## 1. Introduction

Although incoming solar radiation is the main external energy source for the planet, the terrestrial components (atmosphere, hydrosphere, biosphere, and lithosphere) manage the energy input and define both the fast (weather scale) and the slow (climate scale) responses. Moreover, among the terrestrial components of Earth’s system, the atmosphere and ocean (a subcomponent of the hydrosphere) are the leading contributors (Gabites 1950; Fritz 1958). For the weather time scale, there are outstanding variabilities from microscales to the intradiurnal (<24 h), the mesoscale (<2 days), and at synoptic scales (3–7 days). On the other hand, for the climate time scale, vigorous spectral peaks are found for the intraseasonal (30–180 days), from the interannual to El Niño (1.5–7 yr), and the multidecadal (>10 yr) time scales.

Recent studies highlight that the persistent deficiency in modeling the slow climate response can be associated with a misrepresentation of the fast weather-scale variability (Inness 2002; Stevens and Bony 2013; Bony et al. 2015). There are also both observational (e.g., Johnson et al. 1999) and general circulation modeling (e.g., Inness et al. 2001) evidence of the modulation of weather-scale phenomena by climate variability. In addition, because of the large gap between weather and climate time scales, if the weather affects the climate, this connection ought to be through multiscale interaction mechanisms. Thus, a renewed interest in systematic methods to develop simplified multiscale atmospheric models for scale interactions can be noted (e.g., Majda and Klein 2003; Majda and Biello 2003; Biello and Majda 2005; Raupp and Silva Dias 2005, 2006, 2009, 2010). The scale interactions can be responsible for the connection between weather and climate responses and involve either upscale or downscale cascade fluxes (Torrence and Webster 1999; Biello and Majda 2005) or discrete wave interactions (Raupp et al. 2008; Raupp and Silva Dias 2009, 2010). In this context, the wave–wave interactions have been used to explain the generation of low-frequency El Niño variability (e.g., Zebiak 1982; Zebiak and Cane 1987; Suarez and Schopf 1988; Battisti 1988) and the intraseasonal atmospheric variability (Raupp and Silva Dias 2009). In evoking nonlinear wave interaction theory, the rigorous constraints in discrete resonant wave–wave interactions have been used to explain how certain interactions are favored over others (Longuet-Higgins et al. 1967; Domaracki and Loesch 1977; Majda et al. 1999; Holm and Lynch 2002; Raupp and Silva Dias 2009; Ripa 1982, 1983a,b).

The present study applies both multiscale methods and nonlinear wave interaction theory to formulate a model capable of describing scale interactions in a simplified coupled atmosphere–ocean system. The multiscale method adopted here is similar to that adopted by Majda and Klein (2003) for the atmosphere. Thus, our approach can be regarded as an extension of Majda and Klein’s systematic multiscale method by including atmosphere–ocean coupling. Perhaps the most interesting feature of our approach is to retain the eigenvectors and atmospheric and oceanic wave modes as leading-order solutions, which in turn allows these modes to interact through the nonlinearity associated with atmosphere–ocean coupling fluxes.

Two types of nonlinearity are included: the intrinsic advective nonlinearity and the nonlinearity related to the physical processes. The latter includes both the coupling between large-scale waves and moist convection and the heat and momentum fluxes associated with the atmosphere–ocean coupling.

As the focus of the present paper is on the nonlinear interactions in the tropical region, we use the equatorial *β*-plane approximation. Once the model is scaled by suitable multitime and multispace scalings, a perturbation theory is adopted to further simplify the equations and to obtain a reduced and more tractable system describing the interactions involving synoptic, intraseasonal, and interannual time scales in the atmosphere–ocean coupled system.

The paper is organized as follows. In section 2, the basic model equations are introduced and the outlines for the atmosphere–ocean coupling are provided. Suitable scalings to represent the synoptic–intraseasonal–interannual–El Niño (SInEN) regime are reviewed at the end of section 2. In section 3, parameterizations for the mass and momentum fluxes in the SInEN regime are discussed. In section 4, the dynamics and physics are joined to formulate the SInEN model equations. The SInEN model evolves in three time scales, from the equatorial synoptic up to the interannual through the air–sea coupling intraseasonal time scale. The explicit equations for the scale interactions are obtained by asymptotic perturbation methods. In section 5, analytic solutions of the reduced SInEN equations are illustrated for the case of a discrete resonant triad composed of an oceanic Kelvin mode interacting with an atmospheric Rossby mode and an atmospheric Kelvin mode through the parameterized atmosphere–ocean coupling fluxes. The analytic solutions demonstrate the potential of the physical parameterization terms (“physics”) to yield slow-frequency variability by making synoptic and intraseasonal scale waves to exchange energy in interannual time scales. In addition, according to our theoretical model, other effects such as the wave–convection coupling in the atmosphere can also play an important role in the excitation of low-frequency variability. In section 5 we also analyze the spatial patterns of the involved waves and the resulting atmosphere–ocean coupling fluxes. Then we discuss a possible configuration, based on observed features of both the Madden–Julian oscillation (MJO) and El Ninõ–Southern Oscillation (ENSO) phenomena, which makes the interaction associated with the selected triad plausible. In section 6 we summarize the mechanisms that allow the multiscale atmosphere–ocean interactions in the novel nonlinear multiscale model developed here and discuss how this model can be used to explain the slow climate variability.

