Neste trabalho estudamos a interação feixe-plasma aplicando a teoria de turbulência fraca. Primeiramente fazemos uma introdução `a teoria cinética de plasmas, estudando aspectos fundamentais como: a abordagem estatística das equações do plasma, o tratamento a ser dado `as funções de correlação que aparecem nessa abordagem, e o sistema de equações de Vlasov-Maxwell. Em seguida estudamos a aproximação de Vlasov e a solução do sistema de Vlasov- Maxwell na aproximação linear, enfocando a descrição de ondas no plasma e o amortecimento de Landau. Ao longo desse desenvolvimento fazemos uma apresentação relativamente detalhada dos procedimentos introduzidos por Landau ao tratar o sistema Vlasov-Maxwell como um problema de valor inicial, incluindo uma discussão sobre a resolução de integrais no plano complexo, com polos no denominador, e a solução da relação de dispersão para encontrar os modos normais de oscilação no plasma. Apresentamos também uma breve revisão a respeito da aproximação quase-linear do sistema de Vlasov- Maxwell, na qual abordamos a obtenção da equação quase-linear de difusão no espaço das velocidades para partículas e estudamos suas propriedades de conservação. Nesse contexto da teoria quase-linear, apresentamos uma revisão a respeito da evolução temporal do amortecimento de Landau no caso de distribuição de velocidades Maxwelliana e da instabilidade que pode ocorrer no caso de uma função distribuição com um feixe de partículas. Depois dessa introdução `a aproximação linear e `a teoria quase-linear, abordamos a teoria de turbulência fraca num plasma não magnetizado, incluindo interações não lineares entre ondas e partículas. O formalismo apresentado inclui efeitos como emissão espontânea e induzida, decaimento e espalhamento de ondas. O sistema de equações acopladas da teoria de turbulência fraca é então reduzido a uma aplicação a um sistema considerando duas dimensões, depois reescrito em termos de coordenadas polares e adaptado para solução numérica. Apresentamos então uma descrição do código numérico desenvolvido usando linguagem Fortran, abordando um plasma com elétrons descritos por uma função distribuição Maxwelliana com um feixe tênue, e íons descritos por uma função distribuição Maxwelliana. Finalmente, comparamos nossos resultados com outros obtidos por outros autores em trabalhos anteriores, desenvolvidos usando coordenadas cartesianas, avaliando nosso trabalho. Por último, discutimos algumas perspectivas para o desenvolvimento futuro do trabalho. / In the present work we study the beam-plasma interaction using the weak turbulence theory. We start with an introduction to the kinetic theory of plasmas, studying fundamental features, like the statistical approach to the plasma equations, the procedures to be employed to deal with the correlation functions appearing in the statistical approach, and the system of Vlasov-Maxwell equations. In the sequence we discuss the Vlasov approximation and the solution of the Vlasov- Maxwell system in the linear approximation, emphasizing the description of waves in the plasma and the Landau damping. Along the development we present a relatively detailed description of the procedures introduced by Landau to treat the Vlasov-Maxwell system as an initial value problem, including a discussion about the resolution of integrals in the complex plane, with poles in the denominator, and the solution of the dispersion relation to find the normal mode of oscillations in the plasma. We also present a short review about the quasilinear approximation of the Vlasov-Maxwell system, in which we discuss the derivation of the quasilinear diffusion equation in the space of particle velocities, and study its properties of conservation. In the context of the quasilinear theory, we present a short review about the time evolution of the Landau damping in the case of Maxwellian velocity distribution, and about the instability which can occur in the case of a distribution function with a beam of particles. After the introduction to the linear approximation and to the quasilinear theory, we present the equations of weak turbulence theory for a unmagnetized plasma, including non-linear interactions between waves and particles. The formalism which is presented includes effects and spontaneous and induced emission, decay and scattering of waves. The system of coupled equations of the weak turbulence theory is then reduced to application to a bi-dimensional case, and then re-written in terms of polar coordinates and adapted to numerical solution. We then present a description of the numerical code developed using Fortran language, suitable to describe a plasma with electrons described by a Maxwellian distribution function with a tenuous beam, and ions described by a Maxwellian distribution. Finally, we compare our results with results obtained by other authors in previous works, developed using cartesian coordinates, as a validation of our work. Lastly we discuss some perspectives for future developments.

