In recent years, the discontinuous Galerkin (DG) method has been successfully applied to solving hyperbolic conservation laws. Due to its compactness, high order accuracy, and versatility, the DG method has been extensively applied to convection-diffusion problems. In this dissertation, a numerical package, texttt{PHORCE}, is introduced to solve a number of convection-diffusion problems in neutral fluids and plasmas. Unstructured grids are used in order to randomize grid errors, which is especially important for complex geometries. texttt{PHORCE} is written in texttt{C++} and fully parallelized using the texttt{MPI} library. Memory optimization has been considered in this work to achieve improved efficiency. DG algorithms for hyperbolic terms are well studied. However, an accurate and efficient diffusion solver still constitutes ongoing research, especially for a nodal representation of the discontinuous Galerkin (NDG) method. An affine reconstructed discontinuous Galerkin (aRDG) algorithm is developed in this work to solve the diffusive operator using an unstructured NDG method. Unlike other reconstructed/recovery algorithms, all computations can be performed on a reference domain, which promotes efficiency in computation and storage. In addition, to the best of the authors' knowledge, this is the first practical guideline that has been proposed for applying the reconstruction algorithm on a nodal discontinuous Galerkin method. TVB type and WENO type limiters are also studied to deal with numerical oscillations in regions with strong physical gradients in state variables. A high-order positivity-preserving limiter is also extended in this work to prevent negative densities and pressure. A new interface tracking method, mass of fluid (MOF), along with its bound limiter has been proposed in this work to compute the mass fractions of different fluids over time. Hydrodynamic models, such as Euler and Navier-Stokes equations, and plasma models, such as ideal-magnetohydrodynamics (MHD) and two-fluid plasma equations, are studied and benchmarked with various applications using this DG framework. Numerical computations of Rayleigh-Taylor instability growth with experimentally relevant parameters are performed using hydrodynamic and MHD models on planar and radially converging domains. Discussions of the suppression mechanisms of Rayleigh-Taylor instabilities due to magnetic fields, viscosity, resistivity, and thermal conductivity are also included. This work was partially supported by the US Department of Energy under grant number DE-SC0016515. The author acknowledges Advanced Research Computing at Virginia Tech for providing computational resources and technical support that have contributed to the results reported within this work. URL: http://www.arc.vt.edu / Doctor of Philosophy / High-energy density (HED) plasma science is an important area in studying astrophysical phenomena as well as laboratory phenomena such as those applicable to inertial confinement fusion (ICF). ICF plasmas undergo radial compression, with an aim of achieving fusion ignition, and are subject to a number of hydrodynamic instabilities that can significantly alter the implosion and prevent sufficient fusion reactions. An understanding of these instabilities and their mitigation mechanisms is important allow for a stable implosion in ICF experiments. This work aims to provide a high order accurate and robust numerical framework that can be used to study these instabilities through simulations. The first half of this work aims to provide a detailed description of the numerical framework, texttt{PHORCE}. texttt{PHORCE} is a high order numerical package that can be used in solving convection-diffusion problems in neutral fluids and plasmas. Outstanding challenges exist in simulating high energy density (HED) hydrodynamics, where very large gradients exist in density, temperature, and transport coefficients (such as viscosity), and numerical instabilities arise from these region if there is no intervention. These instabilities may lead to inaccurate results or cause simulations to fail, especially for high-order numerical methods. Substantial work has been done in texttt{PHORCE} to improve its robustness in dealing with numerical instabilities. This includes the implementation and design of several high-order limiters. An novel algorithm is also proposed in this work to solve the diffusion term accurately and efficiently, which further enriches the physics that texttt{PHORCE} can investigate. The second half of this work involves rigorous benchmarks and experimentally relevant simulations of hydrodynamic instabilities. Both advection and diffusion solvers are well verified through convergence studies. Hydrodynamic and plasma models implemented are also validated against results in existing literature. Rayleigh-Taylor instability growth with experimentally relevant parameters are performed on both planar and radially converging domains. Although this work is motivated by physics in HED hydrodynamics, the emphasis is placed on numerical models that are generally applicable across a wide variety of fields and disciplines.

