Dans cette thèse, nous considérons les chemins possibles pour donner une mesure quantitative du fait que deux espaces ne sont pas quasi-isométriques. De ce point de vue quantitatif, on reprend la définition de quasi-isométrie et on propose une notion de “croissance de distorsion quasi-isométrique” entre deux espaces métriques. Nous révisons notre article [32] où une borne supérieure optimale pour le lemme de Morse est donnée, avec la variante duale que nous appelons Anti-Morse Lemma, et leurs applications.Ensuite, nous nous concentrons sur des bornes inférieures sur la croissance de distorsion quasi-isométrique pour des espaces métriques hyperboliques. Dans cette classe, les espaces de $L^p$-cohomologie fournissent des invariants de quasi-isométrie utiles et les constantes de Poincaré des boules sont leur incarnation quantitative. Nous étudions comment les constantes de Poincaré sont transportées par quasi-isométries. Dans ce but, nous introduisons la notion de transnoyau. Nous calculons les constantes de Poincaré pour les métriques localement homogènes de la forme $dt^2+\sum_ie^{2\mu_it}dx_i^2$, et donnons une borne inférieure sur la croissance de distorsion quasi-isométrique entre ces espaces.Cela nous permet de donner des exemples présentant différents type de croissance de distorsion quasi-isométrique, y compris un exemple sous-linéaire (logarithmique). / In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ``quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincar\'e constants of balls are their quantitative incarnation. We study how Poincar\'e constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincar\'e constants for locally homogeneous metrics of the form $dt^2+\sum_ie^dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).

A esfera é utilizada como domínio computacional na modelagem de diversos fenômenos físicos, como em previsão numérica do tempo. Sua discretização pode ser feita de diversas formas, sendo comum o uso de malha regulares em latitude/longitude. Recentemente, também para melhor uso de computação paralela, há uma tendência ao uso de malhas mais isotrópicas, dentre as quais a icosaédrica. Apesar de já existirem modelos atmosféricos que usam malhas icosaédricas, não há consenso sobre as metodologias mais adequadas a esse tipo de malha. Nos propusemos, portanto, a estudar em detalhe diversos fatores envolvidos no desenvolvimento de modelos atmosféricos globais usando malhas geodésicas icosaédricas. A discretização usual por volumes finitos para divergente de um campo vetorial utiliza como base o Teorema da Divergência e a regra do ponto médio nas arestas das células computacionais. A distribuição do erro obtida com esse método apresenta uma forte relação com características geométricas da malha. Definimos o conceito geométrico de alinhamento de células computacionais e desenvolvemos uma teoria que serve de base para explicar interferências de malha na discretização usual do divergente. Destacamos os impactos de certas relações de alinhamento das células na ordem da discretização do método. A teoria desenvolvida se aplica a qualquer malha geodésica e também pode ser usada para os operadores rotacional e laplaciano. Investigamos diversos métodos de interpolação na esfera adequados a malhas icosaédricas, e abordamos o problema de interpolação e reconstrução vetorial na esfera em malhas deslocadas. Usamos métodos alternativos de reconstrução vetorial aos usados na literatura, em particular, desenvolvemos um método híbrido de baixo custo e boa precisão. Por fim, utilizamos as técnicas de discretização, interpolação e reconstrução vetorial analisadas em um método semi-lagrangiano para o transporte na esfera em malhas geodésicas icosaédricas. Realizamos experimentos computacionais de transporte, incluindo testes de deformações na distribuição do campo transportado, que mostraram a adequação da metodologia para uso em modelos atmosféricos globais. A plataforma computacional desenvolvida nesta tese, incluindo geração de malhas, interpolações, reconstruções vetoriais e um modelo de transporte, fornece uma base para o futuro desenvolvimento de um modelo atmosférico global em malhas icosaédricas. / Spherical domains are used to model many physical phenomena, as, for instance, global numerical weather prediction. The sphere can be discretized in several ways, as for example a regular latitude-longitude grid. Recently, also motivated by a better use of parallel computers, more isotropic grids have been adopted in atmospheric global circulation models. Among those, the icosahedral grids are promising. Which kind of discretization methods and interpolation schemes are the best to use on those grids are still a research subject. Discretization of the sphere may be done in many ways and, recently, to make better use of computational resources, researchers are adopting more isotropic grids, such as the icosahedral one. In this thesis, we investigate in detail the numerical methodology to be used in atmospheric models on icosahedral grids. The usual finite volume method of discretization of the divergence of a vector field is based on the divergence theorem and makes use of the midpoint rule for integration on the edges of computational cells. The error distribution obtained with this method usually presents a strong correlation with some characteristics of the icosahedral grid. We introduced the concept of cell alignment and developed a theory which explains the grid imprinting patterns observed with the usual divergence discretization. We show how grid alignment impacts in the order of the divergence discretization. The theory developed applies to any geodesic grid and can also be used for other operators such as curl and Laplacian. Several interpolation schemes suitable for icosahedral grids were analysed, including the vector interpolation and reconstruction problem on staggered grids. We considered alternative vector reconstruction methods, in particular, we developed a hybrid low cost and good precision method. Finally, employing the discretizations and interpolations previously analysed, we developed a semi-Lagrangian transport method for geodesic icosahedral grids. Several tests were carried out, including deformational test cases, which demonstrated that the methodology is suitable to use in global atmospheric models. The computational platform developed in this thesis, including mesh generation, interpolation, vector reconstruction and the transport model, provides a basis for future development of global atmospheric models on icosahedral grids.