Este trabalho tem como objetivo principal fazer uma revisão da teoria de turbulência fraca em plasmas e caracterizar o efeito dos diferentes termos associados aos efeitos eletrostáticos e eletromagnéticos sobre a evolução temporal das intensidades das ondas e da função distribuição das partículas que compõem o plasma. Para tanto, será apresentada uma revisão da teoria cinética de plasmas, desde seus aspectos fundamentais. A seguir será discutido o sistema de equações Vlasov-Maxwell na abordagem quase-linear, enfatizando quais as principais características da teoria. Depois, será feita uma revisão de uma formulação relativamente recente, apresentando as bases para a teoria de turbulência fraca, mencionando termo a termo as principais características. Primeiramente será apresentado com detalhe o caso eletrostático, apresentando as equações cinéticas para as ondas tanto para os modos lineares, quanto para os modos não lineares de excitação. A seguir, a generalização da teoria, incluindo os efeitos das ondas eletromagnéticas nas equações cinéticas das ondas e das partículas. Por fim serão apresentados alguns resultados obtidos de uma análise numérica do sistema de equações acopladas que leva em conta tanto ondas eletrostáticas quanto eletromagnéticas, com o objetivo de caracterizar, para os tempos iniciais da evolução, os efeitos associados a cada um dos termos que contribuem para a equação de evolução temporal das ondas eletromagnéticas. A ênfase será dada a esses termos, uma vez que os efeitos associados à evolução das ondas eletrostáticas já têm sido bastante investigados na literatura recente. Para o futuro imediato, a intenção é continuar desenvolvendo o código numérico, visando aplicação a situações em que ocorrem plasmas não térmicos, como é o caso da interação feixe-plasma. A ideia é utilizar o programa bidimensional para a turbulência fraca para investigar a geração de ondas transversas por efeitos não lineares associados com a instabilidade feixe-plasma, para diferentes valores dos parâmetros que caracterizam o plasma de fundo e os feixes de partículas. / This work has as main objective to review the theory of weak turbulence in plasmas and characterize the effect of various terms associated with electrostatic and electromagnetic effects on the time evolution of the wave intensities and of the distribution function of particles composing the plasma. With this objective, a review of plasmas kinetic theory will be presented, starting from fundamental aspects. The sequence will discuss the system of Vlasov-Maxwell equations in quasilinear approach, with emphasis on the main features of theory. Then, a review will be made of a relatively recent formulation, presenting the foundations for the theory of weak turbulence, describing the main characteristics of each term. First will be presented in detail the kinetic equations for the electrostatic case, including mechanisms of excitation of waves in linear modes as well as waves in nonlinear modes. Following, there is a generalization of the theory, including the effects of electromagnetic waves in the kinetic equations of waves and particles. Finally, some results of a numerical analysis of the system of coupled equations including both electrostatic and electromagnetic waves will be presented, with the objective of characterization of the effects associated to each of the terms contributing to the time evolution of electromagnetic waves, for the initial stages of time evolution. Emphasis will be given to such terms, since the effects associated with the evolution of electrostatic waves have been quite investigated in recent literature. For the immediate future, the intention is to continue developing the numerical code, aiming at application to situations where non thermal plasmas occur, as is the case of the beam-plasma interaction. The idea is to use the two dimensional program for weak turbulence to investigate the generation of transverse waves by non-linear effects associated with the beam-plasma instability, for different values of the parameters that characterize the background plasma and the particle beams.

