Carlos F. M. Raupp

Dynamics of resonantly interacting equatorial waves

In this paper we explore some dynamical features on the non-linear interactions among equatorial waves. The shallowwater equation model with the equatorial β-plane approximation is used for this purpose. The Galerkin method is applied to the governing equations with the basis functions given by the eigensolutions of the linear problem. From the phase space expansion of two particular integrals of motion of the system, quadratic to lowest order, some constraints are obtained which the coupling coefficients must satisfy in order to ensure the invariance of such integrals. From the numerical evaluation of the coupling coefficients, these constraints are used to determine the possible resonant triads among equatorial waves. Numerical integrations of the resonant three-wave problem show that the energy of the waves in a resonant triad evolves periodically in time, with the period and amplitude of the energy oscillations dependent on the magnitude of the initial amplitudes of the waves and the way in which the initial energy is distributed among the triad components. The high-frequency modes are found to be energetically more active than the low-frequency modes. The latter tend to act as ‘catalytic’ components in a resonant triad. Integrations of the problem of two resonant triads coupled by a single mode point out the importance of gravity waves in the intertriad energy exchanges, suggesting the significance of these modes in the redistribution of energy throughout the atmospheric motion spectrum. The results also show that the intertriad energy exchanges provided by the highest frequency mode of two triads occur in a longer time-scale than the intratriad interactions. Therefore, these results also suggest the importance of the high-frequency modes in the generation of the low-frequency variability (intraseasonal and even longer term) of the atmospheric flow.

Resonant Wave Interactions in the Presence of a Diurnally Varying Heat Source

Abstract Resonant interactions among equatorial waves in the presence of a diurnally varying heat source are studied in the context of the diabatic version of the equatorial β-plane primitive equations for a motionless, hydrostatic, horizontally homogeneous and stably stratified background atmosphere. The heat source is assumed to be periodic in time and of small amplitude [i.e., O(e)] and is prescribed to roughly represent the typical heating associated with deep convection in the tropical atmosphere. In this context, using the asymptotic method of multiple time scales, the free linear Rossby, Kelvin, mixed Rossby–gravity, and inertio-gravity waves, as well as their vertical structures, are obtained as leading-order solutions. These waves are shown to interact resonantly in a triad configuration at the O(e) approximation, and the dynamics of these interactions have been studied in the presence of the forcing. It is shown that for the planetary-scale wave resonant triads composed of two first baroclinic e...

Resonant Wave Interactions in the Equatorial Waveguide

Weakly nonlinear interactions among equatorial waves have been explored in this paper using the adiabatic version of the equatorial � -plane primitive equations in isobaric coordinates. Assuming rigid lid vertical boundary conditions, the conditions imposed at the surface and at the top of the troposphere were expanded in a Taylor series around two isobaric surfaces in an approach similar to that used in the theory of surface–gravity waves in deep water and capillary–gravity waves. By adopting the asymptotic method of multiple time scales, the equatorial Rossby, mixed Rossby–gravity, inertio-gravity, and Kelvin waves, as well as their vertical structures, were obtained as leading-order solutions. These waves were shown to interact resonantly in a triad configuration at the O(�) approximation. The resonant triads whose wave components satisfy a resonance condition for their vertical structures were found to have the most significant interactions, although this condition is not excluding, unlike the resonant conditions for the zonal wavenumbers and meridional modes. Thus, the analysis has focused on such resonant triads. In general, it was found that for these resonant triads satisfying the resonance condition in the vertical direction, the wave with the highest absolute frequency always acts as an energy source (or sink) for the remaining triad components, as usually occurs in several other physical problems in fluid dynamics. In addition, the zonally symmetric geostrophic modes act as catalyst modes for the energy exchanges between two dispersive waves in a resonant triad. The integration of the reduced asymptotic equations for a single resonant triad shows that, for the initial mode amplitudes characterizing realistic magnitudes of atmospheric flow perturbations, the modes in general exchange energy on low-frequency (intraseasonal and/or even longer) time scales, with the interaction period being dependent upon the initial mode amplitudes. Potential future applications of the present theory to the real atmosphere with the inclusion of diabatic forcing, dissipation, and a more realistic background state are also discussed.