In dieser Arbeit studieren wir Lorentz-Mannigfaltigkeiten mit spezieller Holonomie, d.h. ihre Holonomiedarstellung wirkt schwach-irreduzibel aber nicht irreduzibel. Aufgrund der schwachen Irreduzibilität lässt die Darstellung einen ausgearteten Unterraum invariant und damit also auch eine lichtartige Linie. Geometrisch hat dies zur Folge, dass wir zwei parallele Unterbündel (die Linie und ihr orthogonales Komplement) des Tangentialbündels erhalten. Diese Arbeit nutzt diese und weitere Objekte um zu beweisen, dass kompakte Lorentzmannigfaltigkeiten mit Abelscher Holonomie geodätisch vollständig sind. Zudem werden Lorentzmannigfaltigkeiten mit spezieller Holonomie und nicht-negativer Ricci-Krümung auf den Blättern der Blätterung, induziert durch das orthogonale Komplement der parellelen Linie, und maximaler erster Bettizahl untersucht. Schließlich werden vollständige Ricci-flache Lorentzmannigfaltigkeiten mit vorgegebener voller Holonomie konstruiert. / In the present thesis we study dimensional Lorentzian manifolds with special holonomy, i.e. such that their holonomy representation acts indecomposably but non-irreducibly. Being indecomposable, their holonomy group leaves invariant a degenerate subspace and thus a light-like line. Geometrically, this means that, since being holonomy invariant, this line gives rise to parallel subbundles of the tangent bundle. The thesis uses these and other objects to prove that Lorentian manifolds with Abelian holonomy are geodesically complete. Moreover, we study Lorentzian manifolds with special holonomy and non-negative Ricci curvature on the leaves of the foliation induced by the orthogonal complement of the parallel light-like line whose first Betti number is maximal. Finally, we provide examples of geodesically complete and Ricci-flat Lorentzian manifolds with special holonomy and prescribed full holonomy group.

Devido à sua importância em turbulência causada por ondas de deriva e à aplicação com propósitos em diagnósticos de plasma, a investigação de fluxos zonais (ZF) e modos acústicos geodésicos (GAM) tem atraído bastante atenção na literatura em física de plasmas. Nesta tese, primeiramente consideramos efeitos de equilíbrio com rotação poloidal e toroidal nestes modos, posteriormente investigamos efeitos diamagnéticos em GAM a partir de um modelo de dois fluido, no qual incluímos viscosidade paralela de íons e, na parte final, consideramos amortecimento de Landau e efeitos diamagnéticos simultaneamente no estudo de GAM, porém, a partir do modelo girocinético. Efeitos diamagnéticos são causados por termos que envolvem gradientes de densidade e de temperatura provenientes da função Maxwelliana de equilíbrio. O acoplamento entre os harmônicos poloidais, $m = \\pm1$, e as derivadas radiais de quantidades macroscópicas do plasma é responsável pelo aumento no valor da frequência no GAM de alta frequência e pela instabilidade no GAM de baixa frequência. Este tipo de instabilidade, que é proporcional à frequência diamagnética de elétrons e à razão entre os gradientes de temperatura e de densidade, é mais propenso a ocorrer em posições radiais em que o fator segurança é alto. Modos geodésicos são fracamente amortecidos devido a um mecânismo não colisional conhecido por amortecimento de Landau, o qual é causado pela interação entre a onda eletrostática e partículas carregadas, íons no caso, e a taxa de amortecimento é maior próximo ao centro da coluna de plasma, onde o fator de segurança assume valores mais baixos. O equilíbrio MHD com rotação foi investigado em três regimes com relação às superfícies magnéticas: isotérmico, adiabático e isométrico. Foi observado que o gradiente de temperatura possui sentido oposto em relação à velocidade de rotação poloidal apenas no regime isométrico. Ao considerar equilíbrio com rotação e superfícies magnéticas isotérmicas e incluir fluxo de calor na equação da energia, observamos que ZF apresentam frequência não-nula, a qual é proporcional à velocidade de rotação poloidal e inversamente proporcional ao fator de segurança. Como direções futuras ressaltamos que é importante considerar efeitos eletromagnéticos, estudar automodos geodésicos e incluir o efeito de partículas aprisionadas para o desenvolvimento da física de ZF e GAM. Tal desenvolvimento beneficiará tanto a área de transporte em tokamaks como a área de diagnósticos, na qual a obtenção do perfil radial da temperatura de íons e do fator de segurança é um dos objetivos. Nesta área, um