## 2. A basic coupled atmosphere–ocean model for the equatorial region

### a. Model equations

*o*(

*a*) refers the ocean (atmosphere). The vector

### b. Scalings for the SInEN regime

#### 1) Multispace horizontal scalings

Since time and length scales in the atmosphere are different from those in the ocean, we use multiscale methods (Pedlosky 1987; Majda and Klein 2003; Biello and Majda 2005). For instance, in the tropical region, the zonal extension of the Pacific Ocean (*C* referring to the atmospheric first baroclinic gravity wave speed, and

#### 2) Vertical fluctuation scalings

Consistent with previous models, the oceanic shallow-water equations represent the active layer of the ocean, with mean thickness

#### 3) Multitime scalings

*t*, and slow

*τ*changes. The intermediate scale is the referential time-scale

*ε*:For the SInEN regime considered here,

*U*defined by

Reported values of ^{−1} (e.g., Reed and Recker 1971). Holton (2004), in his scale analysis of large-scale motions in the tropics, adopted 10 m s^{−1} as the atmospheric horizontal velocity scale. Here, we have selected ^{−1} (see Table 1). However, the same qualitative results are obtained if we select the large values for *U* used by Holton (2004). The oceanic gravity wave speed of the first baroclinic mode ^{−1} (Ripa 1982; Battisti 1988), and here the reference value of ^{−1} is used. With these referential length and velocity scales, it follows that the referential time-scale *U*. The model obtained with the above scalings spans from the synoptic to the interannual time scales, with the intraseasonal scale as the coupling time scale. It is noteworthy that, as *C* is the baroclinic gravity wave speed) in the atmosphere,

Typical values of the model parameters.

### c. Scaled model for the SInEN regime

Now, considering the above discussions, the following scalings are utilized:

^{−1},

^{−1}s

^{−1}, and

*ε*can be used as the small parameter in our formal asymptotic development; that is, the reduced model for synoptic–intraseasonal–interannual–El Niño interactions in the coupled atmosphere–ocean system is obtained for

*ε*and thus to guarantee the regularity of the solution (Majda 2002).

It is important to note that in the ocean the quasigeostrophic balance regulates the dynamics at leading order in *ε*, so that the nonlinearity is weak and the oceanic Strouhal number

In the atmosphere, for the case of a purely Rayleigh friction in the momentum forcing *ε*, the atmosphere is rapidly adjusted to the ocean. In such a case, the memory of the system is in the ocean, and the dynamical component of (6) is basically the same as that used by Battisti (1988) and Philander (1999). The advantage of our approach is that (6) allows for linear waves in both the atmosphere and ocean, along with nonlinear effects coupling these modes.

Moreover, for the atmosphere, the scalings [(5)] are consistent with those used by Biello and Majda (2005) to obtain the intraseasonal planetary equatorial synoptic dynamics (IPESD) model. Likewise, the scalings for the ocean are consistent with those used by Battisti (1988), Philander (1999), and Dijkstra (2000).

In the real atmosphere–ocean coupled system, there are indications that the MJO can trigger El Niño (McPhaden 1999). However, not all MJO events trigger an El Niño event, and, consequently, there might exist a nontrivial selection rule. As we shall see later, a possible selection rule that might lead the MJO to excite interannual El Niño variability refers to wave triad resonance associated with the mass and momentum forcings that couple atmosphere and ocean.

## 3. Physical parameterizations for the SInEN model

### a. Physics of the coupling

According to Dijkstra (2000), the sea surface temperature anomalies (SSTA) force changes in the low-level winds through pressure differences directly induced by the temperature gradients or through pressure gradients associated with sensible and latent heat fluxes controlled by the sea surface temperature (SST). As a result, the wind changes result in modifications of the wind stress, which induce changes in the currents and drive further changes in the SST [see also Wang and Weisberg (1994) and Philander (1999)]. Thus, it is necessary to include an equation for the SST to close the model [(1)]. Other processes and some limitations of the parameterizations here used are discussed in section 6.

### b. Momentum flux

#### Momentum flux strength

*ε*at which the momentum forcing must contribute, we first estimate the strength of the momentum flux

^{−2}, and the nondimensional scaled strength

### c. Mass flux

*E*and deep convective precipitation

*P*; that is,With the sign convention adopted in (13), it is assumed that evaporation supplies mass to the atmosphere, whereas precipitation removes mass from it. Although this assumption is adequate for the basinwide zonal scale considered, it can break down for smaller scales—for example, when the effects of the water vapor on the density must be considered.

