A teoria da perturbação baseada em múltiplas escalas temporais é usada neste trabalho para estudar as interações não lineares entre ondas na atmosfera no contexto das equações primitivas no plano beta-equatorial em coordenadas isobáricas. As ondas equatoriais de Rossby, mista Rossby-gravidade, gravidade-inerciais e Kelvin, com diferentes estruturas verticais, são obtidas como soluções de ordem dominante. A partir da condição de solvabilidade do problema de ordem superior, foi obtido um modelo assintótico reduzido que governa a interação fracamente não linear entre as ondas numa determinada tríade ressonante. A conservação da energia total para a solução de ordem dominante do modelo de equações original implica que a soma dos coeficientes de interação numa tríade ressonante qualquer deve ser nula. Usando um método gráfico, foram encontradas algumas tríades ressonantes envolvendo ondas baroclínicas equatorialmente confinadas e ondas de Rossby barotrópicas possuindo significativa influência em médias e altas latitudes. Duas tríades ressonantes receberam especial atenção devido ao significativo acoplamento entre os modos e à possível relação desses modos com alguns aspectos observados acerca dos fenômenos de baixa freqüência na atmosfera. A primeira tríade é composta por uma onda de Rossby barotrópica possuindo o segundo modo meridional interagindo ressonantemente com uma onda mista de Rossby-gravidade com a estrutura vertical do primeiro modo baroclínico, ambas possuindo o mesmo número de onda zonal, por meio de um modo geostrófico zonalmente simétrico (k = 0) possuindo a mesma estrutura vertical da onda mista e o modo meridional n = 1. A segunda tríade é constituída por uma onda mista de Rossby-gravidade com número de onda-2, uma onda de Kelvin com número de onda-1, ambas associadas ao primeiro modo baroclínico, e uma onda de Rossby barotrópica com número de onda-3 e modo meridional n = 2. A integração das equações reduzidas para essas duas tríades ressonantes mostra que para as amplitudes iniciais dos modos caracterizando magnitudes típicas de anomalias atmosféricas observadas, os modos na primeira tríade, em geral, trocam energia na escala intra-sazonal, enquanto os modos na segunda tríade trocam energia na escala intra-sazonal ou numa escala semi-anual, dependendo da amplitude inicial do modo de Rossby barotrópico. São discutidas as implicações dos resultados para a dinâmica da interação trópicos-extratrópicos na escala de baixa-freqüência do espectro dos movimentos atmosféricos. / The asymptotic perturbation theory based on multiple scales in time is used to investigate the nonlinear interactions among equatorial waves in an equatorial beta-plane adiabatic primitive equation atmospheric model in isobaric coordinates. The equatorial Rossby, mixed Rossby-gravity, Kelvin and inertio-gravity waves, with several vertical structures, are obtained as the leading-order solution. From the solvability condition of the problem, a reduced model governing the weakly nonlinear interaction of the waves in a particular resonant triad was obtained. The total energy conservation of the leading-order solution of the original model equations implies that the sum of the coupling coefficients in any resonant triad must be zero. Using a graphical approach, we determined some resonant triads involving equatorially trapped baroclinic waves and barotropic Rossby waves having large mid-latitude amplitude. Two particular interactions deserve special attention because of their strong coupling and their possible relation to observed features in the atmospheric circulation. One is characterized by a first baroclinic mode structure mixed Rossby-gravity wave interacting with a barotropic Rossby wave with the second gravest meridional mode, both having the same wavenumber, through a zonally symmetric geostrophic mode with the same vertical structure as the Yanai wave and having the n = 1 meridional mode. The other triad is composed of a zonal wavenumber-1 Kelvin wave, a zonal wavenumber-2 mixed Rossby-gravity wave, both with the first baroclinic mode vertical structure, and a barotropic zonal wavenumber-3 Rossby wave having the second gravest meridional mode. The barotropic Rossby waves in these two triad interactions have significant projection onto middle and higher latitudes. The integration of the triad equations for these particular interactions shows that, for the initial mode amplitudes characterizing typical magnitudes of atmospheric flow perturbations, the modes in the first triad usually exchange energy on intraseasonal time-scales, while the modes in the second triad exchange energy on either intraseasonal or semi-annual time-scale, depending on the initial condition. The implications of the results for the dynamics of tropics-extratropics interaction on low-frequency time-scales are discussed.