Cette thèse a pour objectif d'étudier la dynamique spatio-temporelle des atomes refroidis par laser dans un piège magnéto-optique (PMO). Il a été montré qu'un nuage d'atomes froids dans le régime de diffusion multiple peut présenter un comportement instable sans modulation externe du système. Cependant, ces instabilités n'ont pas encore été modélisées de façon satisfaisante. Une nouvelle configuration du PMO a été mise en oeuvre pour tenter d'étudier des instabilités dans une seule direction. Ce PMO, qualifié d'anisotrope, n'utilise pas des lasers de mêmes fréquences dans chaque direction de l'espace. Il met en évidence les forts couplages existants entre les directions du piège, si bien qu'il n'est pas possible de l'utiliser pour réduire le nombre de dimensions dans lesquelles les instabilités s'établissent. Toutefois, cette étude constitue un premier pas vers une meilleure description tridimensionnelle du piège. Elle nous a notamment permis de mesurer la probabilité pour qu'un photon diffusé soit réabsorbé à l'intérieur du nuage. Cette quantité est caractéristique du PMO mais elle n'avait jusqu'à là jamais été mesurée. Nous avons également établi un modèle spatio-temporel unidimensionnel du PMO. Il est constitué d'un système d'équations non-linéaires couplées reliant la densité atomique et les intensités des faisceaux lasers. Ce système contient notamment une équation de Vlasov-Fokker-Planck, rencontrée dans de nombreux domaines de la physique. Des simulations numériques ont été effectuées dans un cas simple. D'un point de vue expérimental, l'utilisation d'une caméra rapide nous a permis de mettre en évidence la structure spatiale d'instabilités de type stochastique. / The aim of this thesis is to study the spatio-temporal dynamics of laser cooled atoms in a magneto-optical trap (MOT). Recent works have shown that in the multiple scattering regime, an atomic cloud can have an unstable behavior without external modulation of the system. Nevertheless, these instabilities have not yet been modeled in a satisfactory way. A new configuration of the MOT has been built up as a possible way to study instabilities in only one direction. This trap, called anisotropic MOT, is not made of laser beams with the same laser frequencies along each direction of space. It exhibits the strong couplings between the directions of the trap, with the result that it cannot be used to reduce the number of directions in which instabilities grow up. However, this study can be considered as a new step to a better 3D description of the MOT physics. In particular, it gives us a way to measure the probability that a scattered photon is reabsorbed inside the atomic cloud. This quantity is a characteristic of the MOT but it has never been measured so far. We also develop a 1D spatio-temporal model of the MOT. It consists in a set of coupled nonlinear equations linking the atomic density and the laser intensities. This set contains a Vlasov-Fokker-Planck equation which is used to model a lot of systems in various fields and not only in physics. Numerical simulations have been done in a simple case. In the experiment, the use of a fast video camera allows us to observe the spatial structure of one type of instabilities, the so-called stochastic instabilities.

We study a stochastic particle system with a logarithmically-singular inter-particle interaction potential which allows for inelastic particle collisions. We relate the squared Bessel process to the evolution of localized clusters of particles, and develop a numerical method capable of detecting collisions of many point particles without the use of pairwise computations, or very refined adaptive timestepping. We show that when the system is in an appropriate parameter regime, the hydrodynamic limit of the empirical mass density of the system is a solution to a nonlinear Fokker-Planck equation, such as the Patlak-Keller-Segel (PKS) model, or its multispecies variant. We then show that the presented numerical method is well-suited for the simulation of the formation of finite-time singularities in the PKS, as well as PKS pre- and post-blow-up dynamics. Additionally, we present numerical evidence that blow-up with an increasing total second moment in the two species Keller-Segel system occurs with a linearly increasing second moment in one component, and a linearly decreasing second moment in the other component.