#### 1) Evaporation

*p*is a referential pressure, and

*T*is split into basic-state temperature

*E*contains a nonlinear term yielding a coupling between temperature and wind.

#### 2) Thermodynamic equation

*r*, radiative forcing

*S*, and horizontal and vertical advection is given bySimplified versions of (19) can be obtained for the case of weak horizontal advection (

#### 3) Evaporation strength

^{−2}, or in meters per secondThe dimensionless evaporation strength using

*E*(m s

^{−1}) results in

#### 4) Precipitation

*P*is given bywhere

#### 5) Precipitation strength

In the tropics, several hierarchies of the organization of clouds and precipitation are found (Nakazawa 1988). In general, *P* is confined to a region whose length scale *P* represents planetary-scale precipitation as in the intertropical convergence zone (ITCZ) or the MJO envelope (Nakazawa 1988). Smaller hierarchies of clouds lead to smaller values of

#### 6) Mass flux strength

*E*and

*P*and its strength is determined by

## 4. Multiscale SInEN model

*τ*indicating synoptic, intraseasonal, and interannual time scales, respectively. To ensure the uniform validity of the expansion [(34)], the solvability condition imposes that “If there is any growth of the highest order terms, the growth must be slower than the linear growth” (Kevorkian and Cole 1996); that is,Physically, (35) means that even after a long period

The *ε* powers in parentheses to the right of their correspondent equations in (38) indicate the order in which the equations are balanced.

The leading-order perturbations of each subsystem are governed by the so-called linear equatorial long-wave equations, whose eigenvectors are the anisotropic nondispersive Kelvin and Rossby waves (e.g., Gill 1980; Schubert et al. 2009; Ramírez et al. 2011b). Consequently, the solvability condition [(35)] applied in (38) implies that the source terms for the

## 5. Integration of the multiscale SInEN model: The case of a single resonant triad interaction

As discussed above, in the multiscale SInEN model [(38)] wave modes are allowed in both the atmosphere and ocean subsystems, and the leading-order solution corresponds to the anisotropic nondispersive Kelvin and long Rossby waves. Furthermore, nonlinear mode interactions are due to either the advective nonlinearity or the parameterized mass and momentum forcings. The former allows for interactions of waves that belong to the same subsystem, whereas the latter allows for across-subsystem mode interactions (see Fig. 1). In this section, we deal with the nonlinear wave interaction of atmospheric and oceanic waves through wind stress and evaporation and how these interactions can connect the atmosphere and ocean from synoptic to interannual time scales through the intraseasonal time scale.

### a. Solvability condition and resonant triad equations

*ε*in the nonlinear interaction coefficient of the oceanic mode reflects the slower time scale associated to the oceanic mode amplitude compared with the atmospheric waves. The inner product

The nonlinear interactions through physical parameterizations in (47) allow for the coupling of waves that belong to different fluid flows (subsystems) and have distinctive temporal and spatial scales. Precisely, the distinctive nature of the atmospheric and oceanic fluid flows prevents a direct resonant atmosphere–ocean coupling through advection. Therefore, (47) represents a simplified mechanism by which the resonant interaction illustrated in Fig. 2 might occur.

### b. Parametric interactions

As *ε*; both of them in combination with *ε* for *ε* (e.g.,

In the parametric oscillatory regime of (51) (i.e.,

To further test the sensitivity of the system to the initial condition (i.e., the energy level at which the triad was established), Figs. 5 and 6 depict the integration of (51) for two different values of the initial oceanic wave amplitude:

### c. Resonant triad interactions

We now analyze the dynamics of the same resonant triad discussed in the previous subsection, but considering a nonlinear wind stress parameterization (*ε* small but finite. Although the coupling coefficient

A representative example of the solution to (54) for the case of the full resonant triad interactions is illustrated in Fig. 7. The initial amplitudes for the modes 2 and 3 were chosen to fall into the

The resonant triad in Fig. 7 has a behavior typical of conservative resonant interactions through advective nonlinearity; that is, the highest absolute frequency mode of the triad (mode 1—atmospheric Kelvin wave) always grows or declines at the expense of the other modes. Furthermore, the lowest absolute frequency mode of the triad (mode 3—oceanic Kelvin mode) exhibits the weakest energy modulation. However, a difference between the present triad and a triad associated with advective nonlinearities is that the total energy of the present triad is no longer conserved. As a consequence, the total energy is also strongly modulated in the slow time scale. In addition, even though the atmospheric Kelvin wave is initiated with zero energy, this mode attains a much higher energy level than the remaining triad components and, therefore, is responsible for almost all the energy of the system during the periods of its maximum energy level. These aspects are confirmed by the numerical integration of the system [(47)], which agrees with the analytic solution [(54)] in all the correspondent parameter regimes.