novo tipo de diagnóstico conhecido como espectroscopia em modos acústicos geodésicos está sendo desenvolvido baseado no estudo de automodos. / Due to the important role in drift wave turbulence and applications for plasma diagnostic purposes, the investigation of zonal flows (ZF) and associated geodesic acoustic modes (GAM) has arisen much attention in the plasma physics literature. In this thesis, first we consider equilibrium poloidal and toroidal rotation effects on these modes using the ideal MHD model, then we investigate diamagnetic effects on GAM using a two fluid model that includes parallel ion viscosity, and, in the final step, we include both Landau damping and diamagnetic effects on the study of GAM within the framework of the gyrokinetic model. By diamagnetic effects we mean the density and temperature radial gradients terms coming from the equilibrium Maxwellian distribution function. The effects caused by the coupling between the $m = \\pm1$ poloidal harmonics and the radial derivatives of equilibrium macroscopic quantities are responsible for an increase in the frequency value of the high frequency GAM and for an instability in the low frequency GAM. This instability, which is proportional to the electron drift frequency and the ratio between ion temperature and density gradients, are more likely to occur in radial positions where the safety factor is high. We observe that geodesic modes are slowly damped by a collisionlees mechanism known as Landau damping which is caused by the wave particle interaction between the eletrostatic potential and the íons. This damping is enhanced near the center of the plasma column, where the safety factor has lower values. Equilibrium MHD with plasma rotation were investigated in three regimes regarding the magnetic surfaces: isotherm, adiabatic and isometric. It is found that the temperature gradient has opposite directions compared to the poloidal rotation only for the isometric regime. By considering equilibrium rotation with isotherm magnetic surfaces and including heat flux we observed that ZF has a non-zero frequency which is proportional to the poloidal velocity and the inverse of the safety factor. For future directions we point out that electromagnetic effects, geodesic eigenmodes and trapped particles physics should be important for the development of the ZF and GAM physics, either in the area of anomalous transport caused by drift wave turbulence or for diagnostic purposes for obtaining the radial profile of the ion temperature and the safety factor. In this area, a new kind of diagnostic known as geodesic acoustic mode spectroscopy is being developing based on the study of eigenmodes.

Elastography, which is based on applying pressure and estimating the resulting deformation, involves the forward problem to obtain the strain distributions and inverse problem to construct the elastic distributions consistent with the obtained strains on observation points. This thesis focuses on the former problem whose solution is used as an input to the latter problem. The aim is to provide the inverse problem community with accurate strain estimates of a coronary artery vessel wall. In doing so, a new ultrasonic image-based elastography approach is developed. Because the accuracy and quality of the estimated strain fields depend on the resolution level of the ultrasound image and to date best resolution levels obtained in the literature are not enough to clearly see all boundaries of the artery, one of the main goals is to acquire high-resolution coronary vessel wall ultrasound images at different pressures. For this purpose, first an experimental setup is designed to collect radio frequency (RF) signals, and then image formation algorithm is developed to obtain ultrasound images from the collected signals. To segment the noisy ultrasound images formed, a geodesic active contour-based segmentation algorithm with a novel stopping function that includes local phase of the image is developed. Then, region-based information is added to make the segmentation more robust to noise. Finally, elliptical deformable template is applied so that a priori information regarding the shape of the arteries could be taken into account, resulting in more stable and accurate results. The use of this template also implicitly provides boundary point correspondences from which high-resolution, size-independent, non-rigid and local strain fields of the coronary vessel wall are obtained.

