• 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, 28892919, doi:10.1175/1520-0469(1988)045<2889:DATOAW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Biello, J. A., and A. J. Majda, 2005: A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci., 62, 16941721, doi:10.1175/JAS3455.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bony, S., and et al. , 2015: Clouds, circulation and climate sensitivity. Nat. Geosci., 8, 261268, doi:10.1038/ngeo2398.

  • Boyd, J., 1980: The nonlinear equatorial Kelvin wave. J. Phys. Oceanogr., 10, 111, doi:10.1175/1520-0485(1980)010<0001:TNEKW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Busbridge, I. W., 1948: Some integrals involving hermite polynomials. J. London Math. Soc., s1-23, 135141, 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, 629665.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Export Citation
  • Domaracki, A., and A. Z. Loesch, 1977: Nonlinear interactions among equatorial waves. J. Atmos. Sci., 34, 486498, doi:10.1175/1520-0469(1977)034<0486:NIAEW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Donguy, J. R., and G. Meyers, 1987: Observed and modelled topography of the 20°C isotherm in the tropical Pacific. Oceanol. Acta, 10, 4148.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K., 1987: An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci., 44, 23242340, doi:10.1175/1520-0469(1987)044<2324:AASIMO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ferreira, R., W. Schubert, and J. Hack, 1996: Dynamical aspects of twin tropical cyclones associated with the Madden–Julian oscillation. J. Atmos. Sci., 53, 929945, doi:10.1175/1520-0469(1996)053<0929:DAOTTC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fritz, S., 1958: Seasonal heat storage in the ocean and heating of the atmosphere. Arch. Meteor. Geophys. Bioklimatol., 10A, 291300, doi:10.1007/BF02247144.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 447462, doi:10.1002/qj.49710644905.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 6991, doi:10.1002/qj.49711247105.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., 1986: Unstable and damped equatorial modes in simple coupled ocean-atmosphere models. J. Atmos. Sci., 43, 606630, doi:10.1175/1520-0469(1986)043<0606:UADEMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., and K.-M. Lau, 1990: Intraseasonal and interannual oscillations in coupled ocean-atmosphere models. J. Climate, 3, 713725, doi:10.1175/1520-0442(1990)003<0713:IAIOIC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holm, D. D., and P. Lynch, 2002: Stepwise precession of the resonant swinging spring. SIAM J. Appl. Dyn. Syst., 1, 4464, doi:10.1137/S1111111101388571.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 777793, doi:10.1007/s003820000148.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Johnson, R., T. Rickenbach, S. Rutledge, P. Ciesielski, and W. Schubert, 1999: Trimodal characteristics of tropical convection. J. Climate, 12, 23972418, doi:10.1175/1520-0442(1999)012<2397:TCOTC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kevorkian, J. K., and J. D. Cole, 1996: Multiple Scale and Singular Perturbation Methods. Springer-Verlag, 634 pp.

    • Crossref
    • Export Citation
  • Klein, R., and A. Majda, 2006: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn., 20, 525551, doi:10.1007/s00162-006-0027-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Krishnamurti, T. N., H. S. Bedi, and V. M. Hardiker, 1998: Physical processes. An Introduction to Global Spectral Modeling, Oxford University Press, 138–182.

    • Crossref
    • Export Citation
  • Liu, F., and B. Wang, 2013: An air–sea coupled skeleton model for the Madden–Julian oscillation. J. Atmos. Sci., 70, 31473156, doi:10.1175/JAS-D-12-0348.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., A. E. Gill, and K. Kenyon, 1967: Resonant interactions between planetary waves. Proc. Roy. Soc. London, 299A, 120144, doi:10.1098/rspa.1967.0126.

    • Search Google Scholar
    • Export Citation
  • Lord, R. D., 1948: Some integrals involving hermite polynomials. J. London Math. Soc., 24, 101112.

