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## Abstract

Influence functions (IFs) of a linear barotropic vorticity equation model are computed in order to determine the regions in which anomalous divergence at upper levels, related to tropical heating anomalies, has the largest impact on certain prominent low-frequency anomalies. The present computation differs from that of Branstator in two aspects: (a) the model includes the effects of the basic-flow divergence and the advection by anomalous divergent wind and (b) the influence functions directly assess the influence of upper-level divergence anomalies rather than sources of vorticity. The IFs are applied to the study of low-frequency tropicalâ€“extratropical interactions at the interannual (ENSO) and intraseasonal (30â€“60-day oscillation) timescales. The origin of well-known teleconnection patterns is explored through the identification of comma influence regions in the Tropics and subtropics for their main action centers. The subtropical west and central-east Pacific, north of the equator, is an important source region for the Pacific North America (PNA) pattern, and the South Atlantic convergence zone (SACZ) for the Eurasian and North Atlantic patterns. The IFs and the barotropic model results, as well as the evolution of the anomalous OLR fields associated with the 30â€“60-day oscillation, indicate the existence of a chain of connections. This chain constitutes a link between the South Pacific convergence zone (SPCZ) and the SACZ, as well as the control of the SACZ over the Atlantic and Eurasian pattern, which possibly connects back to the western Pacific. This connection can also occur at international timescales during ENSO events. An explanation of the relative insensitivity of the PNA pattern to the longitudinal position of the Pacific equatorial convection anomaly, reported by Geisler et al., is proposed.

## Abstract

Influence functions (IFs) of a linear barotropic vorticity equation model are computed in order to determine the regions in which anomalous divergence at upper levels, related to tropical heating anomalies, has the largest impact on certain prominent low-frequency anomalies. The present computation differs from that of Branstator in two aspects: (a) the model includes the effects of the basic-flow divergence and the advection by anomalous divergent wind and (b) the influence functions directly assess the influence of upper-level divergence anomalies rather than sources of vorticity. The IFs are applied to the study of low-frequency tropicalâ€“extratropical interactions at the interannual (ENSO) and intraseasonal (30â€“60-day oscillation) timescales. The origin of well-known teleconnection patterns is explored through the identification of comma influence regions in the Tropics and subtropics for their main action centers. The subtropical west and central-east Pacific, north of the equator, is an important source region for the Pacific North America (PNA) pattern, and the South Atlantic convergence zone (SACZ) for the Eurasian and North Atlantic patterns. The IFs and the barotropic model results, as well as the evolution of the anomalous OLR fields associated with the 30â€“60-day oscillation, indicate the existence of a chain of connections. This chain constitutes a link between the South Pacific convergence zone (SPCZ) and the SACZ, as well as the control of the SACZ over the Atlantic and Eurasian pattern, which possibly connects back to the western Pacific. This connection can also occur at international timescales during ENSO events. An explanation of the relative insensitivity of the PNA pattern to the longitudinal position of the Pacific equatorial convection anomaly, reported by Geisler et al., is proposed.

## Abstract

The response of planetary waves to stationary tropical heating in a stratified global atmosphere linearized with respect to a basic zonal mean flow is investigated. The basic zonal wind has meridional and vertical shear. The basic equations are solved by using the method of three-dimensional normal-mode expansion. Forced solutions to a prescribed tropospheric equatorial heating distribution with a specific wavenumber in longitude are examined.

Without the basic zonal flow, the internal vertical modes whose equivalent depths are on the order of a few hundred meters are favorably excited, but the response of the external mode (â€śbarotropicâ€ť mode) is relatively small. With the inclusion of a zonal flow, the vertical shear of the zonal wind permits the coupling of the external mode with the internal vertical modes. As a result of the coupling, a significant response occurs in the external mode due to the excitation of the â€śbaroclinicâ€ť internal modes by tropical heating. The meridional structures of internal vertical modes are equatorially trapped and their intensities are less affected by the basic zonal flow. Since the meridional structures of the external mode is global, a significant response of the external mode to tropospheric tropical heating is no longer confined to the tropics. The direction of the basic zonal flow and its meridional shear have a profound influence on the intensity of planetary waves in the mid- to higher latitudes generated by stationary tropical heating. The present findings may fill in a missing link in the dynamical theory of atmospheric teleconnections.

