Solutions of the magnetohydrodynamic (MHD) equations are very important for modelling laboratory, space and astrophysical plasmas, for example the solar and stellar coronae, as well as for modelling many of the dynamic processes that occur in these different plasma environments such as the fundamental process of magnetic reconnection. Our previous understanding of the behavior of plasmas and their associated dynamic processes has been developed through two-dimensional (2D) models. However, a more realistic model should be three-dimensional (3D), but finding 3D solutions of the MHD equations is, in general, a formidable task. Only very few analytical solutions are known and even calculating solutions with numerical methods is usually far from easy. In this thesis, 3D solutions which model magnetic reconnection and rigidly rotating magnetized coronae are presented. For magnetic reconnection, a 3D stationary MHD model is used. However, the complexity of the problem meant that so far no generic analytic solutions for reconnection in 3D exist and most work consists of numerical simulations. This has so far hampered progress in our understanding of magnetic reconnection. The model used here allows for analytic solutions at least up to a certain order of approximation and therefore gives some better insight in the significant differences between 2D and 3D reconnection. Three-dimensional numerical solutions are also obtained for this model. Rigidly rotating magnetized coronae, on the other hand, are modeled using a set of magnetohydrostatic (MHS) equations. A general theoretical framework for calculating 3D MHS solutions outside massive rigidly rotating central bodies is presented. Under certain assumptions, the MHS equations are reduced to a single linear partial differential equation referred to as the fundamental equation of the theory. As a first step, an illustrative case of a massive rigidly rotating magnetized cylinder is considered, which somehow allows for analytic solutions in a certain domain of validity. In general, the fundamental equation of the theory can only be solved numerically and hence numerical example solutions are presented. The theory is then extended to include a more realistic case of massive rigidly rotating spherical bodies. The resulting fundamental equation of the theory in this case is too complicated to allow for analytic solutions and hence only numerical solutions are obtained using similar numerical methods to the ones used in the cylindrical case.

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There are three main wave types present in the Sun’s atmosphere: Alfvén waves and fast and slow magnetoacoustic waves. Alfvén waves are purely magnetic and would not exist if it was not for the Sun’s magnetic field. The fast and slow magnetoacoustic waves are so named due to their relative phase speeds. As the magnetic field tends to zero, the slow wave goes to zero as the fast wave becomes the sound wave. When a resonance occurs energy may be transferred between the different modes, causing one to increase in amplitude whilst the other decreases. This is known as mode conversion. Mode conversion of fast and slow magnetoacoustic waves takes place when the characteristic wave speeds, the sound and Alfvén speeds, are equal. This occurs in regions where the ratio of the gas pressure to the magnetic pressure, known as the plasma β, is approximately unity. In this thesis we investigate the conversion of fast and slow magnetoacoustic waves as they propagate from low- to high-β plasma. This investigation uses a combination of analytical and numerical techniques to gain a full understanding of the process. The MacCormack finite-difference method is used to model a wave as it undergoes mode conversion. Complementing this analytical techniques are employed to find the wave behaviour at, and distant from, the mode-conversion region. These methods are described in Chapter 2. The simple, one-dimensional model of an isothermal atmosphere permeated by a uniform magnetic field is studied in Chapter 3. Gravitational acceleration is included to ensure that mode conversion takes place. Driving a slow magnetoacoustic wave on the upper boundary conversion takes place as the wave passes from low- to high-β plasma. This is expanded upon in Chapter 4 where the effects of a non-isothermal temperature profile are examined. A tanh profile is selected to mimic the steep temperature gradient found in the transition region. In Chapter 5 the complexity is increased by allowing for a two-dimensional model. For this purpose we choose a radially-expanding magnetic field which is representative of a coronal hole. In this instance the slow magnetoacoustic wave is driven upwards from the surface, again travelling from low to high β. Finally, in Chapter 6 we investigate mode conversion near a two-dimensional, magnetic null point. At the null the plasma β becomes infinitely large and a wave propagating towards the null point will experience mode conversion. The methods used allow conversion of fast and slow waves to be described in the various model atmospheres. The amount of transmission and conversion are calculated and matched across the mode-conversion layer giving a full description of the wave behaviour.

