The nonlinear interactions between long surface waves and interfacial waves in a two-layer fluid are studied theoretically. The fluid is density-stratified and the thicknesses of the top and bottom layers are both assumed to be shallow relative to the length of a typical surface wave and interfacial wave, respectively. A set of Boussinesq-type equations are derived for potential flow in this system. The equations are then analyzed for the dynamics of the nonlinear resonant interactions between a monochromatic surface wave and two oblique interfacial waves. The analysis uses a second order perturbation approach. Consequently, a set of coupled transient evolution equations of wave amplitudes is derived. Moreover, the effect of weak viscosity of the lower layer is incorporated in the problem and the influences of important parameters on surface and interfacial wave evolution (namely the directional angle of interfacial waves, density ratio of the layers, thickness of the fluid layers, surface wave frequency, surface wave amplitude, and lower layer viscosity) are investigated. The results of the parametric study are discussed and are generally in qualitative agreement with previous studies. In shallow water, a triad formed of surface waves (or interfacial waves) can be considered in near-resonant interaction. In contrast to the previous studies which limited the study to a triad (one surface wave and two interfacial waves or one interfacial and two surface waves), the problem is generalized by considering the nonlinear interactions between a triad of surface waves and three oblique pairs of interfacial waves. In this system, each surface wave is in near-resonance interaction with other surface waves and in exact resonance with a pair of oblique interfacial waves. Similarly, each interfacial wave is in near-resonance interaction with other interfacial waves which are propagating in the same direction. Inclusion of all the interactions considerably changes the pattern of evolution of waves and highlights the necessity of accounting for several wave harmonics. Effects of density ratio, depth ratio, and surface wave frequency on the evolution of waves are discussed. Finally, a formulation is derived for spatial evolution of one surface wave spectrum in nonlinear interaction with two oblique interfacial wave spectra. The two-layer Boussinesq-type equations are treated in frequency domain to study the nonlinear interactions of time-harmonic waves. Based on weakly two-dimensional propagation of each wave train, a parabolic approximation is applied to derive the formulation.

Internal waves are prominent fluid phenomena in both the atmosphere and ocean. Because internal waves have the ability to transfer a large amount of energy, they contribute to the global distribution of energy. This causes internal waves to influence global climate patterns and critical ocean mixing. Therefore, studying internal waves provides additional insight in how to model geophysical phenomena that directly impact our lives. There is a myriad of fluid phenomena with which internal waves can interact, including other internal waves. Equipment and processes are developed to perform laboratory experiments analyzing the interaction of two colliding nonresonant internal waves. Nonresonant interactions have not been a major focus in previous research. The goal of this study is to visualize the flow field, compare qualitative results to Tabaei et al., and determine the energy partition to the second-harmonic for eight unique interaction configurations. When two non-resonant internal waves collide, harmonics are formed at the sum and difference of multiples of the colliding waves' frequencies. In order to create the wave-wave interaction, two identical wave generators were designed and manufactured. The interaction flow field is visualized using synthetic schlieren and the energy entering and leaving the interaction region is analyzed. It is found that the energy partitioned to the harmonics is much more dependent on the general direction the colliding waves approach each other than on the angle. Depending on the configurations, between 0.5 and 7 percent of the energy within the colliding waves is partitioned to the second-harmonics. Interactions in which the colliding waves have opposite signed vertical wavenumber partition much more energy to the harmonics. Most of the energy entering the interaction is dissipated by viscous forces or leaves the interaction within the colliding waves. For all eight configurations studied, 5 to 8 percent of the energy entering the interaction has an unknown fate.

Cette étude s'inscrit dans l'effort de réduction des nuisances sonores des avions au décollage. Une des principales composantes est le bruit de jet, dont la partie à basse fréquence peut notamment être imputée au rayonnement acoustique directif des structures cohérentes de grande échelle engendrées par les instabilités dans la couche de mélange du jet. L'évolution de ces ondes d'instabilité peut être décrite au moyen des équations de Stabilité Parabolisées (PSE). Un premier objectif a été de déterminer si dans le cas d'un jet turbulent naturel, les interactions non linéaires entre les ondes d'instabilité ont un impact significatif sur sa dynamique et sur son rayonnement acoustique. À cet effet, une modélisation PSE non linéaire a été développée et appliquée à une configuration réaliste. La possibilité de manipuler ces ondes d'instabilité par non linéarité a ensuite été étudiée en vue d'une réduction du rayonnement acoustique. Pour cela, une analyse PSE a été menée pour déterminer l'effet sur le bruit de jet de l'excitation d'un ou plusieurs modes instables. Ces travaux de thèse ont permis de montrer, d'une part, que les non linéarités semblent avoir un impact mineur sur la dynamique des ondes d'instabilité dans le cas des jets turbulents naturels, et d'autre part, qu'il est possible de réduire le rayonnement acoustique des modes dominants par interactions non linéaires. / This study is part of the effort to reduce aircraft noise during take-off. Jet noise is oneof the main contributors, of which lower frequency component can be attributed to thedirective acoustic field generated by the large-scale coherent structures arising from jetmixing-layer instabilities. The development of these instability waves can be describedusing Parabolized Stability Equations (PSE). A first objective was to determine if inthe case of a natural turbulent jet, nonlinear interactions between instability waveshave a significant impact on its dynamic and acoustic behaviour. For this purpose,a nonlinear PSE model has been developed and applied to a realistic configuration.Then, the possibility to manipulate these instability waves by means of nonlinearity wasinvestigated with a view to reduce noise. To this end, a PSE analysis has been carried outto assess the impact on jet noise of exciting one or more unstable modes. The findingsof this doctoral work demonstrate a minor impact of nonlinearities on the dynamics ofinstability waves for natural turbulent jets on the one hand, and the possibility to makethe initially dominant instability acoustically ineffective using nonlinear interactions onthe other hand.