In Fig. 8 the low-level patterns of the dynamical fields associated with the atmospheric branch of the resonant triad—that is, the atmospheric Kelvin wave (mode 1) and the

The zonal wind stress produced by the interaction between modes 1 and 2 is displayed in Fig. 9. The strong wind stress over the western Pacific Ocean may represent the westerly wind burst that precedes a typical El Niño development [see McPhaden (1999)]. Over the eastern Pacific Ocean, the westerly wind stress is relatively weak; however, it can contribute to weaken the climatological trade winds and to relax the pressure gradient that maintains the warm waters to the west in the Pacific Ocean.

Furthermore, the coupling of the atmospheric Kelvin–Rossby waves and the oceanic Kelvin wave yields the modulated evaporation pattern depicted in Fig. 9. The evaporation envelope is about 6000 km of zonal extension (over the western Pacific), whereas its internal spatial structure is of synoptic or meso-*γ* spatial scale (≈2000 km). The up and down synoptic-scale pattern of the evaporation may allow eastward propagation of the synoptic-scale convective anomalies that are part of the MJO envelope [see Zhang (2005)]. Thus, the spatial patterns of the waves that constitute the resonant triad analyzed here are consistent with mechanisms that may lead to the interaction between synoptic, intraseasonal, and interannual space–time scales.

Therefore, the results presented here for the special case of a single resonant triad interacting through parameterized atmosphere–ocean fluxes demonstrate the potential of the multiscale SInEN model to connect the atmosphere and ocean from synoptic to interannual time scales through the intraseasonal time scale.

## 6. Summary and final remarks

*β*-plane shallow-water equations: one representing the ocean and the other the atmosphere. The reduced multiscale SInEN model is obtained as a distinguished limit of the original coupled shallow-water equations. This limit represents a balanced regime in the atmosphere–ocean system, where the atmospheric Froude number, the oceanic Froude number, the nondimensional strength of atmospheric height and oceanic thickness fluctuations, as well as the ratio between meridional and zonal length scales for both atmosphere and ocean are all small parameters and on the same order of magnitude; that is,This balance assumption is required for the mathematical consistence of the limiting dynamics and is physically coherent with the typical strengths for winds, currents, and thermal anomalies associated with the scales centered around the intraseasonal variability. The selected equations [(1)] are compatible with the commonly adopted framework of applying shallow-water equations to describe the first baroclinic mode of either the troposphere or the ocean active layer.

To bring about the SInEN regime, the mass and momentum forcings for the atmosphere and ocean are also expanded in terms of the small nondimensional parameter of the system. The forcing strengths have been estimated in the context of the commonly held physical parameterizations for air–sea mass and momentum fluxes and deep convection in the atmosphere. For instance, the momentum forcing is represented through atmospheric wind stress, whereas the mass forcing is represented as the difference between evaporation *E* and deep convective precipitation *P*. In turn, evaporation is formulated according to the wind-induced surface heat exchange (WISHE) mechanism, while precipitation is formulated according to the wave–conditional instability of the second kind (CISK) hypothesis, where *P* is proportional to lower-troposphere moisture convergence.

Although the flux formulation is recognized to be rather simplistic (Dijkstra 2000; Philander 1999; Hirst and Lau 1990; Battisti 1988), some other drawbacks can be discussed. For example, in the SInEN model the atmosphere and ocean are not fully thermally coupled, since the impact of heat fluxes does not affect back the ocean thermodynamics and currents. The radiation–SST feedback and the evaporation feedback due to changing latent heat flux might also be considered. Further, the ocean thermohaline dynamics does not fully affect the ocean dynamics since

The scalings used to obtain the SInEN model imply a referential intraseasonal time scale connected to the fast equatorial synoptic and slow interannual time scales through the nondimensional parameter *ε*. Consequently, to obtain solutions of the SInEN equations, a perturbation theory with multiple time scales has been adopted, with the atmospheric variables being assumed to evolve on the fastest two time scales (synoptic and the referential intraseasonal) and the oceanic variables being assumed to evolve on the slowest two time scales (the referential intraseasonal and the interannual). The leading-order perturbations of each subsystem in the SInEN model are governed by the so-called equatorial *β*-plane linear long-wave equations, whose eigenvectors are the anisotropic nondispersive Kelvin and Rossby waves.

