Limitations of stationary Vlasov-Poisson solvers in probe theory

Physical and numerical limitations of stationary Vlasov-Poisson solvers based on backward Liouville methods are investigated with five solvers that combine different meshes, numerical integrators, and electric field interpolation schemes. Since some of the limitations arise when moving from an integrable to a non-integrable configuration, an elliptical Langmuir probe immersed in a Maxwellian plasma was considered and the eccentricity (ep) of its cross-section used as integrability-breaking parameter. In the cylindrical case, ep=0, the energy and angular momentum are both conserved. The trajectories of the charged particles are regular and the boundaries that separate trapped from non-trapped particles in phase space are smooth curves. However, their computation has to be done carefully because, albeit small, the intrinsic numerical errors of some solvers break these conservation laws. It is shown that an optimum exists for the number of loops around the probe that the solvers need to classify a particle trajectory as trapped. For ep≠0, the angular momentum is not conserved and particle dynamics in phase space is a mix of regular and chaotic orbits. The distribution function is filamented and the boundaries that separate trapped from non-trapped particles in phase space have a fractal geometry. The results were used to make a list of recommendations for the practical implementation of stationary Vlasov-Poisson solvers in a wide range of physical scenarios.This work was supported by the European Union's Horizon 2020 Research and Innovation Programme under grant agreement No 828902 (E.T.PACK project). GSA work is supported by the Ministerio de Ciencia, Innovación of Spain under the Grant RYC-2014-15357. The authors thank the Reviewers for their valuable comments and suggestions about the use of energy-conserving numerical integrators

Emissive Langmuir Probe Theory with Application to Low Work Function Electrodynamic Tethers

