MAST-U Baseline Scenario The numerical models described in section 3.1 are used to estimate the FILD signal in a MAST-U baseline scenario [213]. BBNBI calculates the NBI ionization profile that is then traced with the ASCOT code. Subsequently, 6.2. FILD SIGNAL FOR THE MAST-U BASELINE SCENARIO 85 a) NBI1 020 40 60 80 100 120 140 0.00 0.25 0.50 0.75 1.00 Particles [a.u.] Energy [keV] T0 T /2 0 T /3 0 70 80 90 100 110 120 0.00 0.01 0.02 0.03 0.04 Particles [a.u.] Energy [keV] c) accelerated population b) NBI4 020 40 60 80 100 120 140 0.00 0.25 0.50 0.75 1.00 Particles [a.u.] Energy [keV] T0 T /2 0 T /3 0 0.00 0.01 0.02 0.03 0.04 Particles [a.u.] Energy [keV] d) accelerated population 70 80 90 100 110 120 Figure 6.7: (a, b) Energy distribution of NBI1 and NBI4 after the ELM perturbation. (c, d) Magnification of the NBI distributions, where an accelerated population can be seen. d) FILD2 NBI4c) FILD2 NBI1 b) FILD1 NBI4a) FILD1 NBI1 -0.6 -0.5 -0.4 -0.3 -0.1-0.2 0.0 Pitch 140 Energy [keV] 120 100 80 60 40 20 0 1.0 0.5 0.0 Particles [a.u.] -0.6 -0.5 -0.4 -0.3 -0.1-0.2 0.0 Pitch 140 Energy [keV] 120 100 80 60 40 20 0 1.0 0.5 0.0 Particles [a.u.] -0.6 -0.5 -0.4 -0.3 -0.1-0.2 0.0 Pitch 140 Energy [keV] 120 100 80 60 40 20 0 1.0 0.5 0.0 Particles [a.u.] -0.6 -0.5 -0.4 -0.3 -0.1-0.2 0.0 Pitch 140 Energy [keV] 120 100 80 60 40 20 0 1.0 0.5 0.0 Particles [a.u.] Figure 6.8: Velocity space distribution of the ELM-induced fast-ion losses reaching FILD1 and FILD2 from NBI1 and NBI4. 86 CHAPTER 6. NUMERICAL RESULTS 0 ×1019 0 1 2 3 4 5 6 7 8 ρpol T [eV] e 0 200 400 600 800 1000 1200 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 ρpol n [m ] e -3 a) b) Figure 6.9: Electron density (a) and temperature (b) profiles for the MAST-U baseline scenario. the fast-ion distribution that collides with the FILD probe is computed by FILDSIM, which creates a synthetic frame of the FILD plate. The baseline scenario consists of a MHD-quiescent plasma with a conventional double-null divertor configuration. The plasma current and toroidal magnetic field are Ip= 2 MA, Bt= 0.75 T at the magnetic axis. The target kinetic profiles can be observed in figure 6.9 and the plasma shape is shown in figure 6.10(a). The MAST-U NBI ionization profile is modelled with BBNBI. The NBI main injection energy is 75 keV. The 2.5 MW power of each NBI is injected in a proportion of 62 % at the main energy, 27 % at the half energy (37.5 keV) and 11 % at the third energy (25 keV). The MAST-U beamlines have been implemented in BBNBI based on the construction models, whose poloidal and toroidal geometries can be observed in figure 6.10(a) and 6.10(b), respectively. The on-axis NBI (South, SS) is placed on the midplane (z= 0 mm) and the off-axis NBI (South-West, SW) is placed at z= 650 mm above the midplane, both with horizontal beamlines. The NBI beamlets are evenly distributed in a diamond-shaped rectangular grid with an horizontal focal length of 14 m and a vertical focal length of 6 m. The beamlets divergence is 0.6◦. Figures 6.10(a) and 6.10(b) show the estimated beam deposition of the SS and the SW NBI. Figure 6.10(c) illustrates the ionization profile of each injector as a function of the radial coordinate ρpol. It can be observed that the off-axis injector increases the fast-ion density at the edge of the plasma (ρpol >0.6) while the on-axis injector provides a more evenly distributed ionization profile up to the plasma core. The fraction of neutrals that goes through the plasma without ionizing, so-called shine-through, is estimated to be 695 W from the SS injector and 7120 W from the SW injector. Therefore, the shine-through power is a 0.16 % of the total injected power, suggesting a good NBI absorption. The shine-through creates hot spots on the tokamak 6.2. FILD SIGNAL FOR THE MAST-U BASELINE SCENARIO 87 R [m] z [m] 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 1 2 3 4 5 6 7 x 1021 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 ρpol counts [a.u.] [m ] -3 SS injector SW injector R [m] z [m] −2 −1 0 1 2 −2 −1 0 1 2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 x 1021 [m ] -3 SW injector SS injector a) b) c) 200 mm FILD inj SS inj SW total FILD I t B t Figure 6.10: (a) Poloidal and (b) toroidal cross-section of the beam deposition in MAST-U. NBI beamlines are marked in green. (c) Densities of ionized beam neutrals from the on-axis (blue) and the off-axis (red) NBI and total fast-ion profile (black). wall, mainly in sector 2 and the upper part of sector 4, as it is shown in figure 6.11(a), where further reinforcements are installed to guarantee the vessel integrity. It can be observed that FILD is out of the range of the hot spots. BBNBI produces an ensemble of 5 million test particles following the distributions in figure 6.10. The particle orbits are traced with ASCOT, using the baseline magnetic field and a 3D model of the tokamak wall, including the FILD probe. The particles are followed for 10−4s. If the test particle does not collide with the wall during this period, the particle is considered confined. The distribution of lost particles allows to estimate the power load on the wall. The simulation is repeated for different FILD radial positions along its operating range, R= [1.40,1.60] m. It is observed that FILD intersects fastion orbits in a range between [1.40,1.50] m, thus suggesting a wide operating 88 CHAPTER 6. NUMERICAL RESULTS Figure 6.11: (a) Power deposition of the NBI shine-through in MAST-U. (b) Fast-ion power load on the FILD probe (R = 1.4 m). 6.2. FILD SIGNAL FOR THE MAST-U BASELINE SCENARIO 89 Figure 6.12: (a) Poloidal view of the two orbit topologies reaching the FILD probe. (b) Synthetic FILD frame for the fast ions impinging on the FILD probe. range while a good shielding in its retracted position. The fast-ion power impinging on the FILD probe is 357 W when it is fully inserted, as it is shown in figure 6.11(b). The fast ions reaching the FILD head can be considered an estimation of the NBI prompt losses. Each NBI produces a different fastion distribution that impinges on the FILD probe. The fast ions from the SS injector have a pitch angle of Λ = 73◦when they collide with the FILD probe, whereas the fast ions from the SW injector are Λ = 54◦. The orbit topologies of the different fast-ion distributions reaching the FILD probe are shown in figure 6.12(a). The FILDSIM code uses a weighting function that takes into account the finite resolution in velocity space of the FILD detector. It is used to distort the fast-ion distribution on the FILD probe and project it onto a synthetic scintillator plate, where a strike map is also constructed using a field-aligned B = 0.63 T. The synthetic frame, illustrated in Fig. 6.12(b), shows two separate spots on the strike map, each corresponding to a different NBI distribution. This shows that the MAST-U FILD is designed with enough velocity space resolution to distinguish the fast-ion distributions coming from each NBI. This result is aimed to be reproduced during the upcoming diagnostic commissioning. 96 CHAPTER 7. CONCLUSION •The drift in pitch angle is not observed experimentally. This suggests that an electric field with both parallel and perpendicular component should be causing the fast-ion acceleration, as it was proposed by Marchenko et al. [104]. The simulations in AUG have qualitatively reproduced the main experimental observations. However, the important differences between the ELM perturbation simulated with MEGA and the experimental data must be considered, such as the higher dominant toroidal mode numbers and the higher growth rate of the simulated perturbation. Besides, the actual experimental observation of ELM-induced fast-ion acceleration with NBI7 and NBI8 in FILD1 could not be reproduced, probably because of the different topologies of the ELM perturbation. In place, the numerical results are obtained with NBI1 and NBI4, which populates the most resonant regions in the modelled perturbation. The fast-ion simulations in MAST-U aim to foresee the effect of ELMs on the fast-ion population using an ELM perturbation modelled with JOREK. The MAST-U simulations show ELM-induced losses near the midplane on the low field side of the tokamak wall, depicting the strong effect of the ballooning perturbation. They reveal an acceleration mechanism very sensitive to the fast-ion velocity space, similarly to AUG. Besides, the synthetic FILD signal shows accelerated fast-ion losses spread in energy rather than localized, in contrast to the tomographic inversion of the FILD signal in AUG. This could be caused by the larger rate of fast-ion losses during the ELM in MAST-U, that increases the losses along a larger range of energies. The pitch angle of the accelerated losses is aligned with the NBI pitch angle, in agreement with the AUG experiments. As the electric field parallel and perpendicular components are of the same order of magnitude in the MAST-U model, it corroborates the need to invoke a parallel and a perpendicular electric perturbation to keep the pitch-angle structure. 