  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Export Citation
  • Majda, A., and M. Shefter, 2001a: Models for stratiform instability and convectively coupled waves. J. Atmos. Sci., 58, 15671584, doi:10.1175/1520-0469(2001)058<1567:MFSIAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and M. Shefter, 2001b: Waves and instabilities for model tropical convective parametrizations. J. Atmos. Sci., 58, 896914, doi:10.1175/1520-0469(2001)058<0896:WAIFMT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and J. Biello, 2003: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci., 60, 18091821, doi:10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and R. Klein, 2003: Systematic multiscale models for the Tropics. J. Atmos. Sci., 60, 393408, doi:10.1175/1520-0469(2003)060<0393:SMMFTT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and S. Stechmann, 2009: The skeleton of tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA, 106, 84178422, doi:10.1073/pnas.0903367106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 41184133, doi:10.1175/1520-0469(1999)056<4118:IOLSEW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McPhaden, M. J., 1999: Genesis and evolution of the 1997-98 El Niño. Science, 283, 950954, 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, 5182, doi:10.1090/conm/283/04714.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteor. Soc. Japan, 66, 823839, doi:10.2151/jmsj1965.66.6_823.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—Formulation. J. Atmos. Sci., 57, 17411766, doi:10.1175/1520-0469(2000)057<1741:AQETCM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer-Verlag, 710 pp., doi:10.1007/978-1-4612-4650-3.

    • Crossref
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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, 14461462, doi:10.1175/JAS3412.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Raupp, C. F. M., and P. L. Silva Dias, 2006: Dynamics of resonantly interacting equatorial waves. Tellus, 58A, 263279, doi:10.1111/j.1600-0870.2006.00151.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 31653183, doi:10.1175/2009JAS2899.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 706718, doi:10.1111/j.1600-0870.2010.00463.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 33983418, doi:10.1175/2008JAS2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 11171133, doi:10.1175/1520-0469(1971)028<1117:SAPOSS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1982: Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial β plane. J. Phys. Oceanogr., 12, 97111, doi:10.1175/1520-0485(1982)012<0097:NWIIAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1983a: Weak interactions of equatorial waves in a one-layer model. Part I: General properties. J. Phys. Oceanogr., 13, 12081226, doi:10.1175/1520-0485(1983)013<1208:WIOEWI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1983b: Weak interactions of equatorial waves in a one-layer model. Part II: Applications. J. Phys. Oceanogr., 13, 12271240, doi:10.1175/1520-0485(1983)013<1227:WIOEWI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. R., 1976: Water vapor and its thermodynamic effects. A Short Course in Cloud Physics, Pergamon Press, 1526.

  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 32833287, doi:10.1175/1520-0469(1988)045<3283:ADAOFE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thual, S., and A. J. Majda, 2016: A skeleton model for the MJO with refined vertical structure. Climate Dyn., 46, 27732786, doi:10.1007/s00382-015-2731-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torrence, C., and P. J. Webster, 1999: Interdecadal changes in the ENSO–monsoon system. J. Climate, 12, 26792690, doi:10.1175/1520-0442(1999)012<2679:ICITEM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, C., and R. Weisberg, 1994: Equatorial trapped waves of a coupled ocean–atmosphere system. J. Phys. Oceanogr., 24, 19781998, doi:10.1175/1520-0485(1994)024<1978:ETWOAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 17231742, doi:10.1175/1520-0469(1995)052<1723:HTCSIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zdunkowski, C., and A. Bott, 2004: Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 268 pp.

    • Crossref
    • Export Citation
  • Zebiak, S., 1982: A simple atmospheric model of relevance to El Niño. J. Atmos. Sci., 39, 20172027, doi:10.1175/1520-0469(1982)039<2017:ASAMOR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zebiak, S., and M. Cane, 1987: A model El Niño–Southern Oscillation. Mon. Wea. Rev., 115, 22622278, doi:10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, C., 2005: Madden-Julian Oscillation. Rev. Geophys., 43, RG2003, doi:10.1029/2004RG000158.