These wave packets may undergo their own self-mode interactions through the intrinsic advective nonlinearity, and the parameterized mass and momentum fluxes can yield interactions between atmospheric and oceanic wave packets through resonant triads of specific Fourier modes. Therefore, our model might accommodate several dynamical mechanisms contained in other theoretical models—namely, the role of the intrinsic advective nonlinearity in the generation of low-frequency variability (Ripa 1982, 1983a,b), the role of heating forcings in generating low-frequency variability by atmospheric-only wave interactions (Raupp and Silva Dias 2009, 2010), the role of oceanic wave interactions with a diagnostic atmosphere in the excitation of El Niño (Battisti 1988), the role of interactions of linear modes through thermodynamics in the generation of low-frequency variability in simple linear coupled ocean–atmosphere models (Hirst 1986; Hirst and Lau 1990), and the excitation of intraseasonal variability through atmospheric equatorial synoptic-scale turbulence (Biello and Majda 2005).

To illustrate the potential of the SInEn model to connect synoptic, intraseasonal, and interannual time scales in the atmosphere–ocean system, we have considered the special case of a single resonant triad involving an oceanic Kelvin wave, and an atmospheric Kelvin wave, and an atmospheric

Furthermore, the low-level spatial patterns of the triad members reinforce the potential of the resonant wave interaction mechanism through atmosphere–ocean coupling fluxes to connect synoptic, intraseasonal, and interannual variabilities. In fact, for the atmospheric branch of the resonant triad, the low-level winds over the Pacific Ocean due to Kelvin wave activity are superimposed on the pattern produced by the

The next step to investigate the potential of the SInEn model in a more realistic scenario should be to restore the advective nonlinearities of each subsystem. The advection may couple each of the individual Fourier harmonics of the resonant triad analyzed here with all the wavenumbers of their corresponding wave packets. In fact, the model is weakly nonlinear in the ocean but is fully nonlinear in the atmosphere.

Moreover, in the atmosphere, prognostic equations for the moisture field and interaction between different vertical modes should be considered in order to properly represent the cloud-radiation–SST feedback, as well as the intensification of the MJO through vertical tilting of the heating (a crucial aspect in multiscale models for the MJO; e.g., Biello and Majda 2005; Thual and Majda 2016; references therein). The variability of the solar radiative forcing may act as another forcing mechanism to enhance low-frequency atmospheric variability. Recently, by including linearized versions of some of the physical mechanisms described above, the reproduction of certain observed features of the MJO has been achieved (Majda and Stechmann 2009; Liu and Wang 2013). In principle, we believe that the theory constructed here can be generalized to include some of those more complex parameterizations described above, as far as the linear eigenvectors may still constitute the leading-order solutions in the new scenarios.

Thus, despite the aforementioned limitations, the advantage of the SInEn model is that it can be solved analytically, while keeping wave solutions in both the atmosphere and ocean. The SInEn model suggests that the resonant atmosphere–ocean coupling can be a possible mechanism for the generation of low-frequency variability in the climate system. The various mechanisms involved, which determine the conditions for the establishment of the atmosphere–ocean resonant coupling, can be viewed as selection rules for the excitation of intraseasonal variability like in the MJO or even slower variability like the interannual El Niño variability.

## Acknowledgments

Enver Ramirez wishes to thank the São Paulo Research Foundation (FAPESP) for its support under Grant FAPESP 2006/60488-3 and CAPES IAG/USP, PROEX 0531/2017. The work of Carlos Raupp was supported by FAPESP through Grant 2009/11643-4. Pedro L. Silva Dias received support from the CNPq in Brazil, under two separate grants: 309395/2013-5 “Tropical Atmospheric Dynamics: Theory and Application” and INCT/CNPq National Institute for Science and Technology for Climate Change. This research was also funded by CAPES Project PALEOCENE, IBM under the Open Source Cooperation Agreement B1258534, and FAPESP Grant PACMEDY Project 2015/50686. This paper is dedicated to my family; special mention to my wife Rosio—her support during the long working hours and effort in reading and improving several versions of the manuscript was priceless—and to Diego, Andrea, Guillermo, Susana, Nadia, and Efrain, who were truly a source of motivation and are in the warmest place of my heart.

## APPENDIX

### Integrals Involving Hermite Functions

*ε*is a measure of the meridional confinement. The Hermite functions

*p*nodes and different decays can be expressed by

#### a. Reduction in pairs for same meridional decays

Recursive application of the Busbridge identity [(A3)] and the parity condition [(A2)] allows the computation of the turbulent fluxes yielding slow time-scale modulation.

#### b. Reduction in pairs for different meridional decays

## REFERENCES

Abramowitz, M., and I. A. Stegun, Eds., 1972: Parabolic cylinder function.

*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*, 9th ed. Dover, 685–700.Arfken, G. B., H. J. Weber, and F. E. Harris, 1995:

*Mathematical Methods for Physicists: A Comprehensive Guide.*7th ed. Academic Press, 1220 pp.Battisti, D. S., 1988: Dynamics and thermodynamics of a warming event in a coupled tropical atmosphere–ocean model.

,*J. Atmos. Sci.***45**, 2889–2919, doi:10.1175/1520-0469(1988)045<2889:DATOAW>2.0.CO;2.Biello, J. A., and A. J. Majda, 2005: A new multiscale model for the Madden–Julian oscillation.