## Abstract

The response of planetary waves to stationary tropical heating in a stratified global atmosphere linearized with respect to a basic zonal mean flow is investigated. The basic zonal wind has meridional and vertical shear. The basic equations are solved by using the method of three-dimensional normal-mode expansion. Forced solutions to a prescribed tropospheric equatorial heating distribution with a specific wavenumber in longitude are examined.

Without the basic zonal flow, the internal vertical modes whose equivalent depths are on the order of a few hundred meters are favorably excited, but the response of the external mode (â€śbarotropicâ€ť mode) is relatively small. With the inclusion of a zonal flow, the vertical shear of the zonal wind permits the coupling of the external mode with the internal vertical modes. As a result of the coupling, a significant response occurs in the external mode due to the excitation of the â€śbaroclinicâ€ť internal modes by tropical heating. The meridional structures of internal vertical modes are equatorially trapped and their intensities are less affected by the basic zonal flow. Since the meridional structures of the external mode is global, a significant response of the external mode to tropospheric tropical heating is no longer confined to the tropics. The direction of the basic zonal flow and its meridional shear have a profound influence on the intensity of planetary waves in the mid- to higher latitudes generated by stationary tropical heating. The present findings may fill in a missing link in the dynamical theory of atmospheric teleconnections.

## Abstract

The dynamics of convectively coupled equatorial waves (CCEWs) is analyzed in an idealized model of the large-scale atmospheric circulation. The model is composed of a linear rotating shallow-water system with a variable equivalent height, or equivalent gravity wave speed, which varies in space. This model is based on the hypothesis that moist convection acts to remove convective instability, therefore modulating the equivalent height of a shallow-water system. Asymptotic solutions are derived in the case of a small perturbation around a constant coefficient, which is assumed to be a mean moist equivalent height derived from satellite observations. The first-order solutions correspond to the free normal modes of the linear shallow-water system and the second-order flow is derived solving a perturbation eigenvalue problem. The asymptotic solutions are documented in the case of a zonally varying equivalent height and for wavenumbers and frequencies that are consistent with observations of CCEWs. This analysis shows that the dynamics of the secondary divergence and its impact on the full divergence varies mode by mode. For instance, for a negative equivalent height anomaly, which is interpreted as a moister background, the secondary divergence is nearly in phase with the primary divergence in the case of Kelvin wavesâ€”in contrast to mixed Rossbyâ€“gravity waves where the secondary divergence acts to attenuate the primary divergence. While highly idealized, the modeled waves share some features with observations, providing a mechanism for the relationship between CCEWs phase speed, amplitude, and horizontal structure.

## Abstract

The dynamics of convectively coupled equatorial waves (CCEWs) is analyzed in an idealized model of the large-scale atmospheric circulation. The model is composed of a linear rotating shallow-water system with a variable equivalent height, or equivalent gravity wave speed, which varies in space. This model is based on the hypothesis that moist convection acts to remove convective instability, therefore modulating the equivalent height of a shallow-water system. Asymptotic solutions are derived in the case of a small perturbation around a constant coefficient, which is assumed to be a mean moist equivalent height derived from satellite observations. The first-order solutions correspond to the free normal modes of the linear shallow-water system and the second-order flow is derived solving a perturbation eigenvalue problem. The asymptotic solutions are documented in the case of a zonally varying equivalent height and for wavenumbers and frequencies that are consistent with observations of CCEWs. This analysis shows that the dynamics of the secondary divergence and its impact on the full divergence varies mode by mode. For instance, for a negative equivalent height anomaly, which is interpreted as a moister background, the secondary divergence is nearly in phase with the primary divergence in the case of Kelvin wavesâ€”in contrast to mixed Rossbyâ€“gravity waves where the secondary divergence acts to attenuate the primary divergence. While highly idealized, the modeled waves share some features with observations, providing a mechanism for the relationship between CCEWs phase speed, amplitude, and horizontal structure.