  • View in gallery

    Schematic diagram for the SInEN regime. The terms A() and O() represent the slowly variant atmospheric and ocean energies, respectively. Similarly, a() and o() are the leading-order atmospheric, and ocean perturbations and a() and o() are higher-order atmospheric and ocean perturbations. The term adv is the advective nonlinearity, EP is the mass flux, and Qsynoptic is the external synoptic forcing. The scale separation in both the speed and time is given by .

  • View in gallery

    Dispersion wavenumber–frequency diagram depicting the possibility of a resonant triad involving an atmospheric dry Kelvin wave, an atmospheric dry Rossby wave, and an oceanic Kelvin wave. The term “dry” refers to the eigenfrequency of the equatorial wave mode not modified by coupling with deep convection. The wavenumbers/frequencies are nondimensional; however, the dimensional phase speeds (m s−1) of the corresponding waves are also displayed. To determine the resonant triad, the origin of the dispersion curve of the atmospheric Rossby wave is symmetrically displaced to a point over the oceanic Kelvin wave dispersion curve. Thus, the intersection between the dispersion curves of the atmospheric Kelvin and Rossby waves determines a set of three wave modes satisfying the resonance conditions [(39)].

  • View in gallery

    Shown is as a function of the ocean to atmosphere meridional decay ratio for .

  • View in gallery

    Energy modulation of the resonant triad interaction illustrated in Fig. 2 for the unstable regime depicted in Fig. 3. The thick line is the total energy, and the thin line is the energy of the oceanic wave, which remains bound. The dotted line is the energy of the atmospheric Rossby wave, and the long dashed line is the atmospheric Kelvin wave energy. The log10 of the energy scale indicates the exponential growth at an intraseasonal time scale.

  • View in gallery

    Time evolution of the mode energies for the resonant triad interaction illustrated in Fig. 2, but for (stable regime). The triad is composed of an atmospheric Kelvin wave (mode 1—long dashed line), an atmospheric Rossby wave (mode 2—short dashed line), and an oceanic Kelvin wave (mode 3—thin continuous line). The total energy of the triad is also displayed (thick continuous line). The initial amplitudes of the modes are set to . Other parameters are set to

  • View in gallery

    As in Fig. 5, but for .

  • View in gallery

    Similar to Fig. 5, but for the general case where the wave amplitudes are governed by (47). The triad is made of an atmospheric Kelvin mode (mode 1—long dashed line), an atmospheric Rossby mode (mode 2—short dashed line), and an oceanic Kelvin mode (mode 3—continuous line). Total energy of the triad is also displayed (thick continuous line). The initial amplitudes are set as and . The model parameter values considered in this integration are and .

  • View in gallery

    Horizontal structure of the low-level horizontal wind and height fields for (a) the atmospheric first baroclinic equatorial Kelvin wave with the dimensionless zonal wavenumber and frequency and (b) the atmospheric first baroclinic equatorial Rossby wave, with dimensionless zonal wavenumber and frequency . Both of these waves constitute the resonant triad illustrated in Figs. 57.

  • View in gallery

    (a) Zonal wind stress produced by the interaction of the atmospheric Kelvin wave and the Rossby wave (Figs. 8a,b). Over the Pacific Ocean, the wind stress is associated with westerlies. The intense westerlies over the western Pacific Ocean may trigger the oceanic Kelvin wave propagation, which is important for onset of El Niño. Westerly winds over the western Pacific and Indian Oceans are sometimes associated with the MJO. Over the eastern Pacific, low-level westerlies may lead to the weakening of the trade winds and relax the pressure gradient that maintains warm waters and deep convection to the west of the Pacific Ocean; (b) spatial pattern of the evaporation field produced by the coupling between the atmospheric Kelvin–Rossby waves and the oceanic Kelvin mode. The spatial modulation over the western Pacific is about 60° (≈6000 km), and an internal structure of synoptic or meso-γ scale (≈2000 km) is noted. Over the western Pacific and Indian Oceans the internal structure may lead to the propagation of synoptic-scale convection to the east along with the MJO signal. This configuration also reinforces the atmospheric configuration displayed in (a) and Fig. 8.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 92 92 12
PDF Downloads 57 57 12