,*J. Atmos. Sci.***62**, 1694–1721, doi:10.1175/JAS3455.1.Bony, S., and et al. , 2015: Clouds, circulation and climate sensitivity.

,*Nat. Geosci.***8**, 261–268, doi:10.1038/ngeo2398.Boyd, J., 1980: The nonlinear equatorial Kelvin wave.

,*J. Phys. Oceanogr.***10**, 1–11, doi:10.1175/1520-0485(1980)010<0001:TNEKW>2.0.CO;2.Busbridge, I. W., 1948: Some integrals involving hermite polynomials.

,*J. London Math. Soc.***s1-23**, 135–141, doi:10.1112/jlms/s1-23.2.135.Cane, M. A., and E. S. Sarachik, 1976: Forced baroclinic ocean motions. I. The linear equatorial unbounded case.

,*J. Mar. Res.***34**, 629–665.Craik, A. D. D., 1985:

*Wave Interactions and Fluid Flows*. Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, 334 pp.Dijkstra, H. A., 2000:

*Nonlinear Physical Oceanography: A Dynamical Systems Approach to the Large Scale Ocean Circulation and El Niño.*Atmospheric and Oceanographic Sciences Library, Vol. 22, Springer, 458 pp., doi:10.1007/978-94-015-9450-9.Domaracki, A., and A. Z. Loesch, 1977: Nonlinear interactions among equatorial waves.

,*J. Atmos. Sci.***34**, 486–498, doi:10.1175/1520-0469(1977)034<0486:NIAEW>2.0.CO;2.Donguy, J. R., and G. Meyers, 1987: Observed and modelled topography of the 20°C isotherm in the tropical Pacific.

,*Oceanol. Acta***10**, 41–48.Emanuel, K., 1987: An air-sea interaction model of intraseasonal oscillations in the tropics.

,*J. Atmos. Sci.***44**, 2324–2340, doi:10.1175/1520-0469(1987)044<2324:AASIMO>2.0.CO;2.Ferreira, R., W. Schubert, and J. Hack, 1996: Dynamical aspects of twin tropical cyclones associated with the Madden–Julian oscillation.

,*J. Atmos. Sci.***53**, 929–945, doi:10.1175/1520-0469(1996)053<0929:DAOTTC>2.0.CO;2.Fritz, S., 1958: Seasonal heat storage in the ocean and heating of the atmosphere.

,*Arch. Meteor. Geophys. Bioklimatol.***10A**, 291–300, doi:10.1007/BF02247144.Gabites, J. F., 1950: Seasonal variations in the atmospheric heat balance. Sc.D. thesis, Massachusetts Institute of Technology, 192 pp.

Gill, A. E., 1980: Some simple solutions for heat-induced tropical circulation.

,*Quart. J. Roy. Meteor. Soc.***106**, 447–462, doi:10.1002/qj.49710644905.Gill, A. E., 1982:

*Atmosphere–Ocean Dynamics.*Academic Press, 662 pp.Gill, A. E., and P. J. Phlips, 1986: Nonlinear effects on heat-induced circulation of the tropical atmosphere.

,*Quart. J. Roy. Meteor. Soc.***112**, 69–91, doi:10.1002/qj.49711247105.Hirst, A. C., 1986: Unstable and damped equatorial modes in simple coupled ocean-atmosphere models.

,*J. Atmos. Sci.***43**, 606–630, doi:10.1175/1520-0469(1986)043<0606:UADEMI>2.0.CO;2.Hirst, A. C., and K.-M. Lau, 1990: Intraseasonal and interannual oscillations in coupled ocean-atmosphere models.

,*J. Climate***3**, 713–725, doi:10.1175/1520-0442(1990)003<0713:IAIOIC>2.0.CO;2.Holm, D. D., and P. Lynch, 2002: Stepwise precession of the resonant swinging spring.

,*SIAM J. Appl. Dyn. Syst.***1**, 44–64, doi:10.1137/S1111111101388571.Holton, J. R., 2004:

*An Introduction to Dynamic Meteorology*. 4th ed. Academic Press, 535 pp.Inness, P. M., 2002: Representation of the Madden-Julian oscillation in general circulation models. Ph.D. thesis, University of Reading, 150 pp. [Available online at http://www.met.rdg.ac.uk/~pete/petes_thesis.pdf.]

Inness, P. M., J. M. Slingo, S. J. Woolnough, R. B. Neale, and V. D. Pope, 2001: Organization of tropical convection in a GCM with varying vertical resolution: Implications for the simulation of the Madden-Julian Oscillation.

,*Climate Dyn.***17**, 777–793, doi:10.1007/s003820000148.Johnson, R., T. Rickenbach, S. Rutledge, P. Ciesielski, and W. Schubert, 1999: Trimodal characteristics of tropical convection.

