Development of advanced linearized gyrokinetic collision operators using a moment approach

The derivation and numerical implementation of a linearized version of the gyrokinetic (GK) Coulomb collision operator (Jorge et al., J. Plasma Phys., vol. 85, 2019, 905850604) and of the widely used linearized GK Sugama collision operator (Sugama et al., Phys. Plasmas, vol. 16, 2009, 112503) is reported. An approach based on a Hermite-Laguerre moment expansion of the perturbed gyrocentre distribution function is used, referred to as gyromoment expansion. This approach allows the considering of arbitrary perpendicular wavenumber and expressing the two linearized GK operators as a linear combination of gyromoments where the expansion coefficients are given by closed analytical expressions that depend on the perpendicular wavenumber and on the temperature and mass ratios of the colliding species. The drift-kinetic (DK) limits of the GK linearized Coulomb and Sugama operators are also obtained. Comparisons between the gyromoment approach and the DK Coulomb and GK Sugama operators in the continuum GK code GENE are reported, focusing on the ion-temperature-gradient instability and zonal flow damping, finding an excellent agreement. It is confirmed that stronger collisional damping of the zonal flow residual by the Sugama GK model compared with the GK linearized Coulomb (Pan et al., Phys. Plasmas, vol. 27, 2020, 042307) persists at higher collisionality. Finally, we show that the numerical efficiency of the gyromoment approach increases with collisionality, a desired property for boundary plasma applications.

Development of a Millikelvin Scanning Tunneling Microscope for Applications in Ultra High Vacuum and High Magnetic Fields

Maximilian Assig

In this thesis we discuss the realization of a scanning tunneling microscope (STM) operating at temperatures below 20 mK. This is accomplished by attaching the STM to a dilution refrigerator. We can apply high magnetic fields up to 14 T perpendicular and 0.5 T parallel to the sample surface. A full ultra high vacuum (UHV) preparation chamber is attached to the cryostat allowing in situ preparation of the samples to investigate. This setup allows us to investigate low temperature phase transitions in solids and offers the possibility for ultimate energy resolution tunneling spectroscopy measurements on the atomic scale. The concept of realizing the mK-STM is given. A main point of concern was the design of the STM unit. The choice of materials should be well adapted for the use at very low temperatures and in high magnetic fields. A special concern are thermal expansion coefficients and thermal conductivity. In addition, we analyzed the variation of the tip sample distance for a given external excitation for our STM design and two variations of the Besoke-STM by using a one dimensional mass spring model. This gives us the possibility to extract an optimization criterion for our design to limit the response of the tip sample distance to vibrations that are transmitted to the STM. The external damping mechanism, which ensures a low mutual vibration level of the STM as well as the grounding concept for the electronics is explained. To characterize the STM at operating temperatures of 1.5 K, 800 mK and 15 mK we measured topography images of Au(111) and Cu(111) surfaces. This allows us to extract the stability between tip and sample. A statistical analysis of the noise in the tunneling current without activated feedback loop yields a stability of 2.7±0.1 pm in a frequency range f ≈ 0.03 – 25 Hz at 800 mK. To verify if tip and sample are cooled efficiently, we measured the temperature at their positions with additional thermometers. We achieved 17 mK at the tip and 20 mK at the sample position, respectively. The energy resolution of the STM setup was verified by taking tunneling spectra between a superconductor and a normal conducting metal. These spectra could be fitted using the BCS theory of superconductivity. From those ts we can extract the upper limit of the effective temperature of the electrons which is Teff = 87 ± 3 mK for our best measurement. This corresponds to an energy resolution of 3.5 kbTeff = 26 ± 1 µeV. Further, the tunneling conductance between two superconductors was investigated. A sub-gap structure, which can be related to Andreev reflections in asymmetric tunneling junction was observed. In addition, at zero bias voltage a peak due to the Josephson effect can be measured. Side peaks with intensities that scale with this zero bias conductance peak appear most probably due to the interaction of Cooper pairs with the electromagnetical environment in which the junction is placed. Investigating the dependence of the conductance between a superconducting tip and a normal conducting sample on externally applied magnetic fields leads to the appearance of shoulders in the coherence peaks. These shoulders shift linearly with the external magnetic field. Such an effect has been measured for planar tunnel junctions and can be explained with the lifting of the spin degeneracy of the quasi-particle density of states in a superconductor. It was observed for the first time by Meservey, Tedrow and Fulde. The fact, that we are now able to prepare STM tips showing the Meservey-Tedrow-Fulde effect enables us to measure the absolute spin polarization at EF with atomic resolution.