Este trabalho avalia a importância relativa dosmodos dinâmicos na composição da estrutura vertical do fluxo geostrófico e de suas anomalias em relação à média, com o objetivo de fornecer uma descrição das formas de estratificação predominantes na coluna de água e de identificar prováveis regiões de ocorrência de ondas de Rossby planetárias (OR) dos diversos modos dinâmicos. Raios de deformação internos são estimados para avaliar a possibilidade de interações não-lineares entre OR e as alterações da estratificação local forçadas por sua própria passagem. A análise proposta baseia-se em dados de densidade estimados a partir de perfis verticais de salinidade climatológicos aliados a perfis sintéticos de temperatura. Estes últimos são extrapolados verticalmente a partir de dados orbitais da temperatura da superfície do mar (TSM) e da anomalia da altura da superfície livre (h), segundo um método de reconstrução estatística desenvolvido para este estudo. O primeiro modo baroclínico é dominante tanto no fluxo total quanto em suas anomalias, respondendo em média por 30% da estrutura vertical de velocidade, sendo que este valor descresce aproximadamente por uma razão de três em modos superiores. O segundo modo é significativo ou mesmo dominante em algumas regiões, particularmente em latitudes próximas ao equador. O terceiro é evidente em algumas áreas localizadas, mas não assume papéis importantes em escala de bacia. O modo 0 responde por frações de 6 a 9%, mas é provável que estes resultados sejam subestimados pela metodologia aplicada. Anomalias verticais relacionadas ao primeiro modo coincidem com regiões onde OR longas do primeiro modo tem maior expressão, enquanto o segundo modo parece ser um marcador de OR curtas e ondas de instabilidade tropical. Fenômenos transientes associados ao terceiro modo são observados em áreas restritas dos três oceanos. A magnitude das variações dos raios internos em resposta à fenômenos transientes em algumas regiões implicam em variações significativas na velocidade de fase teórica de OR longas lineares, um indício de que efeitos não-lineares podem ser importantes. Por fim, amplificações da importância do modo barotrópico sobre feições topográficas significativas sugere a existência de mecanismos de transferência de energia entre modos dinâmicos induzidas pela interação com a batimetria. / This study evaluates the relative importance of the dynamical modes in the composition of the geostrophic flow and of its anomalies from the long-term average, respectively seeking to determine the dominant vertical stratification structures of the water column and to identify regions where planetary Rossby waves (RW) of different dynamical modes may most likely occur. The baroclinic Rossby radii of deformation are estimated to evaluate the possibility of nonlinear interactions between RWand changes of the local stratification forced by the wave\'s passing. This analysis is based on density data estimated from climatological salinity profiles and synthetic temperature profiles. The latter are vertically extrapolated for sea surface temperature (SST) and sea surface height anomaly (h) satellite data, using a statistical reconstruction method developed in this study. The first baroclinic mode dominates both the total geostrophic flow and its anomalies, accounting for 30% of the velocity\'s vertical structure on average, where this value decreases approximately by a factor of 3 in subsequent baroclinic orders. The second mode is significant or even dominant in some areas, particularly near the equator. The third mode is evident in some localized regions, but can be ignored at basin-scales. The barotropic mode accounts for 6 to 9% fractions on average, however these values are probably underestimated by the used methods. Vertical anomalies related to the first baroclinic mode coincide with regions where long RWs answers for a significant fraction of local variance, while the second mode highlights zones where short RWs and tropical instability waves are reported. Transient phenomena related with the third mode are observed in comparatively small areas on all three oceans. The magnitude of the baroclinic radii\'s variations in response of the transient variability results in significant changes of the theorethical phase speed for long linear RWs, an evidence that nonlinear effects may be important. Lastly, the greater significance of the barotropic mode over proeminent bottom features suggests the existence of energy transfer mechanisms between dynamical modes triggered by the interaction with the bathymetry.