7.2 Outlook MAST-U achieved its first plasma in October 2020. After the commission of the main systems, the first experimental campaign is starting in 2021. During the experimental campaign, FILD will be commissioned, providing measurements of the fast-ion losses for many experiments in MAST-U and testing the design presented in this thesis. Dedicated experiments on the interaction between ELMs and fast ions are being organized in MAST-U. With regards to AUG, further experiments are being planned to compare the fast-ion losses during ELM-mitigated regimes and to investigate the effect on 7.2. OUTLOOK 97 the fast-ion losses of ELM-pacing techniques that enhance the plasma density, such as gas puffing [200] or pellet injection [203]. The dataset presented in this thesis will allow for more statistical analyses, such as the comparison of ELM-synchronized FILD data from different poloidal and toroidal positions. MAST-U, TCV and JET data of ELM-induced fast-ion losses is expected to be added to the dataset, enabling multi-machine studies that will make it possible to construct a scalability diagram of the ELM-induced fast-ion losses. This will help to estimate the fast-ion losses in future devices. Alternative acceleration mechanisms will be investigated, such as wave-particle energy transfer between the fast ions and high-frequency magnetic perturbations (∼ MHz). In this hypothesis, the accelerated fast ions would be merely expelled by the ELMs. High-frequency Alfv´en eigenmodes have already been observed in ASDEX Upgrade with the Ion-Cyclotron Emission (ICE) diagnostic in the presence of beam-injected ions RF-accelerated at the third cyclotron harmonic [216]. Nonetheless, these high frequency modes are suspected to be shear waves, localized in the plasma core near the magnetic axis, thus being very unlikely correlated with the accelerated fast-ion losses during ELMs. The great differences between the ELMs observed experimentally and the ELM perturbation simulated in MEGA motivate the modelling of the fast-ion transport and acceleration with an ELM perturbation closer to the experiments. It is believed that the main differences would be overcome adding diamagnetic effects to the MHD simulations, as it has proven to lower the dominant mode number and growth rate in the ELM simulations [208]. Besides, the effect of fast ions on the ELM stability is currently being adressed with MEGA hybrid kinetic-MHD simulations [206, 207] and will be experimentally investigated in AUG. 98 CHAPTER 7. CONCLUSION List of Publications List of publications as main author •J.F. Rivero-Rodriguez et al. Investigation of beam-ion transport and acceleration during edge localized modes in ASDEX Upgrade. in preparation. •J.F. Rivero-Rodriguez et al. Upgrade and absolute calibration of the JET scintillator-based Fast-Ion Loss Detector. Rev. Sci. Instrum., 92:043553, 2021. •J.F. Rivero-Rodriguez et al. A fast model to resolve the velocity-space of fast-ion losses detected in ASDEX Upgrade and MAST Upgrade. J. Instrum., 14:C09015, 2019. •J.F. Rivero-Rodriguez et al. A rotary and reciprocating scintillator based fast-ion loss detector for the MAST-U tokamak. Rev. Sci. Instrum., 89:10I112, 2018. List of additional publications •J. Gonzalez-Martin et al. Self-Adaptive Diagnostic of Radial Fast- Ion Loss Measurements on the ASDEX Upgrade Tokamak. Rev. Sci. Instrum., 92:053538, 2021. •P. Oyola et al. Implementation of synthetic fast-ion loss detector and imaging heavy ion beam probe diagnostics in the 3D hybrid kinetic- MHD code MEGA. Rev. Sci. Instrum., 92:043558, 2021. •J. Galdon-Quiroga et al. Observation of accelerated beam ion population during edge localized modes in the ASDEX Upgrade tokamak. Nucl. Fusion, 59:066016, 2019. •J. Gonzalez-Martin et al. First measurements of a magnetically driven fast-ion loss detector on ASDEX Upgrade. J. Instrum., 14:C11005, 2019. 