EPFL2011

Collisions in Global Gyrokinetic Simulations of Tokamak Plasmas using the Delta-f Particle-In-Cell Approach

Thibaut Vernay

The present work takes place within the general context of research related to the development of nuclear fusion energy. More specifically, this thesis is mainly a numerical and physical contribution to the understanding of turbulence and associated transport phenomena occuring in tokamak plasmas, the most advanced and promising form of magnetically confined plasmas. The complexity of tokamak plasma phenomena and related physical models, either fluid or kinetic, requires the development of numerical codes to perform simulations of the plasma behaviour under given conditions defined by the magnetic geometry as well as density and temperature profiles. The studies presented in this work are based on electrostatic kinetic simulations, taking advantage of a reduced kinetic model (the gyrokinetic model) which is particularly suitable for studying turbulent transport in magnetically confined plasmas, in effect solving an approximate form of the Vlasov equation for the distribution function of each species (electrons, ions) along with a reduced form of the Poisson equation providing the self-consistent electric fields. The main tool of this work, the gyrokinetic ORB5 code making use of numerical particles according to the Particle-In-Cell (PIC) method, has been upgraded during this thesis with different linearized collision operators related to both ions and electrons. The BIRDIE code, enabling to study collisional effects on the evolution of Langmuir waves in an unmagnetized plasma, has been written in order to serve as a test-bed for the collision operators ultimately implemented in ORB5. Some essential algorithms related to collisional simulations have been jointly implemented, such as the two-weight scheme which is extensively described in this work. The collision operators in ORB5 have been further carefully tested through neoclassical simu- lations and benchmarked against other codes, providing reliable levels of collisional transport. Together with different procedures controlling the numerical noise, the collision operators have then been applied to the study of collisional turbulent transport in two different regimes, the Ion-Temperature-Gradient (ITG) regime and the Trapped-Electron-Mode (TEM) regime re- quiring a trapped electron kinetic response. Although not dominant in core tokamak plasmas, collisional effects nevertheless lead to interesting modifications in the turbulence behaviour which are not captured by the often considered collisionless gyrokinetic models. The so-called coarse-graining procedure, a noise-control algorithm which is suitable for collisional gyrokinetic simulations with particles, is shown to enable carrying out relevant simulations over many col- lision times. Consequently, reliable conclusions regarding turbulent transport in the presence of collisions could be drawn in this thesis. Namely, the turbulent transport in the ITG regime is found to be enhanced by ion collisions through interactions with so-called zonal flows as- sociated to axisymmetric modes, while it is reduced by electron collisions in the TEM regime through electron detrapping processes. The zonal flow dynamics in collisionless and collisional ITG turbulence simulations is studied, emphasizing the limitation of the zonal flow level due to Kelvin-Helmoltz-type instabilities. Additionally, some purely collisionless issues related to tokamak physics are discussed, such as the finite plasma size effects in TEM-dominated regime which are found to be important in non-linear simulations but unimportant in linear simu- lations. The role of zonal flows in temperature-gradient-driven TEM turbulence saturation is confirmed to be weak, in agreement with previous studies. Finally, a realistic global gy- rokinetic simulation, accounting for a proper TCV tokamak magnetic equilibrium and related experimental profiles, has been successfully carried out thus demonstrating the relevance of the ORB5 code for predictions related to physics of real tokamaks. A good agreement with GAM experimental measurements is indeed obtained.

EPFL2013

A moment-based model for plasma dynamics at arbitrary collisionality

Rogério Manuel Cabete De Jesus Jorge

Despite significant development over the last decades, a model able to describe the periphery region of magnetic confinement fusion devices, extending from the edge to the far scrape-off layer, is still missing. This is because this region is characterized by the presence of turbulent fluctuations at scales ranging from the Larmor radius to the size of the machine, the presence of strong flows, comparable amplitudes of background and fluctuating components, and a large range of collisionality regimes. The lack of a proper model has undermined our ability to properly simulate the plasma dynamics in this region, which is necessary to predict the heat flux to the vessel wall of future fusion devices, the L-H transition, and ELM dynamics. These are some of the most important issues on the way to a fusion reactor. In the present thesis, a drift-kinetic and a gyrokinetic model able to describe the plasma dynamics in the tokamak periphery are developed, which take into account electrostatic fluctuations at all relevant scales, allowing for comparable amplitudes of background and fluctuating components. In addition, the models implement a full Coulomb collision operator, and are therefore valid at arbitrary collisionality regimes. For an efficient numerical implementation of the models, the resulting kinetic equations are projected onto a Hermite-Laguerre velocity-space polynomial basis, obtaining a moment-hierarchy. The treatment of arbitrary collisionalities is performed by expressing the full Coulomb collision operator in guiding-center and gyrocentre coordinates, and by providing a closed formula for its gyroaverage in terms of the moments of the plasma distribution function, therefore filling a long standing gap in the literature. The use of systematic closures to truncate the moment-hierarchy equation, such as the semi-collisional closure, allows for the straightforward adjustment of the kinetic physics content of the model. In the electrostatic high collisionality regime, our models are therefore reduced to an improved set of drift-reduced Braginskii equations, which are widely used in scrape-off layer simulations. The first numerical studies based on our models are carried out, shedding light on the interplay between collisional, using the Coulomb collision operator, and collisionless mechanisms. In particular, the dynamics of electron-plasma waves and the drift-wave instability are studied at arbitrary collisionality. A comparison is made with the collisionless limit and simplified collision operators used in state-of-the-art simulation codes, where large deviations in the growth rates and eigenmode spectra are found, especially at the levels of collisionality relevant for present and future magnetic confinement fusion devices.

EPFL2019