A teoria da perturbação baseada em múltiplas escalas temporais é usada neste trabalho para estudar as interações não lineares entre ondas na atmosfera no contexto das equações primitivas no plano beta-equatorial em coordenadas isobáricas. As ondas equatoriais de Rossby, mista Rossby-gravidade, gravidade-inerciais e Kelvin, com diferentes estruturas verticais, são obtidas como soluções de ordem dominante. A partir da condição de solvabilidade do problema de ordem superior, foi obtido um modelo assintótico reduzido que governa a interação fracamente não linear entre as ondas numa determinada tríade ressonante. A conservação da energia total para a solução de ordem dominante do modelo de equações original implica que a soma dos coeficientes de interação numa tríade ressonante qualquer deve ser nula. Usando um método gráfico, foram encontradas algumas tríades ressonantes envolvendo ondas baroclínicas equatorialmente confinadas e ondas de Rossby barotrópicas possuindo significativa influência em médias e altas latitudes. Duas tríades ressonantes receberam especial atenção devido ao significativo acoplamento entre os modos e à possível relação desses modos com alguns aspectos observados acerca dos fenômenos de baixa freqüência na atmosfera. A primeira tríade é composta por uma onda de Rossby barotrópica possuindo o segundo modo meridional interagindo ressonantemente com uma onda mista de Rossby-gravidade com a estrutura vertical do primeiro modo baroclínico, ambas possuindo o mesmo número de onda zonal, por meio de um modo geostrófico zonalmente simétrico (k = 0) possuindo a mesma estrutura vertical da onda mista e o modo meridional n = 1. A segunda tríade é constituída por uma onda mista de Rossby-gravidade com número de onda-2, uma onda de Kelvin com número de onda-1, ambas associadas ao primeiro modo baroclínico, e uma onda de Rossby barotrópica com número de onda-3 e modo meridional n = 2. A integração das equações reduzidas para essas duas tríades ressonantes mostra que para as amplitudes iniciais dos modos caracterizando magnitudes típicas de anomalias atmosféricas observadas, os modos na primeira tríade, em geral, trocam energia na escala intra-sazonal, enquanto os modos na segunda tríade trocam energia na escala intra-sazonal ou numa escala semi-anual, dependendo da amplitude inicial do modo de Rossby barotrópico. São discutidas as implicações dos resultados para a dinâmica da interação trópicos-extratrópicos na escala de baixa-freqüência do espectro dos movimentos atmosféricos. / The asymptotic perturbation theory based on multiple scales in time is used to investigate the nonlinear interactions among equatorial waves in an equatorial beta-plane adiabatic primitive equation atmospheric model in isobaric coordinates. The equatorial Rossby, mixed Rossby-gravity, Kelvin and inertio-gravity waves, with several vertical structures, are obtained as the leading-order solution. From the solvability condition of the problem, a reduced model governing the weakly nonlinear interaction of the waves in a particular resonant triad was obtained. The total energy conservation of the leading-order solution of the original model equations implies that the sum of the coupling coefficients in any resonant triad must be zero. Using a graphical approach, we determined some resonant triads involving equatorially trapped baroclinic waves and barotropic Rossby waves having large mid-latitude amplitude. Two particular interactions deserve special attention because of their strong coupling and their possible relation to observed features in the atmospheric circulation. One is characterized by a first baroclinic mode structure mixed Rossby-gravity wave interacting with a barotropic Rossby wave with the second gravest meridional mode, both having the same wavenumber, through a zonally symmetric geostrophic mode with the same vertical structure as the Yanai wave and having the n = 1 meridional mode. The other triad is composed of a zonal wavenumber-1 Kelvin wave, a zonal wavenumber-2 mixed Rossby-gravity wave, both with the first baroclinic mode vertical structure, and a barotropic zonal wavenumber-3 Rossby wave having the second gravest meridional mode. The barotropic Rossby waves in these two triad interactions have significant projection onto middle and higher latitudes. The integration of the triad equations for these particular interactions shows that, for the initial mode amplitudes characterizing typical magnitudes of atmospheric flow perturbations, the modes in the first triad usually exchange energy on intraseasonal time-scales, while the modes in the second triad exchange energy on either intraseasonal or semi-annual time-scale, depending on the initial condition. The implications of the results for the dynamics of tropics-extratropics interaction on low-frequency time-scales are discussed.