99 100 CHAPTER 7. CONCLUSION •G. Birkenmeier et al. Beam modelling and hardware design of an imaging heavy ion beam probe for ASDEX Upgrade. J. Instrum., 14:C10030, 2019. •J. Ayllon-Guerola et al. Determination of the Fast-Ion Phase-Space Coverage for the FILD Spatial Array of the ASDEX Upgrade Tokamak. J. Instrum., 14:C10032, 2019. •H. Meyer et al. Overview of physics studies on ASDEX Upgrade. Nucl. Fusion, 59:112014, 2019. •E. Joffrin et al. Overview of the JET preparation for deuterium-tritium operation with the ITER like-wall. Nucl. Fusion, 59:112021, 2019. •B. Labit et al. Dependence on plasma shape and plasma fueling for small edge-localized mode regimes in TCV and ASDEX Upgrade. Nucl. Fusion, 59:086020, 2019. •M. Garcia-Munoz et al. Active control of Alfv´en eigenmodes in magnetically confined toroidal plasmas. Plasma Phys. Control. Fusion, 61:054007, 2019. •L. Sanchis et al. Characterisation of the fast-ion edge resonant transport layer induced by 3D perturbative fields in the ASDEX Upgrade tokamak through full orbit simulations. Plasma Phys. Control. Fusion, 61:014038, 2019. •J. Galdon-Quiroga et al. Beam-Ion Acceleration during Edge Localized Modes in the ASDEX Upgrade Tokamak. Phys. Rev. Lett., 121:025002, 2018. •J. Gonzalez-Martin et al. First measurements of a scintillator based fast-ion loss detector near the ASDEX Upgrade divertor. Rev. Sci. Instrum., 89:10I106, 2018. •J. Galdon-Quiroga et al. Velocity-space sensitivity and tomography of scintillator-based fast-ion loss detectors. Plasma Phys. Control. Fusion, 60:105005, 2018. •V. Kiptily et al. Fusion product losses due to fishbone instabilities in deuterium JET plasmas. Nucl. Fusion, 58:014003, 2018. •J. Galdon-Quiroga et al. Conceptual design of a scintillator based Imaging Heavy Ion Beam Probe for the ASDEX Upgrade tokamak. J. Instrum., 12:C08023, 2017. 7.2. OUTLOOK 101 •M. Rodriguez-Ramos et al. First absolute measurements of fast-ion losses in the ASDEX Upgrade tokamak. Plasma Phys. Control. Fusion, 59:105009, 2017. •J. Ayllon-Guerola et al. Dynamic and thermal simulations of a fast-ion loss detector for ITER. Fusion Eng. Des., 123:807-810, 2017. •M. Kocan et al. The impact of the fast ion fluxes and thermal plasma loads on the design of the ITER fast ion loss detector. J. Instrum., 12:C12027, 2017. •J. Ayllon-Guerola et al. A fast feedback controlled magnetic drive for the ASDEX Upgrade fast-ion loss detectors. Rev. Sci. Instrum, 87:11E705, 2016. •M. Garcia-Munoz et al. Conceptual design of the ITER fast-ion loss detector. Rev. Sci. Instrum, 87:11D829, 2016. 102 CHAPTER 7. CONCLUSION Bibliography [1] Intergovernmental Panel on Climate Change. 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(b) Magnetic field gradient due to the field line curvature. 28 2.3 Schematic drawing of a linear magnetic mirror caused by the magnetic field gradient in the direction of the field line [13]. . 30 2.4 Poloidal (a) and top (b) view of a passing (blue) and a trapped orbit(red)[67]. .......................... 31 2.5 (a) Timetraces of the electron density (blue), the loop voltage (red) and the divertor current (black). (b) Timetraces of the two toroidally displaced FILDs. (c,d) Close-up of a single ELM for figure (a) and (b), respectively [56]. . . . . . . . . . . 35 2.6 (a) Intra-ELM velocity-space measurements of the fast-ion losses [56]. (b) Gyroradius profile of the intra-ELM measurements between Λ = [58◦,65◦]. The blue crosses are the experimental data, the red curves correspond to the tomographic inversion and the black curve is the reconstructed signal [56]. . 35 3.1 ASCOT5 scalability diagram in MARCONI SkyLake. . . . . . 40 3.2 (a) FILD probe head and escaping fast-ion orbit reaching the aperture. (b) Scintillator plate where the collimated ions are dispersed using the tokamak magnetic field. . . . . . . . . . . 41 121 128 LIST OF TABLES experiments. I thank Antti for showing me that a researcher’s life is much richer than simply experiments and modelling. Finalmente agradezco a mi familia, el gran pilar de mi vida. A mi madre por su paciencia y su coraje. Ella siempre tiene un o´ıdo para escuchar a sus hijos y con eso ya se ha sacado tres ingenier´ıas y un doctorado. A mi padre por su esfuerzo incansable. A mis hermanos Pepe y Alejandra porque han sido un ejemplo a seguir y los mejores amigos que uno puede imaginar. A mi sobrina Paula, la ni˜na de la familia, que con cada cent´ımetro que crece, el resto nos sentimos un poco m´as viejitos. Y crece mucho. A Mari Carmen, que como le dije en su boda, lleva m´as tiempo siendo de la familia del que puedo recordar. Y por ´ultimo a Ana, “He aqu´ı mi secreto. Es muy simple: no se ve bien sino con el coraz´on. Lo esencial es invisible a los ojos.” [217]. Te debo la portada de mi tesis doctoral.