If sufficiently persistent, semi-volatile organic contaminants (SVOCs) can travel long distances through the atmosphere from their points of release and become concentrated in cold, remote regions. As air is sampled for SVOCs to establish both their presence and the success of emission reduction efforts, it becomes helpful to determine sampling site proximity to sources and the origin of the sampled air masses. Comparing three increasingly sophisticated methods for quantifying source proximity of sampling locations, it was judged necessary to account for the actual history of the sampled air through construction of an airshed, especially if wind is highly directional and population distribution is very non-uniform. The airshed concept was improved upon by introducing a ‘geodesic’ grid of equally spaced cells, rather than a simple latitude/longitude grid, to avoid distortion near Earth’s poles and to allow for the comparison of airshed shapes. Assuming that a perfectly round airshed reveals no information about sources allows the significance of each cell of an airshed to be judged based on its departure from roundness. Combining air-mass histories with a 2 year-long series of SVOC air concentrations at Little Fox Lake in Canada’s Yukon Territory did not identify distinct source regions for most analytes, although γ-hexachlorocyclohexane appears to originate broadly in north-eastern Russia and/or Alaska. Based on this remoteness from sources, the site is judged to be well suited to monitor changes in the hemispheric background concentrations of SVOCs. A model-based exploration revealed wet-gaseous deposition as the dominant process responsible for cold-trapping SVOCs in mountain soils. Such cold trapping is particularly effective if precipitation rate increases with altitude and if temperature differences along the mountain are large. Considerable sensitivity of the modeled extent of cold-trapping to parameters as diverse as scale, mean temperature, atmospheric particle concentration and time relative to emission maxima is consistent with the wide variety of observed enrichment behaviour. Concentration gradients of polycyclic aromatic hydrocarbons and polychlorinated biphenyls in air and soil measured on four Western Canadian mountains with variable distance from sources revealed source proximity as the main driver of concentrations at both the whole-mountain scale and along individual mountain transects.

If sufficiently persistent, semi-volatile organic contaminants (SVOCs) can travel long distances through the atmosphere from their points of release and become concentrated in cold, remote regions. As air is sampled for SVOCs to establish both their presence and the success of emission reduction efforts, it becomes helpful to determine sampling site proximity to sources and the origin of the sampled air masses. Comparing three increasingly sophisticated methods for quantifying source proximity of sampling locations, it was judged necessary to account for the actual history of the sampled air through construction of an airshed, especially if wind is highly directional and population distribution is very non-uniform. The airshed concept was improved upon by introducing a ‘geodesic’ grid of equally spaced cells, rather than a simple latitude/longitude grid, to avoid distortion near Earth’s poles and to allow for the comparison of airshed shapes. Assuming that a perfectly round airshed reveals no information about sources allows the significance of each cell of an airshed to be judged based on its departure from roundness. Combining air-mass histories with a 2 year-long series of SVOC air concentrations at Little Fox Lake in Canada’s Yukon Territory did not identify distinct source regions for most analytes, although γ-hexachlorocyclohexane appears to originate broadly in north-eastern Russia and/or Alaska. Based on this remoteness from sources, the site is judged to be well suited to monitor changes in the hemispheric background concentrations of SVOCs. A model-based exploration revealed wet-gaseous deposition as the dominant process responsible for cold-trapping SVOCs in mountain soils. Such cold trapping is particularly effective if precipitation rate increases with altitude and if temperature differences along the mountain are large. Considerable sensitivity of the modeled extent of cold-trapping to parameters as diverse as scale, mean temperature, atmospheric particle concentration and time relative to emission maxima is consistent with the wide variety of observed enrichment behaviour. Concentration gradients of polycyclic aromatic hydrocarbons and polychlorinated biphenyls in air and soil measured on four Western Canadian mountains with variable distance from sources revealed source proximity as the main driver of concentrations at both the whole-mountain scale and along individual mountain transects.