The topic of this thesis is a detailed investigation of different aspects of the particle acceleration mechanisms operating in Collapsing Magnetic Traps (CMTs), which have been suggested as one possible mechanism for particle acceleration during solar flares. The acceleration processes in CMTs are investigated using guiding centre test particle calculations. Results including terms of different orders in the guiding centre approximation are compared to help identify which of the terms are important for the acceleration of particles. For a basic 2D CMT model the effects of different initial conditions (position, kinetic energy and pitch angle) of particles are investigated in detail. The main result is that the particles that gain most energy are those with initial pitch angles close to 90° and start in weak field regions in the centre of the CMT. The dominant acceleration mechanism for these particles is betatron acceleration, but other particles also show signatures of Fermi acceleration. The basic CMT model is then extended by (a) including a magnetic field component in the invariant direction and (b) by making it asymmetric. It is found that the addition of a guide field does not change the characteristics of particle acceleration very much, but for the asymmetric models the associated energy gain is found to be much smaller than in symmetric models, because the particles can no longer remain very close to the trap centre throughout their orbit. The test particle method is then also applied to a CMT model from the literature which contains a magnetic X-line and open and closed field lines and the results are compared with the previous results and the findings in the literature. Finally, the theoretical framework of CMT models is extended to 2.5D models with shear flow and to fully 3D models, allowing the construction of more realistic CMT models in the future.

Significant X-ray variability and flaring has been observed from Sgr A* but is poorly understood from a theoretical standpoint. We perform general relativistic magnetohydrodynamic simulations that take into account a population of non-thermal electrons with energy distributions and injection rates that are motivated by PIC simulations of magnetic reconnection. We explore the effects of including these non-thermal electrons on the predicted broadband variability of Sgr A* and find that X-ray variability is a generic result of localizing non-thermal electrons to highly magnetized regions, where particles are likely to be accelerated via magnetic reconnection. The proximity of these high-field regions to the event horizon forms a natural connection between IR and X-ray variability and accounts for the rapid timescales associated with the X-ray flares. The qualitative nature of this variability is consistent with observations, producing X-ray flares that are always coincident with IR flares, but not vice versa, i.e., there are a number of IR flares without X-ray counterparts.