## Abstract

We consider the problem of the linear response of a stratified, equatorial, Î˛-plane model atmosphere to specified transient sources of heat and momentum. The method of solution involves transforms in all three spatial coordinates. A finite StĂĽrm-Liouville transform is used in *z*, a Fourier transform in *x*, and a generalized Hermite transform in *y*. The resulting spectral equations can then be solved analytically for a specified forcing. Of particular interest is the case of a Gaussian-shaped heat source centered at latitude *y _{o}
* and with

*e*-folding radius

*a*. The heat source is transient and has time scale 1/Î±. Using the Parceval relation we compute how the forced energy is partitioned between Kelvin, mixed Rossby-gravity, Rossby and gravity modes as a function of

*a*,

*y*, Î±. Model results using a heat source centered at 11Â°S with an

_{o}*e*-folding radius of 750 km and a time scale of about a day indicate that many aspects of the summertime upper tropospheric circulation over South America can be explained by the dispersive properties of Rossby and mixed Rossby-gravity waves. These results also show that the transient heat source excites Kelvin waves which propagate rapidly eastward as a nondispersive wave group. The existence of the Kelvin waves has implications for the initialization of tropical forecast models. By applying a nonlinear normal mode initialization procedure in the middle of a model simulation we investigate how the initialization distorts the subsequent evolution. Much of the distortion is associated with the Kelvin wave response.

## Abstract

We consider the problem of the linear response of a stratified, equatorial, Î˛-plane model atmosphere to specified transient sources of heat and momentum. The method of solution involves transforms in all three spatial coordinates. A finite StĂĽrm-Liouville transform is used in *z*, a Fourier transform in *x*, and a generalized Hermite transform in *y*. The resulting spectral equations can then be solved analytically for a specified forcing. Of particular interest is the case of a Gaussian-shaped heat source centered at latitude *y _{o}
* and with

*e*-folding radius

*a*. The heat source is transient and has time scale 1/Î±. Using the Parceval relation we compute how the forced energy is partitioned between Kelvin, mixed Rossby-gravity, Rossby and gravity modes as a function of

*a*,

*y*, Î±. Model results using a heat source centered at 11Â°S with an

_{o}*e*-folding radius of 750 km and a time scale of about a day indicate that many aspects of the summertime upper tropospheric circulation over South America can be explained by the dispersive properties of Rossby and mixed Rossby-gravity waves. These results also show that the transient heat source excites Kelvin waves which propagate rapidly eastward as a nondispersive wave group. The existence of the Kelvin waves has implications for the initialization of tropical forecast models. By applying a nonlinear normal mode initialization procedure in the middle of a model simulation we investigate how the initialization distorts the subsequent evolution. Much of the distortion is associated with the Kelvin wave response.

## Abstract

One possible explanation for the relatively high signal of the mixed Rossbyâ€“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 *Î˛* 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 Rossbyâ€“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 Rossbyâ€“gravity waves with zonal wavenumbers 4 and 5, which have periods of the order of 4 days. The phase propagation of these unstable mixed modes leads to a periodic energy exchange between the mixed Rossbyâ€“gravity waves and the quasigeostrophic modes (Rossby and ultralong Kelvin modes). This regular nonlinear energy exchange implies a 4-day-cycle vacillation in the solution, which might be linked to the 4â€“6-day local oscillations in the dynamical field data throughout the Amazon region found in observational studies. Besides the importance of quasigeostrophic modes in the excitation of mixed Rossbyâ€“gravity waves, the numerical results also suggest that the predominance of the slow modes is crucial for maintaining the high signal of the unstable mixed modes, since these waves are strongly suppressed by the inclusion of the fast modes in the integration.

## Abstract

One possible explanation for the relatively high signal of the mixed Rossbyâ€“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 *Î˛* 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 Rossbyâ€“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 Rossbyâ€“gravity waves with zonal wavenumbers 4 and 5, which have periods of the order of 4 days. The phase propagation of these unstable mixed modes leads to a periodic energy exchange between the mixed Rossbyâ€“gravity waves and the quasigeostrophic modes (Rossby and ultralong Kelvin modes). This regular nonlinear energy exchange implies a 4-day-cycle vacillation in the solution, which might be linked to the 4â€“6-day local oscillations in the dynamical field data throughout the Amazon region found in observational studies. Besides the importance of quasigeostrophic modes in the excitation of mixed Rossbyâ€“gravity waves, the numerical results also suggest that the predominance of the slow modes is crucial for maintaining the high signal of the unstable mixed modes, since these waves are strongly suppressed by the inclusion of the fast modes in the integration.