Excitation Mechanism of Mixed Rossby–Gravity Waves in the Equatorial Atmosphere: Role of the Nonlinear Interactions among Equatorial Waves

Abstract One possible explanation for the relatively high signal of the mixed Rossby–gravity waves observed in the tropical atmosphere is explored in this paper. This explanation is based on the nonlinear interactions among equatorial waves, and is made by adopting the nonlinear shallow water equations on the equatorial β plane. These equations are solved by a spectral method that uses the eigensolutions of the linear problem as the expansion basis. Numerical simulations are performed with a specified stationary mass source representative of the tropospheric heating associated with the typical convective activity over the Amazon Basin during the austral summer period. The numerical results show that the mixed Rossby–gravity waves are excited by a nonlinear mechanism in which the slow modes excited by the thermal forcing generate a quasigeostrophic basic state that supplies energy especially to the mixed Rossby–gravity waves with zonal wavenumbers 4 and 5, which have periods of the order of 4 days. The pha...

Interaction of equatorial waves through resonance with the diurnal cycle of tropical heating

Abstract In this work, a new theoretical mechanism is presented in which equatorial Rossby and inertio-gravity wave modes may interact with each other through resonance with the diurnal cycle of tropical deep convection.We have adopted the two-layer incompressible equatorial primitive equations forced by a parametric heating that roughly represents deep convection activity in the tropical atmosphere. The heat source was parametrized in the simplest way according to the hypothesis that it is proportional to the lower-troposphere moisture convergence, with the background moisture state functionmimicking the structure of the ITCZ. In this context, we have investigated the possibility of resonant interaction between equatorially trapped Rossby and inertio-gravity modes through the diurnal cycle of the background moisture state function. The reduced dynamics of a single resonant duo shows that when this diurnal variation is considered, a Rossby wave mode can undergo significant amplitude modulations when interacting with an inertio-gravity wave mode, which is not possible in the context of the resonant triad non-linear interaction. Therefore, the results suggest that the diurnal variation of the ITCZ can be a possible dynamical mechanism that leads the Rossby waves to be significantly affected by high frequency modes.

Asymptotic approach for the nonlinear equatorial long wave interactions

In the present work we use an asymptotic approach to obtain the long wave equations. The shallow water equation is put as a function of an external parameter that is a measure of both the spatial scales anisotropy and the fast to slow time ratio. The values given to the external parameters are consistent with those computed using typical values of the perturbations in tropical dynamics. Asymptotically, the model converge toward the long wave model. Thus, it is possible to go toward the long wave approximation through intermediate realizable states. With this approach, the resonant nonlinear wave interactions are studied. To simplify, the reduced dynamics of a single resonant triad is used for some selected equatorial trios. It was verified by both theoretical and numerical results that the nonlinear energy exchange period increases smoothly as we move toward the long wave approach. The magnitude of the energy exchanges is also modified, but in this case depends on the particular triad used and also on the initial energy partition among the triad components. Some implications of the results for the tropical dynamics are disccussed. In particular, we discuss the implications of the results for El Nino and the Madden-Julian in connection with other scales of time and spatial variability.

Multiscale Atmosphere–Ocean Interactions and the Low-Frequency Variability in the Equatorial Region

AbstractIn the present study a simplified multiscale atmosphere–ocean coupled model for the tropical interactions among synoptic, intraseasonal, and interannual scales is developed. Two nonlinear equatorial β-plane shallow-water equations are considered: one for the ocean and the other for the atmosphere. The nonlinear terms are the intrinsic advective nonlinearity and the air–sea coupling fluxes. To mimic the main differences between the fast atmosphere and the slow ocean, suitable anisotropic multispace/multitime scalings are applied, yielding a balanced synoptic–intraseasonal–interannual–El Nino (SInEN) regime. In this distinguished balanced regime, the synoptic scale is the fastest atmospheric time scale, the intraseasonal scale is the intermediate air–sea coupling time scale (common to both fluid flows), and El Nino refers to the slowest interannual ocean time scale. The asymptotic SInEN equations reveal that the slow wave amplitude evolution depends on both types of nonlinearities. Analytic solution...