Cette thèse est divisée en deux parties. La première est consacrée à la méthode du barycentre, introduite en 1995 par G. Besson, G. Courtois et S. Gallot pour résoudre la conjecture de l'Entropie Minimale. Dans le Chapitre 1 nous décrivons ses développements les plus récents, notamment l'extension de cette méthode au cadre des variétés dont la courbure sectionnelle est de signe quelconque (voir les énoncés 1.2.1 et 1.4.1). Dans le Chapitre 2 et 3 nous présentons des résultats dans lesquels la méthode du barycentre joue un rôle important. Le problème “deux variétés dont les flots géodésiques sont conjugués sont-elles isométriques ?” (problème de la rigidité par conjugaison des flots) est le thème du Chapitre 2. Après avoir montré que deux telles variétés ont la même géométrie à grande échelle, on montre comment on peut utiliser ce résultat et la méthode du barycentre pour donner une nouvelle preuve de la rigidité (par conjugaison des flots) des variétés plates. Dans le Chapitre 3 nous utilisons la méthode du barycentre (en courbure de signe quelconque) et des inégalités de Sobolev itérées pour démontrer un théorème de comparaison entre les spectres de deux variétés riemanniennes (Y , g) et (X , g') de volumes proches, sachant qu'il existe une approximation de Gromov-Hausdorff de degré non nul entre ces deux variétés. Il s'agit d'un résultat d'approximation avec majoration de l'erreur d'approximation (et pas seulement d'un résultat de convergence). Remarquons qu'il n'est fait aucune autre hypothèse géométrique (et en particulier aucune hypothèse de courbure) sur la variété (Y , g), ce qui autorise un grand nombre de contre-exemples prouvant que le résultat est optimal. Dans la deuxième partie de la thèse (chapitre 4), on démontre un Lemme de Margulis sans hypothèse sur la courbure, qui s'applique aux variétés dont les groupes fondamentaux sont des produits libres (et qui ne possèdent pas d'élément de torsion d'ordre 2). Nous donnons également une borne inférieure de la systole des variétés dont le diamètre et l'entropie volumique sont majorés et dont le groupe fondamental est isomorphe à un produit libre sans torsion. Comme conséquences de ce dernier résultat nous obtenons des résultats de précompacité et de finitude topologique ou différentiable pour les variétés riemanniennes et une minoration de leur volume, tout ceci sans faire d'hypothèse de courbure. / This Ph.D. Thesis is divided into two parts. In the first part we present the barycenter method, a technique which has been introduced by G. Besson, G. Courtois and S. Gallot in 1995, in order to solve the Minimal Entropy conjecture. In Chapter 1 we are interested in the more recent developments of this method, more precisely in the recent extension of the method to the case of manifolds having sectional curvature of variable sign. In Chapters 2 and 3 we shall present some new results whose proofs make use of the barycenter method. The Conjugacy Rigidity problem is the theme of Chapter 2. First we show a general result which provide a comparison between the large scale geometry of the Riemannian universal coverings of two compact manifolds whose geodesic flows are conjugates. Then we shall show how we can apply the latter result and the barycenter method in curvature of variable sign in order to give a new proof of the conjugacy rigidity of flat manifolds. In Chapter 3 we shall give a proof of a spectra comparison theorem for a compact Riemannian manifold which admits a Gromov-Hausdorff-approximation of non zero absolute degree on a fixed compact manifold (X,g') and which has volume almost smaller than the one of the reference manifold. The proof relies on the barycenter method in curvature of variable sign and on iterated Sobolev inequalities. We underline that it is an approximation result (and not just a convergence result) and that no curvature assumptions are made or inferred on (Y,g). The second part of the Thesis consists of a single chapter. In this chapter we prove a Margulis Lemma without curvature assumptions for Riemannian manifolds having decomposable 2-torsionless fundamental group. We shall give also a proof of a universal lower bound for the homotopy systole of compact Riemannian manifolds having bounded volume entropy and diameter, and decomposable torsionless fundamental group. As a consequence of the latter result we shall deduce a Precompactness and Finiteness theorem and a Volume estimate without curvature assumptions.