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

View More View Less
  • 1 Centro de Previsão de Tempo e Estudos Climáticos, Instituto Nacional de Pesquisas Espaciais, São Paulo, Brazil
  • | 2 Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo, Brazil
© Get Permissions
Full access

Abstract

In 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 Niño (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 Niño 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 atmosphere–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 wave–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 atmosphere–ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El Niño).

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Enver Ramirez, enver.ramirez@gmail.com

Abstract

In 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 Niño (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 Niño 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 atmosphere–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 wave–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 atmosphere–ocean resonant coupling can be viewed as selection rules for the excitation of intraseasonal variability (MJO) or even slower interannual variability (El Niño).

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Enver Ramirez, enver.ramirez@gmail.com

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

A simplified coupled model to study the tropical multiscale air–sea wave interactions can be obtained by using two nonlinear shallow-water models, one representing the ocean and the other the atmosphere. Although the advective nonlinearities are not directly responsible for the atmosphere–ocean energy exchange, they are preserved in (1). Thus the governing equations are given by
e1a
e1b
e1c
e1d
The subscript o (a) refers the ocean (atmosphere). The vector represents the ocean state (i.e., currents and thickness), while represents the horizontal wind and geopotential height, is the reduced gravity, and the equatorial Coriolis parameter is represented by the -plane approximation (Gill 1982; Pedlosky 1987). The convention used in the atmosphere for its vertical structure is that the shallow-water equations represent the lowest atmospheric portion of the first baroclinic mode [similar to Liu and Wang (2013)]. The source/sink terms are denoted by and the atmospheric height and ocean thickness are
e2
where is the dynamical height/thickness and its time-independent mean value. The term is a nondimensional measure of the amplitude perturbation, and is the height/thickness perturbation per unit of vertical length.

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 () is a distinctive parameter. In the ocean, allows for the delayed oscillator mechanism of the El Niño phenomenon (Philander 1999) and, presumably, is the zonal extension of the Pacific Ocean—one of the causes for El Niño to occur only in this tropical ocean. In the atmosphere, despite teleconnections, significant tropical spatial variability is at and within the scale. Consequently, is taken as the referential zonal planetary scale in our model. On the other hand, the effects of both rotation and latitudinal trapping for large-scale waves near the equator are measured through the equatorial Rossby deformation radius. Thus, two important spatial scales are introduced: the atmospheric (oceanic) Rossby deformation radius (), where , with C referring to the atmospheric first baroclinic gravity wave speed, and , with representing the oceanic first baroclinic gravity wave speed. The parameters relating zonal and meridional length scales for each subsystem are given by the anisotropy parameters and . Since , the horizontal spatial scales are anisotropic (Schubert et al. 2009; Ramírez et al. 2011a,b) and, as , the balance relation is useful to describe the spatial-scale separation.

2) Vertical fluctuation scalings

Consistent with previous models, the oceanic shallow-water equations represent the active layer of the ocean, with mean thickness m (e.g., Battisti 1988). In addition, observational records show that the fluctuations of the oceanic thermocline are of about 30–50 m (Donguy and Meyers 1987), which therefore results in estimates for the oceanic nondimensional height fluctuations . For the atmosphere, the allowed fluctuations in the equivalent height associated with the synoptic-scale temperature fluctuations can be estimated through the hypsometric equation (cf. Emanuel 1987; Klein and Majda 2006), resulting in a nondimensional height fluctuation , which is consistent with the estimates adopted in the atmospheric asymptotic multiscale model of Klein and Majda (2006). Therefore, , is suitable to represent both the height and thickness fluctuations in the atmosphere and ocean, respectively. Thus, herein we shall use to represent the vertical fluctuation in both the atmosphere and ocean.