## 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 *Î˛*-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*(Îµ)] 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 Rossbyâ€“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*(Îµ) 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 equatorially trapped waves and one barotropic Rossby mode, the spectrum of the thermal forcing is such that only one of the triad components is resonant with the heat source. As a result, to illustrate the role of the diurnal forcing in these interactions in a simplified fashion, two kinds of triads have been analyzed. The first one refers to triads composed of a *k* = 0 first baroclinic geostrophic mode, which is resonant with the stationary component of the diurnal heat source, and two dispersive modes, namely, a mixed Rossbyâ€“gravity wave and a barotropic Rossby mode. The other class corresponds to triads composed of two first baroclinic inertio-gravity waves in which the highest-frequency wave resonates with a transient harmonic of the forcing. The integration of the asymptotic reduced equations for these selected resonant triads shows that the stationary component of the diurnal heat source acts as an â€śacceleratorâ€ť for the energy exchanges between the two dispersive waves through the excitation of the catalyst geostrophic mode. On the other hand, since in the second class of triads the mode that resonates with the forcing is the most energetically active member because of the energy constraints imposed by the triad dynamics, the results show that the convective forcing in this case is responsible for a longer time scale modulation in the resonant interactions, generating a period doubling in the energy exchanges. The results suggest that the diurnal variation of tropical convection might play an important role in generating low-frequency fluctuations in the atmospheric circulation through resonant nonlinear interactions.

## 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 *Î˛*-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*(Îµ)] 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 Rossbyâ€“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*(Îµ) 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 equatorially trapped waves and one barotropic Rossby mode, the spectrum of the thermal forcing is such that only one of the triad components is resonant with the heat source. As a result, to illustrate the role of the diurnal forcing in these interactions in a simplified fashion, two kinds of triads have been analyzed. The first one refers to triads composed of a *k* = 0 first baroclinic geostrophic mode, which is resonant with the stationary component of the diurnal heat source, and two dispersive modes, namely, a mixed Rossbyâ€“gravity wave and a barotropic Rossby mode. The other class corresponds to triads composed of two first baroclinic inertio-gravity waves in which the highest-frequency wave resonates with a transient harmonic of the forcing. The integration of the asymptotic reduced equations for these selected resonant triads shows that the stationary component of the diurnal heat source acts as an â€śacceleratorâ€ť for the energy exchanges between the two dispersive waves through the excitation of the catalyst geostrophic mode. On the other hand, since in the second class of triads the mode that resonates with the forcing is the most energetically active member because of the energy constraints imposed by the triad dynamics, the results show that the convective forcing in this case is responsible for a longer time scale modulation in the resonant interactions, generating a period doubling in the energy exchanges. The results suggest that the diurnal variation of tropical convection might play an important role in generating low-frequency fluctuations in the atmospheric circulation through resonant nonlinear interactions.

## Abstract

In the present study a simplified multiscale atmosphereâ€“ocean coupled model for the tropical interactions among synoptic, intraseasonal, and interannual scales is developed. Two nonlinear equatorial *Î˛*-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 airâ€“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 synopticâ€“intraseasonalâ€“interannualâ€“El NiĂ±o (SInEN) regime. In this distinguished balanced regime, the synoptic scale is the fastest atmospheric time scale, the intraseasonal scale is the intermediate airâ€“sea coupling time scale (common to both fluid flows), and El NiĂ±o 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 solutions of the reduced SInEN equations for a single atmosphereâ€“ocean resonant triad illustrate the potential of the model to understand slow-frequency variability in the tropics. The resonant nonlinear wind stress allows a mechanism for the synoptic-scale atmospheric waves to force intraseasonal variability in the ocean. The intraseasonal ocean temperature anomaly coupled with the atmosphere through evaporation forces synoptic and intraseasonal atmospheric variability. The waveâ€“convection coupling provides another source for higher-order atmospheric variability. Nonlinear interactions of intraseasonal ocean perturbations may also force interannual oceanic variability. The constrains that determine the establishment of the atmosphereâ€“ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El NiĂ±o).