京都大学 / 0048 / 新制・課程博士 / 博士(エネルギー科学) / 甲第17913号 / エネ博第285号 / 新制||エネ||59(附属図書館) / 30733 / 京都大学大学院エネルギー科学研究科エネルギー基礎科学専攻 / (主査)教授 岸本 泰明, 教授 中村 祐司, 教授 前川 孝 / 学位規則第4条第1項該当 / Doctor of Energy Science / Kyoto University / DFAM

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第17924号 / 情博第506号 / 新制||情||89(附属図書館) / 30744 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 梅野 健, 教授 中村 佳正, 教授 船越 満明 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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L'origine de champs magnétiques observés dans les plasmas de laboratoire et d'astrophysique est l'un des problèmes récurrents en physique des plasmas. À cet égard, les instabilités de type Weibel sont considérées d'une grande importance. Ces instabilités ont pour origine une anisotropie de température (instabilité de Weibel) et des moments des électrons (instabilité de filamentation de courant). L'objectif principal de cette thèse est l'étude théorique et numérique de ces instabilités dans un plasma non collisionnel en régime relativiste. Le premier aspect de ce travail est l'étude du régime non-linéaire de ces instabilités et du rôle des effets cinétiques et relativistes sur la structure des champs électromagnétiques auto-cohérents. Dans ce cadre, un problème essentiel pour les applications et la théorie, concerne l'identification et l'analyse des structures cohérentes développées spontanément dans le régime non-linéaire sur des échelles cinétiques. Un deuxième aspect du travail est le développement de techniques analytiques et numériques pour l'étude des plasmas non collisionnels. Le modèle mathématique de référence, à la base des études des plasmas chauds, est le modèle Vlasov-Maxwell, où l'équation de Vlasov (théorie des champs moyens) est couplée aux équations de Maxwell de façon auto-cohérente. Un modèle unidimensionnel, le modèle multi-faisceaux, a également été introduit durant cette thèse. Basé sur une technique de réduction en dimension, il est à la fois un modèle analytique "simple" présentant l'avantage de pouvoir résoudre une équation de Vlasov 1D pour chaque faisceau de particules, et un modèle numérique moins coûteux qu'un modèle complet / The origin of magnetic fields observed in laboratory and astrophysical plasmas is one ofthe most challenging problems in plasma physics. In this respect, the Weibel type instabilities are considered of key importance. These instabilities are caused by a temperature anisotropy (Weibel instability) and electron momentum (current filamentation instability). The main objective of this thesis is the theoretical and numerical study of these instabilities in a collisionless plasma in the relativistic regime. The first aspect of this work is to study the nonlinear regime of these instabilities and the role of kinetic and relativistic effects on the structure of self-consistent electromagnetic fields. In this context, a key problem for the theory and applications, is the identification and analysis of coherent structures developed spontaneously in the nonlinear regime of kinetic scales. A second aspect of the work is the development of analytical and numerical techniques for the study of collisionless plasmas. A mathematical model of reference is the Vlasov-Maxwell model, where the Vlasov equation (mean field theory) is coupled to the Maxwell equations in a self-consistent way. A one-dimensional model, the multi-stream model, is also introduced. Based on a dimensional reduction technique, it is both an analytical model "simple" having the advantage of being able to solve a 1D Vlasov equation for each particle beam, and a numerical model less expensive than a complete model

Cette thèse traite de deux sujets: la résolubilité forte d'équations différentielles stochastiques à dérive hölderienne et bruit hypoelliptique et la simulation de processus progressifs-rétrogrades découplés de McKean-Vlasov. Dans le premier cas, on montre qu'un système hypoelliptique, composé d'une composante diffusive et d'une composante totalement dégénérée, est fortement résoluble lorsque l'exposant de la régularité Hölder de la dérive par rapport à la composante dégénérée est strictement supérieur à 2/3. Ce travail étend au cadre dégénéré les travaux antérieurs de Zvonkin (1974), Veretennikov (1980) et Krylov et Röckner (2005). L'apparition d'un seuil critique pour l'exposant peut-être vue comme le prix à payer pour la dégénérescence. La preuve repose sur des résultats de régularité de la solution de l'EDP associée, qui est dégénérée, et est basée sur une méthode parametrix. Dans le second cas, on propose un algorithme basé sur les méthodes de cubature pour la simulation de processus progessifs-rétrogrades découplés de McKean-Vlasov apparaissant dans des problèmes de contrôle dans un environnement de type champ moyen. Cet algorithme se divise en deux parties. Une première étape de construction d'un arbre de particules, à dynamique déterministe, approchant la loi de la composante progressive. Cet arbre peut être paramétré de manière à obtenir n'importe quel ordre d'approximation (en terme de pas de discrétisation de l'intervalle). Une seconde