3) Multitime scalings

For multitime scalings, the time derivative is split into fast , intermediate t, and slow τ changes. The intermediate scale is the referential time-scale , and its neighboring scales are separated by the scale separation parameter ε:
e3
For the SInEN regime considered here, is related to a measure of the air–sea coupling velocity U defined by
e4

Reported values of , in the tropical troposphere, lie in the range of 1–9 m s−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 m s−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 lies in the range of 2.4–2.9 m s−1 (Ripa 1982; Battisti 1988), and here the reference value of m s−1 is used. With these referential length and velocity scales, it follows that the referential time-scale is the intraseasonal time-scale days. Therefore, for a time-scale separation , the neighboring time scales are the equatorial synoptic time scale days and the interannual time scale days. Since , the same qualitative results can be obtained if we use either or instead of 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, is related to the slow nonlinear advective time scale. In contrast, for the ocean is the characteristic time scale for the linear gravity wave propagation. This motivates the ansatz in (34) for the evolution of the model, which establishes that the atmosphere evolves using the fastest two time scales (synoptic and intraseasonal), and the ocean evolves using the slowest two time scales (intraseasonal and interannual).

Table 1.

Typical values of the model parameters.

Table 1.

c. Scaled model for the SInEN regime

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

ocean scalings
e5a
atmospheric scalings
e5b
The referential intraseasonal time scale and the planetary zonal length scale are common for both the atmosphere and ocean, while the other selected scalings are different for the two subsystems. Application of scalings [(5)] into (1) results in
e6a
e6b
e6c
e6d
e6e
e6f
where the scaled forcing terms are given by
e7a
e7b
In (6) and (7) above, is the atmospheric anisotropy parameter, is the atmospheric Froude number, and and refer to their oceanic counterparts. Furthermore, with m, m s−1, m, m, m, m−1s−1, and m, it follows that and . Thus, as the oceanic Rossby deformation radius is one order of magnitude smaller than its atmospheric counterpart (), the ocean is more zonally elongated than the atmosphere, and this is consistent with the observational estimates for the tropical latitudinal extension.
On the other hand, for typical values of the referential currents in the active ocean layer , it follows that . Thus, the scalings considered here yield the balance
e8
Therefore, following Majda and Klein (2003) and Biello and Majda (2005), 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 , and the conservative choice is physically reasonable. The balance relations in (8) are required for the singular terms in (6) to appear in a skew-symmetric form, which allows us to obtain energy estimates independent of 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 (Zdunkowski and Bott 2004) relating the advection to the local derivative is small. In contrast, in the atmosphere the nonlinear terms are of the same order as the local time derivatives and, therefore, the atmospheric Strouhal number . Consequently, if prognostic equations are considered, the advective nonlinearity is not negligible. In both the atmosphere and ocean, the meridional acceleration is weaker than its corresponding zonal acceleration, and the system is slightly dispersive, with the spectrum of normal modes being modified under such conditions. The linear and weakly nonlinear equatorial wave spectra undergoing a continuous transition between a fully nondispersive regime to a dispersive regime, as a function of the anisotropy parameter, have been studied by Ramírez et al. (2011b).