Recent concepts for controlled magneto-inertial fusion (MIF), such as magnetized liner inertial fusion (MagLIF), have suffered from magnetohydrodynamic (MHD) instabilities that lead to degradations in fusion yield. High levels of azimuthally-correlated MHD instability structures have been observed on cylindrical liner experiments without a pre-imposed axial magnetic field (Bz=0) elsewhere in the literature and are believed to be seeded from surface machining roughness. This dissertation uses highly resolved (0.5 μm and less resolution) 1D and 2D resistive magnetohydrodynamics (MHD) arbitrary-Lagrangian-Eulerian (ALE) simulations of electrical wire explosions (EWEs) and liner implosions to show that micrometer-scale surface roughness seeds the electrothermal instability (ETI), which induces early melting in pockets across the conductor and leads to millimeter-scale instability growth. The relationship between the ETI and the MRTI in liner implosions is also described in this dissertation, which shows that the traditional growth rates associated with these modes are coupled together and are not linearly independent. This dissertation also describes the preliminary implementation of a Koopman neural network architecture for learning the nonlinear dynamics of a high energy density (HED) exploding or imploding electrical conductor. / Doctor of Philosophy / Researchers have been working on controlling nuclear fusion and harnessing it as a power source since the discovery that nuclear fusion powers stars. In many of these controlled nuclear fusion concepts the aim is to heat the fuel until it forms a high-temperature plasma state of matter and then compress it to the point that the atoms are close enough and at high enough speeds that they collide fuse together. In the magnetized liner inertial fusion (MagLIF) concept these temperatures, densities, and pressures are achieved by surrounding the fusion fuel with a cylindrical piece of metal called a liner and using magnetic fields to implode the liner inward. Experiments have shown, however, that these liner implosions do not occur smoothly and that the system becomes unstable and can mix liner material into the fuel, which disrupts the fusion process. This dissertation investigates the stability of liner implosions and electrical wire explosions. In particular, this dissertation shows that surface roughness imparted on the surface of a solid fusion target by a machining process can grow into a millimeter-scale perturbation. It also describes the relationship between two common types of instabilities found in current-driven nuclear fusion: the magneto-Rayleigh-Taylor instability and the electrothermal instability. Finally, it looks at using neural networks to better understand the dynamics of electrical wire explosions.

In this thesis we investigate current sheets in the solar corona. The well known 1D model for the tearing mode instability is presented, before progressing to 2D where we introduce a non-uniform resistivity. The effect this has on growth rates is investigated and we find that the inclusion of the non-uniform term in η cause a decrease in the growth rate of the dominant mode. Analytical approximations and numerical simulations are then used to model current sheet formation by considering two distinct experiments. First, a magnetic field is sheared in two directions, perpendicular to each other. A twisted current layer is formed and we find that as we increase grid resolution, the maximum current increases, the width of the current layer decreases and the total current in the layer is approximately constant. This, together with the residual Lorentz force calculated, suggests that a current sheet is trying to form. The current layer then starts to fragment. By considering the parallel electric field and calculating the perpendicular vorticity, we find evidence of reconnection. The resulting temperatures easily reach the required coronal values. The second set of simulations carried out model an initially straight magnetic field which is stressed by elliptical boundary motions. A highly twisted current layer is formed and analysis of the energetics, current structures, magnetic field and the resulting temperatures is carried out. Results are similar in nature to that of the shearing experiment.