This research investigates the realization of parametric amplification in superconducting circuits and structures where nonlinearity is provided by Josephson junction (JJ) elements. We aim to develop a systematic analysis over JJ-based devices toward design of novel traveling-wave Josephson parametric amplifiers (TW-JPA). Chapters of this thesis fall into three categories: lumped JPA, superconducting periodic structures and discrete Josephson transmission lines (DJTL). The unbiased Josephson junction (JJ) is a nonlinear element suitable for parametric amplification through a four-photon process. Two circuit topologies are introduced to capture the unique property of the JJ in order to efficiently mix signal, pump and idler signals for the purpose of signal amplification. Closed-form expressions are derived for gain characteristics, bandwidth determination, noise properties and impedance for this kind of parametric power amplifier. The concept of negative resistance in the gain formulation is observed. A design process is also introduced to find the regimes of operation for gain achievement. Two regimes of operation, oscillation and amplification, are highlighted and distinguished in the result section. Optimization of the circuits to enhance the bandwidth is also carried out. Moving toward TW-JPA, the second part is devoted to modelling the linear wave propagation in a periodic superconducting structure. We derive closed-form equations for dispersion and s-parameters of infinite and finite periodic structures, respectively. Band gap formation is highlighted and its potential applications in the design of passive filters and resonators are discussed. The superconducting structures are fabricated using YBCO and measured, illustrating a good correlation with the numerical results. A novel superconducting Transmission Line (TL), which is periodically loaded by Josephson junctions (JJ) and assisted by open stubs, is proposed as a platform to realize a traveling-wave parametric device. Using the TL model, this structure is modeled by a system of nonlinear partial differential equations (PDE) with a driving source and mixed-boundary conditions at the input and output terminals, respectively. This model successfully emulates parametric and nonlinear microwave propagation when long-wave approximation is applicable. The influence of dispersion to sustain three non-degenerate phased-locked waves through the TL is highlighted. A rigorous and robust Finite Difference Time Domain (FDTD) solver based on the explicit Lax-Wendroff and implicit Crank-Nicolson schemes has been developed to investigate the device responses under various excitations. Linearization of the wave equation, under small-amplitude assumption, dispersion and impedance analysis is performed to explore more aspects of the device for the purpose of efficient design of a traveling-wave parametric amplifier. Knowing all microwave characteristics and identifying different regimes of operation, which include impedance properties, cut-off propagation, dispersive behaviour and shock-wave formation, we exploit perturbation theory accompanied by the method of multiple scale to derive the three nonlinear coupled amplitude equations to describe the parametric interaction. A graphical technique is suggested to find three waves on the dispersion diagram satisfying the phase-matching conditions. Both cases of perfect phase-matching and slight mismatching are addressed in this work. The incorporation of two numerical techniques, spectral method in space and multistep Adams-Bashforth in time domain, is employed to monitor the unilateral gain, superior stability and bandwidth of this structure. Two types of functionality, mixing and amplification, with their requirements are described. These properties make this structure desirable for applications ranging from superconducting optoelectronics to dispersive readout of superconducting qubits where high sensitivity and ultra-low noise operation is required.