## Abstract

In the present study a simplified multiscale atmosphereâ€“ocean coupled model for the tropical interactions among synoptic, intraseasonal, and interannual scales is developed. Two nonlinear equatorial *Î˛*-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 airâ€“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 synopticâ€“intraseasonalâ€“interannualâ€“El NiĂ±o (SInEN) regime. In this distinguished balanced regime, the synoptic scale is the fastest atmospheric time scale, the intraseasonal scale is the intermediate airâ€“sea coupling time scale (common to both fluid flows), and El NiĂ±o 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 solutions of the reduced SInEN equations for a single atmosphereâ€“ocean resonant triad illustrate the potential of the model to understand slow-frequency variability in the tropics. The resonant nonlinear wind stress allows a mechanism for the synoptic-scale atmospheric waves to force intraseasonal variability in the ocean. The intraseasonal ocean temperature anomaly coupled with the atmosphere through evaporation forces synoptic and intraseasonal atmospheric variability. The waveâ€“convection coupling provides another source for higher-order atmospheric variability. Nonlinear interactions of intraseasonal ocean perturbations may also force interannual oceanic variability. The constrains that determine the establishment of the atmosphereâ€“ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El NiĂ±o).

## Abstract

A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high.

When the basic state possesses positive relative vorticity in an inner region, the energy partition can be substantially modified, and cloud-cluster-scale heating can become considerably more efficient.

The energy partition results have important implications for the lateral boundary condition used in tropical cyclone models. Faced with the fact that a perfect non-reflecting condition is possible but impractical to implement, one is forced to use an approximate condition which causes some reflection of gravity-inertia waves and hence some distortion of the geostrophic adjustment process. The distortion can be kept small by the use of a suitable radiation condition.

## Abstract

A linearized system of equations for the atmosphere's first internal mode in the vertical is derived. The system governs small-amplitude, forced, axisymmetric perturbations on a basic-state tangential flow which is independent of height. When the basic flow is at rest, solutions for the transient and final adjusted state are found by the method of Hankel transforms. Two examples are considered, one with an initial top hat potential vorticity and one with an initial Gaussian-type potential vorticity. These two examples, which extend the work of Fischer (1963) and Obukhov (1949), indicate that the energetical efficiency of cloud-cluster-scale heating in producing balanced vortex flow is very low, on the order of a few percent. The vast majority of the energy is simply partitioned to gravity-inertia waves. In contrast the efficiency of cloud-cluster-scale vorticity transport is very high.

When the basic state possesses positive relative vorticity in an inner region, the energy partition can be substantially modified, and cloud-cluster-scale heating can become considerably more efficient.

The energy partition results have important implications for the lateral boundary condition used in tropical cyclone models. Faced with the fact that a perfect non-reflecting condition is possible but impractical to implement, one is forced to use an approximate condition which causes some reflection of gravity-inertia waves and hence some distortion of the geostrophic adjustment process. The distortion can be kept small by the use of a suitable radiation condition.

## Abstract

Weakly nonlinear interactions among equatorial waves have been explored in this paper using the adiabatic version of the equatorial *Î˛*-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 surfaceâ€“gravity waves in deep water and capillaryâ€“gravity waves. By adopting the asymptotic method of multiple time scales, the equatorial Rossby, mixed Rossbyâ€“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*(Îµ) 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.

## Abstract

Weakly nonlinear interactions among equatorial waves have been explored in this paper using the adiabatic version of the equatorial *Î˛*-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 surfaceâ€“gravity waves in deep water and capillaryâ€“gravity waves. By adopting the asymptotic method of multiple time scales, the equatorial Rossby, mixed Rossbyâ€“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*(Îµ) 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.

## Abstract

Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertiaâ€“acoustic (IA) and inertiaâ€“gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.

## Abstract

Here the theory of global nonhydrostatic normal modes has been further developed with the analysis of both linear and weakly nonlinear energetics of inertiaâ€“acoustic (IA) and inertiaâ€“gravity (IG) modes. These energetics are analyzed in the context of a shallow global nonhydrostatic model governing finite-amplitude perturbations around a resting, hydrostatic, and isothermal background state. For the linear case, the energy as a function of the zonal wavenumber of the IA and IG modes is analyzed, and the nonhydrostatic effect of vertical acceleration on the IG waves is highlighted. For the nonlinear energetics analysis, the reduced equations of a single resonant wave triad interaction are obtained by using a pseudoenergy orthogonality relation. Integration of the triad equations for a resonance involving a short harmonic of an IG wave, a planetary-scale IA mode, and a short IA wave mode shows that an IG mode can allow two IA modes to exchange energy in specific resonant triads. These wave interactions can yield significant modulations in the dynamical fields associated with the physical-space solution with periods varying from a daily time scale to almost a month long.