In magnetised plasmas, magnetic reconnection is the process of magnetic field merging and recombination through which considerable amounts of magnetic energy may be converted into other forms of energy. Reconnection is a key mechanism for solar flares and coronal mass ejections in the solar atmosphere, it is believed to be an important source of heating of the solar corona, and it plays a major role in the acceleration of particles in the Earth's magnetotail. For reconnection to occur, the magnetic field must, in localised regions, be able to diffuse through the plasma. Ideal locations for diffusion to occur are electric current layers formed from rapidly changing magnetic fields in short space scales. In this thesis we consider the formation and nature of these current layers in magnetised plasmas. The study of current sheets and current layers in two, and more recently, three dimensions, has been a key field of research in the last decades. However, many of these studies do not take plasma pressure effects into consideration, and rather they consider models of current sheets where the magnetic forces sum to zero. More recently, others have started to consider models in which the plasma beta is non-zero, but they simply focus on the actual equilibrium state involving a current layer and do not consider how such an equilibrium may be achieved physically. In particular, they do not allow energy conversion between magnetic and internal energy of the plasma on their way to approaching the final equilibrium. In this thesis, we aim to describe the formation of equilibrium states involving current layers at both two and three dimensional magnetic null points, which are specific locations where the magnetic field vanishes. The different equilibria are obtained through the non-resistive dynamical evolution of perturbed hydromagnetic systems. The dynamic evolution relaxes via viscous damping, resulting in viscous heating. We have run a series of numerical experiments using LARE, a Lagrangian-remap code, that solves the full magnetohydrodynamic (MHD) equations with user controlled viscosity and resistivity. To allow strong current accumulations to be created in a static equilibrium, we set the resistivity to be zero and hence simply reach our equilibria by solving the ideal MHD equations. We first consider the relaxation of simple homogeneous straight magnetic fields embedded in a plasma, and determine the role of the coupling between magnetic and plasma forces, both analytically and numerically. Then, we study the formation of current accumulations at 2D magnetic X-points and at 3D magnetic nulls with spine-aligned and fan-aligned current. At both 2D X-points and 3D nulls with fan-aligned current, the current density becomes singular at the location of the null. It is impossible to be precisely achieve an exact singularity, and instead, we find a gradual continuous increase of the peak current over time, and small, highly localised forces acting to form the singularity. In the 2D case, we give a qualitative description of the field around the magnetic null using a singular function, which is found to vary within the different topological regions of the field. Also, the final equilibrium depends exponentially on the initial plasma pressure. In the 3D spine-aligned experiments, in contrast, the current density is mainly accumulated along and about the spine, but not at the null. In this case, we find that the plasma pressure does not play an important role in the final equilibrium. Our results show that current sheet formation (and presumably reconnection) around magnetic nulls is held back by non-zero plasma betas, although the value of the plasma pressure appears to be much less important for torsional reconnection. In future studies, we may consider a broader family of 3D nulls, comparing the results with the analytical calculations in 2D, and the relaxation of more complex scenarios such as 3D magnetic separators.

Magnetic fields are fundamental to the structure and dynamics of the Sun’s corona. Observations show them to be locally complex, with highly sheared and twisted fields visible in solar filaments/prominences. The free magnetic energy contained in such fields is the primary source of energy for coronal mass ejections, which are important—but still poorly understood drivers of space weather in the near-Earth environment. In this thesis, a new model is developed for the evolution of the large-scale magnetic field in the global solar corona. The model is based on observations of the radial magnetic field on the solar photosphere (visible surface). New active regions emerge, and their transport and dispersal by surface motions are simulated accurately with a surface flux transport model. The 3D coronal magnetic field is evolved in response to these photospheric motions using a magneto-frictional technique. The resulting sequence of nonlinear force-free equilibria traces the build-up of magnetic helicity and free energy over many months. The global model is applied to study two phenomena: filaments and coronal mass ejections. The magnetic field directions in a large sample of observed filaments are compared with a 6-month simulation. Depending on the twist of newly-emerging active regions, the correct chirality is simulated for up to 96% of filaments tested. On the basis of these simulations, an explanation for the observed hemispheric pattern of filament chirality is put forward, including why exceptions occur for filaments in certain locations. Twisted magnetic flux ropes develop in the simulations, often losing equilibrium and lifting off, removing helicity. The physical basis for such losses of equilibrium is demonstrated through 2D analytical models. In the 3D global simulations, the twist of emerging regions is a key parameter controlling the number of lift-offs, which may explain around a third of observed coronal mass ejections.

No description available.