Mención Internacional en el título de doctorMotivated by the need of mitigating the increase in the debris population that has been accumulating in the Low Earth Orbit (LEO) in more than 60 years of intense human activity in space, Electrodynamic Tethers (EDTs) have been proposed as efficient devices to deorbit satellites at the end of life. Consisting in long conductors that are deployed from the satellite at the end of its mission, EDTs exploit the interaction with the ionospheric plasma to create a current that, flowing along the device, interacts with the geomagnetic field giving rise to a magnetic drag that deorbits the satellite. Low Work-function tethers (LWTs) are particularly attractive because no expellant is needed for their operations. Once deployed, the LWT exchanges electrons with the ionospheric plasma, collecting them at one segment and, upon being coated with a material of low-enough work function, emitting them back at the complementary segment. Accurate models of the plasma-LWT interaction are necessary to quantify the performances of the device with software for mission analysis. Since the characteristic length of a space tether is several orders of magnitude larger than the Debye length in LEO, the current distribution along the tether can be computed from the current-voltage characteristic of a two-dimensional probe of same cross-section. This dissertation presents a numerical investigation of the interaction between twodimensional (electron-emitting) objects and Maxwellian plasmas representative of the LEO environment. A kinetic approach is adopted to study the features of the plasma sheath. In particular, a model based on the Orbital Motion Theory (OMT) is applied to study geometries that, although favourable for LWTs applications, received little attention in past works. To this extent, a novel stationary Eulerian Vlasov-Poisson solver based on a backward Liouville method is presented in detail. After a thorough verification procedure versus more mature numerical tools, a discussion of the physical and numerical limitations of stationary Vlasov-Poisson solvers is presented. Its results are used to provide a list of guidelines for their practical use in plasma-material interaction problems. Using the same code, an analysis was carried out in order to characterise deeply the sheath around electron-emitting objects with elliptic cross sections. By varying the size, eccentricity and emission level of the probe, the study assessed the parameter domains for which Orbital-Motion-Limited (OML) current collection and Space-Charge-Limited (SCL) current emission hold. The local curvature of the probe revealed to have an important impact on its operational regime and, as compared with cylindrical ones, elliptic bodies were found to be more likely to meet non-OML and SCL conditions. Electron emission was also shown to be favourable for OML current collection. Regarding LWTs applications, an interesting equivalence between the emitted current in SCL conditions by ellipses and cylinders was found. In the last part of the dissertation the hypothesis about the steady-state of the system is relaxed and a novel semi-Lagrangian Vlasov-Poisson solver developed as an extension of the stationary one is introduced. The impact of the population of trapped particles on the macroscopic magnitudes of the sheath is discussed. The results of a comparison between Eulerian solvers and a Particle-In-Cell (PIC) code for emissive probes are also presented to investigate the importance of the numerical noise of the PIC code. Particle trapping is shown to depend on both the history of the system and on the emission level. For high electron-emission, the trapped population reduces SCL effects.Ante el reto de mitigar el aumento de la población de basura espacial que se ha ido acumulando en la órbita terrestre baja (Low Earth Orbit - LEO) en los más de 60 años de intensa actividad humana en el espacio, las amarras electrodinámicas (Electrodynamic Tether - EDT) surgen como dispositivos eficientes para el desorbitado de satélites al final de su vida útil. Los EDTs, largos cables conductores, aprovechan su interacción con el plasma ionosférico para crear una corriente que, al fluir a los largo del cable, interactúa con el campo geomagnético y generan una fuerza de frenado que desorbita el satélite. Las llamadas amarras con baja función de trabajo (Low Work-function tethers - LWTs) son especialmente atractivas porque no involucran ningún consumible ni elemento activo para su funcionamiento. Una vez desplegado, el LWT intercambia electrones con el plasma ionosférico de manera totalmente pasiva. Los electrones son recogidos en un segmento llamado anódico y se emiten de vuelta al plasma en el segmento complementario catódico gracias a los efectos termoiónico y fotoeléctrico que facilitan el recubrimiento con baja función de trabajo de la propia amarra. Para poder evaluar las prestaciones del dispositivo, se necesitan modelos precisos de la interacción entre el plasma y el LWT. Dado que la longitud de una amarra espacial es varios ´ordenes de magnitud mayor que la longitud de Debye en LEO, los perfiles de corriente y voltaje se pueden calcular a partir de las curvas características de una sonda bi-dimensional con la misma sección transversal. La tesis presenta un análisis numérico de la interacción entre objetos bidimensional que emiten electrones y plasmas representativos del entorno espacial en LEO. Se adopta un enfoque cinético para estudiar las características de la vaina del plasma. En particular, se aplica un modelo basado en la Orbital Motion Theory (OMT) para el estudio de geometrías que, pese a ser ventajosas para aplicaciones de LWTs, recibieron poca atención en el pasado. Para ello se ha desarrollado un nuevo código Vlasov-Poisson euleriano y estacionario basado en el método de backward Liouville. Tras un extenso proceso de verificación frente a resultados obtenidos con código más maduros, se discuten las limitaciones de tipo físico y numérico intrínsecas a los códigos Vlasov-Poisson estacionarios. Los resultados se han utilizado para preparar una lista de recomendaciones prácticas sobre el uso de estos códigos en problemas de interacción plasma-material. El código se ha utilizado para caracterizar en profundidad las vainas que se forman alrededor de objetos con sección transversal elíptica y que emiten electrones. En el análisis se ha variado el tamaño, la excentricidad y el nivel de emisión del objeto, lo cual ha permitido determinar los dominios paramétricos en donde la captura de corriente esta dada por la llamada teoría Orbital Motion Limited (OML) y la emisión ocurre bajo condiciones de Space Charge Limited (SCL). Se ha observado que la curvatura local de la elipse juega un papel importante en determinar el régimen de operación y se encontró que los cuerpos elípticos son más propensos a cumplir con las condiciones de no-OML y SCL que los cilíndricos. También ha permitido concluir que la emisión de electrones favorece la captura de corriente en condiciones OML. Con respecto a los LWTs, se ha encontrado que existe un radio equivalente para calcular la corriente emitida por un cuerpo elíptico bajo condiciones SCL a partir de los resultados de un cuerpo cilíndrico. La última parte de la tesis estudia los transitorios que ocurren entre una condición iniciales dada y el estado estacionario que se alcanza en el equilibrio. Para ello se ha desarrollado un código semilagrangiano para resolver el sistema Vlasov-Poisson no estacionario, el cual constituye una extensión del código estacionario usado en la primera parte de la tesis. El nuevo código ha permitido discutir el impacto de la población de partículas atrapadas en el transitorio sobre las magnitudes macroscópicas de la vaina del plasma en el equilibrio. Se presentan también los resultados de una comparación entre el código euleriano y un código Particle-In-Cell (PIC) para sondas emisivas con el fin de investigar la importancia del ruido numérico del código PIC. Se demuestra que la cantidad de atrapados depende tanto de la historia del sistema como del nivel de emisión. Para una alta emisión de electrones, la población de atrapados reduce los efectos SCL.This work was supported by PIPF Scholarship awarded on a competitive basis by Universidad Carlos III de Madrid. During the thesis I had the opportunity to participate in the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement No 828902 (E.T.PACK project) and the FET Innovation Launchpad project No 101034874 (BMOM).Programa de Doctorado en Ingeniería Aeroespacial por la Universidad Carlos III de MadridPresidente: Eduardo Antonio Ahedo Galilea.- Secretario: Fernando García Rubio.- Vocal: Richard Marchan

Instabilities, anomalous transport, and nonlinear structures in partially and fully magnetized plasmas.