,*J. Climate***12**, 2397–2418, doi:10.1175/1520-0442(1999)012<2397:TCOTC>2.0.CO;2.Kevorkian, J. K., and J. D. Cole, 1996:

*Multiple Scale and Singular Perturbation Methods.*Springer-Verlag, 634 pp.Klein, R., and A. Majda, 2006: Systematic multiscale models for deep convection on mesoscales.

,*Theor. Comput. Fluid Dyn.***20**, 525–551, doi:10.1007/s00162-006-0027-9.Krishnamurti, T. N., H. S. Bedi, and V. M. Hardiker, 1998: Physical processes.

*An Introduction to Global Spectral Modeling*, Oxford University Press, 138–182.Liu, F., and B. Wang, 2013: An air–sea coupled skeleton model for the Madden–Julian oscillation.

,*J. Atmos. Sci.***70**, 3147–3156, doi:10.1175/JAS-D-12-0348.1.Longuet-Higgins, M. S., A. E. Gill, and K. Kenyon, 1967: Resonant interactions between planetary waves.

,*Proc. Roy. Soc. London***299A**, 120–144, doi:10.1098/rspa.1967.0126.Lord, R. D., 1948: Some integrals involving hermite polynomials.

,*J. London Math. Soc.***24**, 101–112.Lynch, P., 2003: Supplement to resonant Rossby wave triads and the swinging spring.

,*Bull. Amer. Meteor. Soc.***84**, 616, doi:10.1175/BAMS-84-5-Lynch.Majda, A., 2002:

*Introduction to P.D.E.’s and Waves for the Atmosphere and Ocean.*Courant Lecture Notes, Vol. 9, American Mathematical Society, 234 pp.Majda, A., and M. Shefter, 2001a: Models for stratiform instability and convectively coupled waves.

,*J. Atmos. Sci.***58**, 1567–1584, doi:10.1175/1520-0469(2001)058<1567:MFSIAC>2.0.CO;2.Majda, A., and M. Shefter, 2001b: Waves and instabilities for model tropical convective parametrizations.

,*J. Atmos. Sci.***58**, 896–914, doi:10.1175/1520-0469(2001)058<0896:WAIFMT>2.0.CO;2.Majda, A., and J. Biello, 2003: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves.

,*J. Atmos. Sci.***60**, 1809–1821, doi:10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.Majda, A., and R. Klein, 2003: Systematic multiscale models for the Tropics.

,*J. Atmos. Sci.***60**, 393–408, doi:10.1175/1520-0469(2003)060<0393:SMMFTT>2.0.CO;2.Majda, A., and S. Stechmann, 2009: The skeleton of tropical intraseasonal oscillations.

,*Proc. Natl. Acad. Sci. USA***106**, 8417–8422, doi:10.1073/pnas.0903367106.Majda, A., R. R. Rosales, E. G. Tabak, and C. V. Turner, 1999: Interaction of large-scale equatorial waves and dispersion of Kelvin waves through topographic resonances.

,*J. Atmos. Sci.***56**, 4118–4133, doi:10.1175/1520-0469(1999)056<4118:IOLSEW>2.0.CO;2.McPhaden, M. J., 1999: Genesis and evolution of the 1997-98 El Niño.

,*Science***283**, 950–954, doi:10.1126/science.283.5404.950.Menzaque, F., R. Rosales, E. Tabak, and C. Turner, 2001: The forced inviscid Burgers equation as a model for nonlinear interactions among dispersive waves.

,*Contemp. Math.***283**, 51–82, doi:10.1090/conm/283/04714.Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the western Pacific.

,*J. Meteor. Soc. Japan***66**, 823–839, doi:10.2151/jmsj1965.66.6_823.Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—Formulation.

,*J. Atmos. Sci.***57**, 1741–1766, doi:10.1175/1520-0469(2000)057<1741:AQETCM>2.0.CO;2.Pedlosky, J., 1987:

*Geophysical Fluid Dynamics*. 2nd ed. Springer-Verlag, 710 pp., doi:10.1007/978-1-4612-4650-3.Philander, S. G., 1999:

*El Niño, La Niña, and the Southern Oscillation*. International Geophysics Series, Vol. 46, Academic Press, 293 pp.Ramírez, E., P. L. Silva Dias, and C. Raupp, 2011a: Asymptotic approach for the nonlinear equatorial long wave interactions.

,*J. Phys.: Conf. Ser.***285**, 012020, doi:10.1088/1742-6596/285/1/012020.Ramírez, E., P. L. Silva Dias, C. Raupp, and J. P. Bonatti, 2011b: The family of anisotropically scaled equatorial waves.