étape, conditionnelle à l'arbre, permettant l'approximation de la composante rétrograde. Deux schémas explicites sont proposés permettant un ordre d'approximation de 1 et 2. / This thesis deals with two subjects: the strong well-posedness of stochastic differential equations with Hölder drift and hypoelliptic noise and the simulation of decoupled forward backward stochastic differential equations of McKean-Vlasov type. In the first work, we study a class of degenerate system with hypoelliptic noise. We prove that strong well-posedness holds for this system when the drift is only H\"{o}lder, with Hölder exponent larger than the critical value 2/3. This work extends to the degenerate setting the earlier results obtained by Zvonkin (1974), Veretennikov (1980) and Krylov and Röckner (2005). The existence of a threshold for the Hölder exponent in the degenerate case may be understood as the price to pay to balance the degeneracy of the noise. Our proof relies on regularization properties of the associated PDE, which is degenerate in the current framework and is based on a parametrix method. In the second work, we propose a new algorithm to approach weakly the solution of a McKean-Vlasov stochastic differential equation. Based on the cubature method, the algorithm is deterministic differing from the usual methods based on interacting particles. It can be parametrized in order to obtain a given order of convergence. Then, we construct implementable algorithms to solve decoupled forward backward stochastic differential equations of McKean-Vlasov type, which appear in some stochastic control problems in a mean field environment. We give two algorithms and show that they have convergence of orders one and two under appropriate regularity conditions.

Les instabilités de type Weibel naissent si la distribution des vitesses du plasma présente une anisotropie. Elles entraînent la génération d’un champ magnétique dû à la formation de filaments de courant ainsi qu’une activité électrostatique importante. Ces phénomènes de base apparaissent dans de nombreuses situations, naturelles (vent solaire, jets relativistes) ou expérimentales (interaction laser-plasma) : les plasmas dans lesquels ils naissent peuvent être relativistes ou non, magnétisés ou non, collisionnels ou non, ce qui pose la question du choix du modèle à utiliser pour les décrire. La théorie cinétique est le cadre le plus complexe dans lequel nous travaillerons. De par sa complexité, il est intéressant de développer des modèles réduits. Un premier travail mené au cours de cette thèse est l’utilisation d’un modèle fluide incluant la dynamique du tenseur de pression pour modéliser la phase linéaire des instabilités de type Weibel. On discute le rôle essentiel joué par les composantes hors diagonale du tenseur dans la génération du champ magnétique, puis la capacité du modèle à reproduire quantitativement ou qualitativement les résultats cinétiques en introduisant la notion de limite hydrodynamique. La seconde partie de la thèse est ciblée sur le développement du code semi-lagrangien relativiste VLEM utilisant une méthode de décomposition de domaine : on présente les principales méthodes mathématiques utilisées dans le code, puis on aborde la problématique de la conservation de la charge à laquelle on apporte une réponse reposant sur une adaptation de la méthode d’Esirkepov. Le code est enfin validé grâce à plusieurs simulations d’instabilités de type Weibel / Weibel-type instabilities occurs when the velocity distribution function of the charged particles displays a pronounced anisotropy. A long-lasting magnetic field is generated due to the formation of current filaments, and it is accompanied by an important electrostatic activity. These ``basic’’ phenomena have been greatly investigated because of their involvement in many physical problems, natural (solar wind, relativistic jets) or experimental (laser-plasma interaction) : they occurs in plasmas which can be collisional or not, magnetised or not, relativistic or not. One needs to choose a suitable model for their description. The kinetic theory is the most complete and somewhat complex theoretical framework which we will consider. Due to its complexity, it may be interesting to develop reduced models. The first work realised during this thesis is the utilisation of a non-relativistic fluid description, including the dynamics of the pressure tensor, in order to model the linear Weibel-type instabilities. We put in evidence the effect of the non-diagonal components of the tensor on the magnetic field generation. We discuss the ability of the model to reproduce quantitatively or qualitatively the kinetic results by introducing the hydrodynamics limit. The second part of this thesis work is dedicated to the development of the relativistic semi-lagrangian code VLEM, using a domain decomposition scheme : we present the main mathematical tools used in the code, then we deal with the problem of the charge conservation and propose a solution for VLEM, based on an adaptation of the Esirkepov method. Finally, we validate the code through simulations of Weibel-type