Plasmas behavior, to a large extent, is determined by collective phenomena such as waves. Wave excitation, turbulence, and formation of quasi-coherent nonlinear structures are defining features of nonlinear multi-scale plasma dynamics. In this thesis, instabilities, anomalous transport, and structures in partially and fully magnetized plasmas were studied with a combination of analytical and numerical tools. The phenomena studied in this thesis are of interest for many applications, e.g., plasma reactors for material processing, electric propulsion, magnetic plasma confinement, and space plasma physics. Large equilibrium flows of ions and electrons exist in many devices with partially magnetized plasmas in crossed electric and magnetic fields. Such flows result in various instabilities and turbulence that produce anomalous electron transport across the magnetic field. We present first principle, self-consistent, nonlinear fluid simulations that predict the level of anomalous current generally consistent with experimental data. We also show that drift waves in partially magnetized plasmas (which we called Hall drift waves), destabilized by the electron drift along with density gradients, tend to form (via inverse energy cascade) shear flows similar to zonal flows in fully magnetized plasmas. These flows become unstable due to a secondary instability (similar to Kelvin–Helmholtz instability) and produce large-scale quasi-stationary vortices. Then, it was shown that in nonlinear regimes, the axial mode instability due to electron and ion flows (along the electric field) forms large-amplitude cnoidal type waves. At the same time, the strong electric field produced by axial modes affects Hall drift waves stability and provides a feedback mechanism on density gradient driven turbulence, creating a complex picture of interacting anomalous transport, zonal flows, vortices, and streamers. In the case where axial modes are destabilized by boundary effects, the nonlinear dynamics result in a new nonlinear equilibrium or standing oscillating waves. The formation of shear flows (zonal flows) was also studied in the framework of the Hasegawa-Mima equation and it was established that zonal flows can saturate due to nonlinear self-interactions. Lastly, a novel approach for high-fidelity numerical simulations of multi-scale nonlinear plasma dynamics is developed which is illustrated with the example of an unmagnetized plasma

Particle-in-cell simulation of plasma-based amplification using a moving window

Current high-power laser amplifiers use chirped-pulse amplification to prevent damage to their solid-state components caused by intense electromagnetic fields. To increase laser power further requires ever larger and more expensive devices. The Raman backscatter instability in plasma facilitates an alternative amplification strategy without the limitations imposed by material damage thresholds. Plasma-based amplification has been experimentally demonstrated, but only with relatively low efficiency. Further progress requires extensive use of numerical simulations, which usually need significant computational resources. Here we present particle-in-cell (PIC) simulation techniques for accurately simulating Raman amplification using a moving window with suitable boundary conditions, reducing computational cost. We show that an analytical model for matched pump propagation in a parabolic plasma channel slightly overestimates amplification as pump laser intensity is increased. However, a method for loading data saved from separate pump-only simulations demonstrates excellent agreement with full PIC simulation. The reduction in required resources will enable parameter scans to be performed to optimize amplification, and stimulate efforts toward developing viable plasma-based laser amplifiers. The methods may also be extended to investigate Brillouin scattering, and for the development of laser wakefield accelerators. Efficient, compact, low-cost amplifiers would have widespread applications in academia and industry

Magnetic Fields and Non-Local Transport in Laser Plasmas

The first Vlasov-Fokker-Planck simulations of nanosecond laser-plasma interactions – including the effects of self-consistent magnetic fields and hydrodynamic plasma expansion – will be presented. The coupling between non-locality and magnetic field advection is elucidated. For the largest (initially uniform) magnetic fields externally imposed in recent long-pulse laser gas-jet plasma experiments (12T) a significant degree of cavitation of the B-field will be shown to occur (> 40%) in under 500ps. This is due to the Nernst effect and leads to the re-emergence of non-locality even if the initial value of the magnetic field strength is sufficient to localize transport. Classical transport theory may also break down in such interactions as a result of inverse bremsstrahlung heating. Although non-locality may be suppressed by a large B-field, inverse bremsstrahlung still leads to a highly distorted distribution. Indeed the best fit for a 12T applied field (after 440ps of laser heating) is found to be a super- Gaussian distribution – f0 α e−vm – with m = 3.4. The effects of such a distribution on the transport properties under the influence of magnetic fields are elucidated in the context of laser-plasmas for the first time. In long pulse laser-plasma interactions magnetic fields generated by the thermoelectric (‘∇ne × ∇Te’) mechanism are generally considered dominant. The strength of B-fields generated by this mechanism are affected, and new generation mechanisms are expected, when non-locality is important. Non-local B-field generation is found to be dominant in the interaction of an elliptical laser spot with a nitrogen gas-jet