,*J. Adv. Model. Earth Syst.***3**, M12002, doi:10.1029/2011MS000078.Raupp, C. F. M., and P. L. Silva Dias, 2005: Excitation mechanism of mixed Rossby–gravity waves in the equatorial atmosphere: Role of the nonlinear interactions among equatorial waves.

,*J. Atmos. Sci.***62**, 1446–1462, doi:10.1175/JAS3412.1.Raupp, C. F. M., and P. L. Silva Dias, 2006: Dynamics of resonantly interacting equatorial waves.

,*Tellus***58A**, 263–279, doi:10.1111/j.1600-0870.2006.00151.x.Raupp, C. F. M., and P. L. Silva Dias, 2009: Resonant wave interactions in the presence of a diurnally varying heat source.

,*J. Atmos. Sci.***66**, 3165–3183, doi:10.1175/2009JAS2899.1.Raupp, C. F. M., and P. L. Silva Dias, 2010: Interaction of equatorial waves through resonance with the diurnal cycle of tropical heating.

,*Tellus***62A**, 706–718, doi:10.1111/j.1600-0870.2010.00463.x.Raupp, C. F. M., P. L. Silva Dias, P. Milewski, and E. Tabak, 2008: Resonant wave interactions in the equatorial waveguide.

,*J. Atmos. Sci.***65**, 3398–3418, doi:10.1175/2008JAS2387.1.Reed, R. J., and E. E. Recker, 1971: Structure and properties of synoptic-scale wave disturbances in the equatorial western Pacific.

,*J. Atmos. Sci.***28**, 1117–1133, doi:10.1175/1520-0469(1971)028<1117:SAPOSS>2.0.CO;2.Ripa, P., 1982: Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial

*β*plane.,*J. Phys. Oceanogr.***12**, 97–111, doi:10.1175/1520-0485(1982)012<0097:NWIIAO>2.0.CO;2.Ripa, P., 1983a: Weak interactions of equatorial waves in a one-layer model. Part I: General properties.

,*J. Phys. Oceanogr.***13**, 1208–1226, doi:10.1175/1520-0485(1983)013<1208:WIOEWI>2.0.CO;2.Ripa, P., 1983b: Weak interactions of equatorial waves in a one-layer model. Part II: Applications.

,*J. Phys. Oceanogr.***13**, 1227–1240, doi:10.1175/1520-0485(1983)013<1227:WIOEWI>2.0.CO;2.Rogers, R. R., 1976: Water vapor and its thermodynamic effects.

*A Short Course in Cloud Physics*, Pergamon Press, 15–26.Schubert, W. H., L. G. Silvers, M. T. Masarik, and A. O. Gonzalez, 2009: A filtered model of tropical wave motions.

,*J. Adv. Model. Earth Syst.***1**(3), doi:10.3894/JAMES.2009.1.3.Stevens, B., and S. Bony, 2013: What are climate models missing.

,*Science***340**, doi:10.1126/science.1237554.Suarez, M. J., and P. S. Schopf, 1988: A delayed action oscillator for ENSO.

,*J. Atmos. Sci.***45**, 3283–3287, doi:10.1175/1520-0469(1988)045<3283:ADAOFE>2.0.CO;2.Thual, S., and A. J. Majda, 2016: A skeleton model for the MJO with refined vertical structure.

,*Climate Dyn.***46**, 2773–2786, doi:10.1007/s00382-015-2731-x.Torrence, C., and P. J. Webster, 1999: Interdecadal changes in the ENSO–monsoon system.

,*J. Climate***12**, 2679–2690, doi:10.1175/1520-0442(1999)012<2679:ICITEM>2.0.CO;2.Wang, C., and R. Weisberg, 1994: Equatorial trapped waves of a coupled ocean–atmosphere system.

,*J. Phys. Oceanogr.***24**, 1978–1998, doi:10.1175/1520-0485(1994)024<1978:ETWOAC>2.0.CO;2.Yano, J.-I., J. C. McWilliams, M. W. Moncrieff, and K. A. Emanuel, 1995: Hierarchical tropical cloud systems in an analog shallow-water model.

,*J. Atmos. Sci.***52**, 1723–1742, doi:10.1175/1520-0469(1995)052<1723:HTCSIA>2.0.CO;2.Zdunkowski, C., and A. Bott, 2004:

*Dynamics of the Atmosphere: A Course in Theoretical Meteorology.*Cambridge University Press, 268 pp.Zebiak, S., 1982: A simple atmospheric model of relevance to El Niño.

,*J. Atmos. Sci.***39**, 2017–2027, doi:10.1175/1520-0469(1982)039<2017:ASAMOR>2.0.CO;2.Zebiak, S., and M. Cane, 1987: A model El Niño–Southern Oscillation.

,*Mon. Wea. Rev.***115**, 2262–2278, doi:10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2.Zhang, C., 2005: Madden-Julian Oscillation.

,*Rev. Geophys.***43**, RG2003, doi:10.1029/2004RG000158.