Réalisé en co-direction avec Antoine Strugarek. / La tachocline solaire est encore aujourd’hui un important sujet de débat dans la communauté. La compréhension de cette mince couche, à l’interface entre les zones radiative et convective, est très importante à la compréhension globale du fonctionnement du Soleil. En effet, l’inclusion d’une tachocline a un impact majeur dans les modèles de dynamo générant le champ magnétique du Soleil. De plus, la rotation différentielle observée dans la zone de convection devrait se propager dans la zone de radiation, où la rotation est uniforme, de sorte que la tachocline devrait être beaucoup plus épaisse que ce que les observations indiquent. Le processus menant au confinement de la tachocline est encore incertain, bien que de nombreuses hypothèses furent apportées pour tenter de l’expliquer. Un des ces scénarios propose que la pénétration du champ magnétique dynamo sous la zone convective mène à la suppression de la rotation différentielle dans la tachocline. Nous présentons ici un modèle MHD simplifié en une dimension afin de tester ce scénario de tachocline rapide. Nous nous intéressons à deux cas particuliers : une tachocline où le transport de moment cinétique est dû à la viscosité, puis une tachocline où l’épaississement radiatif domine la viscosité. Nous avons analysé plusieurs simulations dans le but de déterminer dans quelles conditions physiques le confinement de la tachocline est possible via ce scénario. L’amplitude du champ magnétique pénétrant sous la zone convective, la diffusivité magnétique, la viscosité et la diffusivité thermique ont un impact majeur sur les résultats et nous concluons en déterminant selon quels régimes de paramètres la tachocline pourrait être confinée par un tel champ dynamo. / The solar tachocline remains the subject of vigorous ongoing research efforts. Understanding the dynamics of this thin layer at the interface between the radiative and convective zones is important to the overall understanding the Sun’s inner workings. Indeed, the presence of a tachocline plays a major role in most dynamo models that describe the generation of the solar magnetic field. Moreover, the differential rotation observed in the convection zone should spread in the radiation zone, where the rotation is uniform, so the tachocline should be much thicker than inferred from helioseismic inversions. The physical mechanism(s) responsible for confining the tachocline has not yet been identified with confidence, although many promising hypotheses have been put forth. One of these invokes the penetration of a dynamo magnetic field below the convective zone, leading to the suppression of the differential rotation in the tachocline through the action of magnetic stresses. We present here a simplified MHD model formulated in one spatial dimension, in order to test this fast tachocline scenario. We focus on two specific physical cases : one where the angular momentum transport is due to the viscosity and the other where radiative spreading dominates over viscosity. We carry out and analyze several simulations to determine under which physical conditions the confinement of the tachocline is possible via this scenario. The amplitude of the magnetic field penetrating the convective zone, the magnetic diffusivity, the viscosity and the thermal diffusivity all have a major impact on the results, and we conclude by determining under which parameters the tachocline could be confined by such a dynamo field.

Nous analysons les oscillations torsionnelles se développant dans une simulation magnétohydrodynamique de la zone de convection solaire produisant des champs magnétiques de type solaire (champs axisymétriques subissant des inversions de polarités régulières sur des échelles temporelles décadaires). Puisque ces oscillations sont également similaires à celles observées dans le Soleil, nous analysons les dynamiques zonales aux grandes échelles. Nous séparons donc les termes aux grandes échelles (force de Coriolis exercée sur la circulation méridienne et les champs magnétiques aux grandes échelles) de ceux aux petites échelles (les stress de Reynolds et de Maxwell). En comparant les flux de moments cinétiques entre chacune des composantes, nous nous apercevons que les oscillations torsionnelles sont maintenues par l’écoulement méridien aux grandes échelles, lui même modulé par les champs magnétiques. Une analyse d’échange d’énergie confirme ce résultat, puisqu’elle montre que seul le terme comprenant la force de Coriolis injecte de l’énergie dans l’écoulement. Une analyse de la dynamique rotationnelle ayant lieu à la limite de la zone stable et de la zone de convection démontre que celle-ci est fortement modifiée lors du passage de la base des couches convectives à la base de la fine tachocline s’y formant juste en-dessous. Nous concluons par une discussion au niveau du mécanisme de saturation en amplitude dans la dynamo s’opérant dans la simulation ainsi que de la possibilité d’utiliser les oscillations torsionnelles comme précurseurs aux cycles solaires à venir. / We study torsional oscillations developping in a magnetohydrodynamic simulation of the solar convective layers producing solar-like magnetic cycles (large-scale axisymmetric fields subjected to regular polarity reversals). Since these oscillations are similar to those observed in the Sun, we perform an analysis of large-scale zonal dynamics. We separate the large-scale terms (Coriolis force exerted on the meridional circulation and large-scale magnetic fields) from the small-scale contributions (Reynolds and Maxwell stresses). Upon comparing angular momentum fluxes between each of those components, we find that torsional oscillations are driven by the large-scale meridional flow, itself modulated by magnetic fields. An analysis of energy transfers confirms this result, where we see that only the Coriolis force term directly inputs energy in the flow. An analysis of angular momentum fluxes occuring at the interface between the stable and the convective zones shows that the local dynamics therein undergoes a complete shift in going from the base of the convective layers through the base of the thin tachocline developping just beneath it. We conclude by discussing the mechanism of amplitude saturation in the dynamo operating in the simulation and the possibility of using torsional oscillations as precursors to upcoming solar cycles.

No description available.

No description available.