Made available in DSpace on 2015-05-15T11:46:20Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1326144 bytes, checksum: 8caf7598b3ff31900cccda592a06981f (MD5) Previous issue date: 2014-07-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we investigate monotonicity and symmetry properties of of solutions to equations involving the p-Laplace-Beltrami operator in hyperbolic space and sphere. The main tools used to obtain the result is a variant of the method of moving planes and a careful use of the maximum and comparison principles / Neste trabalho investigamos propriedades de simetria e monotonicidade de soluções para equações envolvendo o operador de p-Laplace-Beltrami no espaço hiperbólico e na esfera. As principais ferramentas empregadas para obtenção do resultado é uma variante do método dos planos móveis e um cuidadoso uso de princípios do máximo e de comparação

Devido à sua importância em turbulência causada por ondas de deriva e à aplicação com propósitos em diagnósticos de plasma, a investigação de fluxos zonais (ZF) e modos acústicos geodésicos (GAM) tem atraído bastante atenção na literatura em física de plasmas. Nesta tese, primeiramente consideramos efeitos de equilíbrio com rotação poloidal e toroidal nestes modos, posteriormente investigamos efeitos diamagnéticos em GAM a partir de um modelo de dois fluido, no qual incluímos viscosidade paralela de íons e, na parte final, consideramos amortecimento de Landau e efeitos diamagnéticos simultaneamente no estudo de GAM, porém, a partir do modelo girocinético. Efeitos diamagnéticos são causados por termos que envolvem gradientes de densidade e de temperatura provenientes da função Maxwelliana de equilíbrio. O acoplamento entre os harmônicos poloidais, $m = \\pm1$, e as derivadas radiais de quantidades macroscópicas do plasma é responsável pelo aumento no valor da frequência no GAM de alta frequência e pela instabilidade no GAM de baixa frequência. Este tipo de instabilidade, que é proporcional à frequência diamagnética de elétrons e à razão entre os gradientes de temperatura e de densidade, é mais propenso a ocorrer em posições radiais em que o fator segurança é alto. Modos geodésicos são fracamente amortecidos devido a um mecânismo não colisional conhecido por amortecimento de Landau, o qual é causado pela interação entre a onda eletrostática e partículas carregadas, íons no caso, e a taxa de amortecimento é maior próximo ao centro da coluna de plasma, onde o fator de segurança assume valores mais baixos. O equilíbrio MHD com rotação foi investigado em três regimes com relação às superfícies magnéticas: isotérmico, adiabático e isométrico. Foi observado que o gradiente de temperatura possui sentido oposto em relação à velocidade de rotação poloidal apenas no regime isométrico. Ao considerar equilíbrio com rotação e superfícies magnéticas isotérmicas e incluir fluxo de calor na equação da energia, observamos que ZF apresentam frequência não-nula, a qual é proporcional à velocidade de rotação poloidal e inversamente proporcional ao fator de segurança. Como direções futuras ressaltamos que é importante considerar efeitos eletromagnéticos, estudar automodos geodésicos e incluir o efeito de partículas aprisionadas para o desenvolvimento da física de ZF e GAM. Tal desenvolvimento beneficiará tanto a área de transporte em tokamaks como a área de diagnósticos, na qual a obtenção do perfil radial da temperatura de íons e do fator de segurança é um dos objetivos. Nesta área, um novo tipo de diagnóstico conhecido como espectroscopia em modos acústicos geodésicos está sendo desenvolvido baseado no estudo de automodos. / Due to the important role in drift wave turbulence and applications for plasma diagnostic purposes, the investigation of zonal flows (ZF) and associated geodesic acoustic modes (GAM) has arisen much attention in the plasma physics literature. In this thesis, first we consider equilibrium poloidal and toroidal rotation effects on these modes using the ideal MHD model, then we investigate diamagnetic effects on GAM using a two fluid model that includes parallel ion viscosity, and, in the final step, we include both Landau damping and diamagnetic effects on the study of GAM within the framework of the gyrokinetic model. By diamagnetic effects we mean the density and temperature radial gradients terms coming from the equilibrium Maxwellian distribution function. The effects caused by the coupling between the $m = \\pm1$ poloidal harmonics and the radial derivatives of equilibrium macroscopic quantities are responsible for an increase in the frequency value of the high frequency GAM and for an instability in the low frequency GAM. This instability, which is proportional to the electron drift frequency and the ratio between ion temperature and density gradients, are more likely to occur in radial positions where the safety factor is high. We observe that geodesic modes are slowly damped by a collisionlees mechanism known as Landau damping which is caused by the wave particle interaction between the eletrostatic potential and the íons. This damping is enhanced near the center of the plasma column, where the safety factor has lower values. Equilibrium MHD with plasma rotation were investigated in three regimes regarding the magnetic surfaces: isotherm, adiabatic and isometric. It is found that the temperature gradient has opposite directions compared to the poloidal rotation only for the isometric regime. By considering equilibrium rotation with isotherm magnetic surfaces and including heat flux we observed that ZF has a non-zero frequency which is proportional to the poloidal velocity and the inverse of the safety factor. For future directions we point out that electromagnetic effects, geodesic eigenmodes and trapped particles physics should be important for the development of the ZF and GAM physics, either in the area of anomalous transport caused by drift wave turbulence or for diagnostic purposes for obtaining the radial profile of the ion temperature and the safety factor. In this area, a new kind of diagnostic known as geodesic acoustic mode spectroscopy is being developing based on the study of eigenmodes.