L'objectif de cette thèse est de clarifier le rôle de la composante rotationnelle de la force de Lorentz dans sa capacité à imposer la topologie, et la dynamique des écoulements turbulents MHD à bas Rm, confinés par des parois rigides et électriquement isolantes. Le travail présenté ici se scinde en deux parties : D'une part une étude théorique effectuée dans un cadre faiblement inertiel, d'autre part une étude expérimentale d’écoulements turbulents pleinement développés. L’étude théorique porte sur un vortex isolé, stationnaire et axisymétrique, confiné entre deux parois rigides et électriquement isolantes, perpendiculaires à un champ magnétique uniforme. Grâce à un développement asymptotique des équations de Navier-Stokes, valable quel que soit le nombre de Hartmann, nous montrons que la dimensionnalité topologique de l’écoulement de base ne dépend que d'un seul paramètre. Ce paramètre en question compare en fait la distance sur laquelle la partie rotationnelle de la force de Lorentz est capable d'agir dans la direction du champ magnétique, avant d’être contrée par les effets visqueux. Cette étude met en lumière deux mécanismes inertiels capables d'engendrer une composante de la vitesse dans la direction du champ magnétique au premier ordre, en introduisant des recirculations dans le plan méridional : du pompage d'Ekman direct ou inverse. Un dispositif expérimental à également été construit durant ce projet, afin d’étudier la dynamique d’écoulements turbulents de métaux liquides soumis à des champs magnétiques intenses. La turbulence stationnaire engendrée par ce dispositif était forcée électriquement en imposant un courant continu à travers une matrice carrée et periodique d’électrodes d'injection. Grâce à ce dispositif, nous avons montré que les statistiques des fluctuations turbulentes étaient raisonnablement homogènes et axisymétriques, malgré un forçage inhomogène et anisotrope. Nous confirmons également, en comparant les densités d’énergie cinétique turbulentes mesurées le long des parois perpendiculaires au champ magnétique, que les processus physiques en jeu dans le domaine inertiel 3D de la turbulence MHD confinée à bas Rm sont bien la composante rotationnelle de la force de Lorentz d'une part, et les transferts inertiels d'autre part. Grâce à une étude statistique dans l'espace des échelles, nous montrons que la cinématique de la turbulence forcée dans notre expérience suit en fait une loi universelle qui ne dépend que de deux longueurs caractéristiques. Premièrement, l’échelle d'injection, dans la direction perpendiculaire au champ magnétique. Deuxièmement, le rayon d'action de la force de Lorentz avant d’être contrée par les effets inertiels, dans la direction parallèle au champ. Nous prouvons que le rapport entre cette dernière longueur caractéristique et la hauteur de l'enceinte expérimentale permet de différencier les structures turbulentes cinématiquement quasi-2D de celles qui sont cinématiquement 3D. En calculant directement le flux d’énergie cinétique turbulente perpendiculaire à travers les échelles horizontales, nous montrons que ce dernier est toujours dirigé vers les grandes échelles. Ce quel que soit la dimensionnalité des échelles en question. Autrement dit, une cascade inverse d’énergie perpendiculaire peut exister sans pour autant que les structures turbulentes associées soient quasi-2D. / This thesis aims at clarifying the role of the solenoidal component of the Lorentz force in fixing the topological dimensionality, and the ensuing dynamics of low-Rm MHD turbulent flows confined between electrically insulating and no-slip Hartmann walls. The work presented here breaks down into two main parts: An analytical investigation carried out in the weakly inertial limit on the one hand, and an experimental study of fully developed turbulence on the other hand. The analytical investigation was performed on a single steady and axisymmetric electrically driven vortex confined between no-slip and electrically insulating walls perpendicular to a uniform magnetic field. Thanks to an asymptotic expansion valid for any Hartmann number, we showed that the topological dimensionality of the leading order is fully imposed by a single parameter, which compares the distance over which the Lorentz force is able to act in the direction of the magnetic field, before it is balanced out by viscous friction. This study highlights two inertial mechanisms capable of introducing a third velocity component in the direction of the field, by means of recirculations in the meridional plane: direct and/or inverse Ekman pumping. An experimental platform was designed and built from the ground up during this project, to investigate the dynamics of liquid metal turbulence subject to extreme magnetic fields. The turbulence sustained in our experiment was forced electrically by imposing a DC current through a square periodic array of electrodes. Thanks to this setup, we showed that the statistics of the turbulent fluctuations were homogeneous and axisymmetric to a satisfactory level, despite the forcing mechanism being inhomogeneous and anisotropic. By comparing the energy densities measured along the walls perpendicular to the magnetic field, we confirm that the physical processes at stake in the 3D inertial range of wall-bounded MHD turbulence at low-Rm are the solenoidal component of the Lorentz force on the one hand, and inertia on the other hand. Thanks to a statistical analysis in scale space, we show that their exists a universal law imposing the kinematics of turbulent structures in our experiment, which turns out to be fully described by only two lenghtscales. First, the forcing scale in the direction perpendicular to the magnetic field. Second, the range of action of the Lorentz force before it is balanced out by inertial transfers, in the direction parallel to the field. We prove that the ratio of this latter scale over the height of the channel in fact segregates kinematically quasi-2D from kinematically 3D turbulent structures. By computing the actual flux of perpendicular turbulent kinetic energy along perpendicular scales, we show that it always flows towards larger turbulent scales regardless of their topology. In other words, we show that the existence of an inverse cascade of perpendicular kinetic energy does not necessarily require perpendicular turbulent scales to be topologically quasi-2D in the inertial range.