Este trabalho avalia a importância relativa dosmodos dinâmicos na composição da estrutura vertical do fluxo geostrófico e de suas anomalias em relação à média, com o objetivo de fornecer uma descrição das formas de estratificação predominantes na coluna de água e de identificar prováveis regiões de ocorrência de ondas de Rossby planetárias (OR) dos diversos modos dinâmicos. Raios de deformação internos são estimados para avaliar a possibilidade de interações não-lineares entre OR e as alterações da estratificação local forçadas por sua própria passagem. A análise proposta baseia-se em dados de densidade estimados a partir de perfis verticais de salinidade climatológicos aliados a perfis sintéticos de temperatura. Estes últimos são extrapolados verticalmente a partir de dados orbitais da temperatura da superfície do mar (TSM) e da anomalia da altura da superfície livre (h), segundo um método de reconstrução estatística desenvolvido para este estudo. O primeiro modo baroclínico é dominante tanto no fluxo total quanto em suas anomalias, respondendo em média por 30% da estrutura vertical de velocidade, sendo que este valor descresce aproximadamente por uma razão de três em modos superiores. O segundo modo é significativo ou mesmo dominante em algumas regiões, particularmente em latitudes próximas ao equador. O terceiro é evidente em algumas áreas localizadas, mas não assume papéis importantes em escala de bacia. O modo 0 responde por frações de 6 a 9%, mas é provável que estes resultados sejam subestimados pela metodologia aplicada. Anomalias verticais relacionadas ao primeiro modo coincidem com regiões onde OR longas do primeiro modo tem maior expressão, enquanto o segundo modo parece ser um marcador de OR curtas e ondas de instabilidade tropical. Fenômenos transientes associados ao terceiro modo são observados em áreas restritas dos três oceanos. A magnitude das variações dos raios internos em resposta à fenômenos transientes em algumas regiões implicam em variações significativas na velocidade de fase teórica de OR longas lineares, um indício de que efeitos não-lineares podem ser importantes. Por fim, amplificações da importância do modo barotrópico sobre feições topográficas significativas sugere a existência de mecanismos de transferência de energia entre modos dinâmicos induzidas pela interação com a batimetria. / This study evaluates the relative importance of the dynamical modes in the composition of the geostrophic flow and of its anomalies from the long-term average, respectively seeking to determine the dominant vertical stratification structures of the water column and to identify regions where planetary Rossby waves (RW) of different dynamical modes may most likely occur. The baroclinic Rossby radii of deformation are estimated to evaluate the possibility of nonlinear interactions between RWand changes of the local stratification forced by the wave\'s passing. This analysis is based on density data estimated from climatological salinity profiles and synthetic temperature profiles. The latter are vertically extrapolated for sea surface temperature (SST) and sea surface height anomaly (h) satellite data, using a statistical reconstruction method developed in this study. The first baroclinic mode dominates both the total geostrophic flow and its anomalies, accounting for 30% of the velocity\'s vertical structure on average, where this value decreases approximately by a factor of 3 in subsequent baroclinic orders. The second mode is significant or even dominant in some areas, particularly near the equator. The third mode is evident in some localized regions, but can be ignored at basin-scales. The barotropic mode accounts for 6 to 9% fractions on average, however these values are probably underestimated by the used methods. Vertical anomalies related to the first baroclinic mode coincide with regions where long RWs answers for a significant fraction of local variance, while the second mode highlights zones where short RWs and tropical instability waves are reported. Transient phenomena related with the third mode are observed in comparatively small areas on all three oceans. The magnitude of the baroclinic radii\'s variations in response of the transient variability results in significant changes of the theorethical phase speed for long linear RWs, an evidence that nonlinear effects may be important. Lastly, the greater significance of the barotropic mode over proeminent bottom features suggests the existence of energy transfer mechanisms between dynamical modes triggered by the interaction with the bathymetry.

Nonlinear four-wave interactions (Snl) are critical for acquiring realistic spectra needed by operational wave models. High computational demand to calculate these interactions led to an approximation method named the Discrete Interaction Approximation (DIA) to be used broadly in the major operational wave models for a long time. However, the accuracy of the DIA has been controversial since it was first introduced and more precise approximations such as the Two Scale Approximation (TSA in short) are now available. The only issue with the initial TSA`s efficiency is performing an order of a magnitude slower than the DIA in speed. This study questions the exactness of the DIA while trying to increase the competence of the TSA by making improvements on its execution time. Particularly, in this thesis, the main effort is on the local scale term of the TSA since it is the part that consumes the most time while running the code. The findings of this work imply that the TSA can improve its operation speed significantly while maintaining its accuracy with making alterations in the code. By decreasing the number of bands in the local scale it is possible to run the TSA up to 7.5 faster than its initial version.