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching

Computational and Numerical Simulations

Computational and Numerical Simulations is an edited book including 20 chapters. Book handles the recent research devoted to numerical simulations of physical and engineering systems. It presents both new theories and their applications, showing bridge between theoretical investigations and possibility to apply them by engineers of different branches of science. Numerical simulations play a key role in both theoretical and application oriented research

GENE-3D - ein globaler gyrokinetischer Turbulenzcode für Stellaratoren und gestörte Tokamaks

This thesis describes the development and application of GENE-3D, a global gyrokinetic turbulence HPC code for stellarators. The gyrokinetic equations as well as their implementation and the use of field-aligned coordinates in non-axisymmetric geometries are discussed. GENE-3D is benchmarked for validity and performance. Different geometries of Wendelstein 7-X are investigated for their influence on turbulent properties. Also the influence of the machine size on linear growth rates is studied.Diese Arbeit beschreibt die Entwicklung und Anwendung von GENE-3D, ein globaler gyrokinetischer Turbulenzcode für Stellaratoren. Die gyrokinetischen Gleichungen sowie deren Implementierung und das am Feld ausgerichtete Koordinatensystem werden für nicht-axisymmetrische Geometrien vorgestellt. GENE-3D wird auf Korrektheit getestet.Der Einfluß unterschiedlicher Wendelstein 7-X Geometrien auf den turbulenten Transport und der Einfluß der Maschinengröße auf die linearen Anwachsraten wird untersucht

Particle in Cell and Hybrid Simulations of the Z Double-Post-Hole Convolute Cathode Plasma Evolution and Dynamics

The Z-accelerator at Sandia National Laboratories (SNL), is a high-current pulsed power machine used to drive a range of high energy density physics (HEDP) experiments [1]. To achieve peak currents of >20MA, in a rise time of ~100ns, the current is split over four levels of transmission line, before being added in parallel in a double-post-hole convolute (DPHC) and delivered to the load through a single inner magnetically insulated transmission line (MITL). The electric field on the cathode electrode, >107Vm-1, drives the desorption and ionisation of neutral contaminants to form a plasma from which electrons are emitted into the anode-cathode (a-k) gap. The current addition path in the DPHC forms magnetic 'null' regions, across which electrons are lost to the anode, shunting current from the inner MITL and load. In experiment, current losses of >10% have been measured within the convolute; this reduces the power delivered to the load, negatively impacting the load performance, as well as complicating the prediction of the Poynting flux used to drive detailed magneto-hydrodynamic (MHD) simulations [2, 3]. In this thesis we develop 3-dimensional (3D) Particle-in-Cell (PIC) and hybrid fluid-PIC computer models to simulate the plasma evolution in the DPHC and inner MITL. The expected experimental current loss at peak current was matched in simulations where Hydrogen plasma was injected from the cathode elec- trode at a rate of 0.0075mlns-1 (1ml=1015cm-2), with an initial temperature of 3eV. The simulated current loss was driven by plasma penetrating the downstream side of the anode posts, reducing the effective a-k gap spacing and enhancing electron losses to the anode. The current loss at early time (<10MA), was matched in simulations where space-charge-limited (SCL) electron emission was allowed directly from the cathode; to match the loss over the entire current pulse, a delay model is motivated. Here, plasma injection was delayed after the start of SCL emission, based on realistic plasma expansion velocities of ~3cmμs-1. The PIC model, which was necessary to accurately simulate the kinetic behaviour of the lower density plasma and charged particle sheaths, was computationally intensive such that the spatial resolutions achieved in the 3D simulations were relatively poor. With the aim of reducing the computational overhead, allowing finer spatial resolutions to be accessed, we investigate the applicability of hybrid techniques to simulating the cathode plasma in the convolute. Our PIC model was both implemented in the resistive MHD code, Gorgon, where part of the plasma was modelled in the single fluid approximation, and extended to include an inertial two-fluid description of the plasma. The hybrid models were applied to the DPHC simulations, the results from which are used to motivate a three component model; here, the densest part of the convolute plasma is modelled using the single fluid MHD approximation, transitioning to a fully kinetic PIC description of the lower density plasma and charged particle sheaths, linked by a two-fluid description