Neste trabalho foram estudadas instabilidades magnetohidrodinâmicas (MHD) utilizando um novo sistema bolométrico que foi instalado no tokamak TCABR para medidas da evolução temporal da potência irradiada. Este novo sistema conta com 24 cordas verticais, capazes de mapear toda uma secção poloidal da coluna de plasma com resolução espacial de aproximadamente 2 cm e uma resolução temporal de 20 µs. Como se sabe, as instabilidades MHD degradam o connamento do plasma e modicam a topologia das superfícies magnéticas, causando a perda da energia do plasma. Por conta disso, compreender essas instabilidades é fundamental para o sucesso dos futuros reatores de fusão nuclear. As perturbações (oscilações) causadas pelas instabilidades MHD modulam diversos parâmetros macroscópicos do plasma como a densidade, a temperatura e a potência irradiada. Então, utilizando o diagnóstico bolométrico, é possível medir as oscilações no perl de potência irradiada e, a partir deles, extrair informações importantes para determinar a origem e as características de tais instabilidades. No tokamak TCABR, as instabilidades foram caracterizadas através da análise espectral dos 24 sinais provenientes do novo sistema bolométrico. Para auxiliar a caracterização das instabilidades, um programa foi desenvolvido em Matlab para simular as medidas das perturbações no perl de potência irradiada. Através do mesmo procedimento de análise espectral, os resultados simulados foram comparados aos experimentais de forma que os parâmetros simulados, como largura e posição das ilhas magnéticas, fossem ajustados aos experimentais. Através dessa metodologia de análise, que combina simulação e experimento, foi possível caracterizar diversas instabilidades como o precursor dos dentes de serra e ilhas magnéticas de modos m = 2 e m = 3. / In this dissertation, magnetohydrodynamic (MHD) instabilities were investigated using a new bolometric system that was installed in the TCABR tokamak for radiation power measurements. This diagnostic is composed by 24 vertical chords that provide a full view of the poloidal cross section of the plasma column and provides spatial and temporal proles with approximately 2 cm space and 20 µs time resolution. As it is well known, the MHD instabilities degrade the plasma connement and modify the magnetic topology, leading to energy loss from the plasma. Therefore, the understanding of these instabilities is essential for the success of the controlled thermonuclear fusion reactors. The MHD instabilities also cause perturbations (oscillations) in various macroscopic parameters, such as plasma density, temperature, and radiated power. Therefore, the oscillations in the radiated power prole measured by the bolometric diagnostic system provide a possibility to investigate the origin and features of the instabilities. In the TCABR tokamak, the instabilities were characterized by spectral analysis of the 24 vertical chords of the bolometric signals. In addition, a Matlab program was developed to simulate the integral characteristic of the oscillations in the radiated power measured by the bolometric system. The spectral analysis of the simulated signals is then compared with the spectral analysis of the bolometric signals. The simulated parameters, island width and radial position, were then adjusted to t the experimental spectrum results. Using this method of analysis, which combines experiment and simulation, it was possible to characterize various instabilities, such as sawtooth precursor and m = 2 and m = 3 magnetic islands.

This diploma thesis is focused on waves and oscillations in the solar corona, which takes place in a large number of phenomena that occurs here. In recent years these waves have been observed in Earth observation same like as cosmic observation.The task of this thesis is to familiarize with the problems of waves and oscillations in solar corona and subsequent interpretation into numerical simulations that are performed by FLASH code. The aim of the work will be to modify numerical simulations and thus to achieve more realistic of the observed phenomena.

Some low-mass stars appear to have larger radii than predicted by standard 1D structure models; prior work has suggested that inefficient convective heat transport, due to rotation and/or magnetism, may ultimately be responsible. In this thesis, we explore this possibility using a combination of 1D stellar models, 2D and 3D simulations, and analytical theory. First, we examine this issue using 1D stellar models constructed using the Modules for Experiments in Stellar Astrophysics (MESA) code. We begin by considering standard models that do not explicitly include rotational/magnetic effects, with convective inhibition modelled by decreasing a depth-independent mixing length theory (MLT) parameter αMLT. We provide formulae linking changes in αMLT to changes in the interior specific entropy, and hence to the stellar radius. Next, we modify the MLT formulation in MESA to mimic explicitly the influence of rotation and magnetism, using formulations suggested by Stevenson (1979) and MacDonald and Mullan (2014) respectively. We find rapid rotation in these models has a negligible impact on stellar structure, primarily because a star’s adiabat, and hence its radius, is predominantly affected by layers near the surface; convection is rapid and largely uninfluenced by rotation there. Magnetic fields, if they influenced convective transport in the manner described by MacDonald and Mullan (2014), could lead to more noticeable radius inflation. Finally, we show that these non-standard effects on stellar structure can be fabricated using a depth-dependent αMLT: a non-magnetic, non-rotating model can be produced that is virtually indistinguishable from one that explicitly parameterises rotation and/or magnetism using the two formulations above. We provide formulae linking the radially-variable αMLT to these putative MLT reformulations. We make further comparisons between MLT and simulations of convection, to establish how heat transport and stellar structure are influenced by rotation and magnetism, by looking at the entropy content of 2D local and 3D global convective calculations. Using 2D “box in a star” simulations, created using the convection code Dedalus, we investigate changes in bulk properties of the specific entropy for increasingly stratified domains. We observe regions stable against convection near the bottom boundary, resulting in the specific entropy in the bulk of the domain exceeding the bottom boundary value: this could be a result of physical effects, such as increased amounts of viscous dissipation for more supercritical, highly stratified cases, but may also be influenced by the artificial boundary conditions imposed by these local simulations. We then turn to 3D global simulations, created using the convection code Rayleigh, and investigate these same properties as a function of rotation rate. We find the average of the shell-averaged specific entropy gradient in the middle third of the domain to scale with rotation rate in a similar fashion to the scaling law derived via MLT arguments in Barker et al. (2014), i.e., |⟨ds/dr⟩| ∝ Ω^4/5.