In the atmosphere, for the case of a purely Rayleigh friction in the momentum forcing with a particularly strong damping days, the Gill-type model is recovered. Thus, to leading order in 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

The low-level surface winds impinge a stress onto the surface that transfers momentum to the ocean. In principle, this flux is parameterized by the bulk formula (Krishnamurti et al. 1998; Rogers 1976). However, as the flux is transferred throughout the active water column, the stress is weighted by the factor (proportional to the depth of the layer). Thus,
e9
where is the air density, the drag coefficient for momentum, and the water density. Furthermore, we consider the case in which is dominated by the zonal wind stress (Cane and Sarachik 1976; Dijkstra 2000). The elimination of the meridional wind stress is also consistent with the dominant geostrophic balance in the meridional momentum equation. Thus, , , and the dimensionless wind stress used in the scaled model equations is given by
e10
where the coefficient for momentum exchange is given by
e11

Momentum flux strength

To access the order in ε at which the momentum forcing must contribute, we first estimate the strength of the momentum flux . Thus, using the values in Table 1, it follows that m s−2, and the nondimensional scaled strength is given by
e12

c. Mass flux

The mass flux is set as the difference between evaporation E and deep convective precipitation P; that is,
e13
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

The moisture flux is given by the bulk formula, which reads where is the air density, the drag coefficient for water vapor flux, the latent heat of vaporization, and and are the saturation and anemometer level moisture, respectively. Analogous to the wind stress forcing we use . Thus,
e14
As in Neelin and Zeng (2000), the anemometer level moisture is split in two parts: namely, , where is a referential time/spatial independent moisture, and represents local departures from . Although several processes can be accommodated in , such as the mesoscale/synoptic-scale structure and evolution, we will set . This shortcoming will somehow be fixed later as the synoptic-scale mass flux will emerge as necessary forcing in order to close the model equations and to excite the lowest-order perturbations in the atmosphere. The saturation moisture can be approximated by
e15
where is the saturation water vapor pressure, p is a referential pressure, and . The temperature T is split into basic-state temperature and anomalous temperature . Then, by using the Clausius–Clapeyron equation and neglecting the temperature dependency of the latent heat of vaporization, we obtain
e16
where , hPa, K, and hPa. Thus, the moisture flux as a function of and is given by
e17
where and are nondimensional coefficients for the linear and nonlinear components of the moisture flux
e18a
e18b
In the equations above, the square brackets in the terms and represent the dimensional strength of moisture and sea surface temperature anomalies, respectively. Differently from Liu and Wang (2013), the expression for E contains a nonlinear term yielding a coupling between temperature and wind.

2) Thermodynamic equation

The approximate thermodynamic equation with dissipation rate r, radiative forcing S, and horizontal and vertical advection is given by
e19
Simplified versions of (19) can be obtained for the case of weak horizontal advection () and no radiative forcing (i.e., ). Moreover, the vertical advection depends on the vertical gradient associated with the difference between surface and the subsurface temperature anomalies. Thus,
e20
where relates the subsurface temperature anomaly to the height perturbation (Battisti 1988).
In addition, in the fast thermodynamic adjustment, , the sea surface temperature anomaly can also be related to the height perturbation through the following expression:
e21
where is the dissipation rate modified by the vertical advection. With the aid of the thermodynamic equation [(21)], the moisture flux can be written as a function of the ocean height perturbation
e22
where
e23
and is related to by
e24

3) Evaporation strength

To access the evaporation strength, temperature fluctuations in the tropical atmosphere are used as proxy of the moisture fluctuations in the same region. In this way, is used to estimate the moisture flux. Following Majda and Shefter (2001b), the typical magnitude of temperature fluctuations in the tropical troposphere is given by for a referential temperature K. This estimate is roughly valid for specific moisture fluctuations
e25
Thus, the evaporation strength is given by W m−2, or in meters per second
e26
The dimensionless evaporation strength using E (m s−1) results in
e27

4) Precipitation

Precipitation is parameterized by the low-level moisture convergence due to anomalous winds according to
e28
where is the precipitation efficiency, is the boundary layer depth given by , is the moisture field, and for and zero otherwise. If the moisture field is approximated by the time/spatial independent referential moisture , then P is given by
e29
where
e30

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 is smaller than , with representing a measure of spatial organization of clouds and precipitation. In the upper limit, , 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 . Examples of these smaller hierarchies of cloud organization are the planetary-scale organization by cloud clusters with km, superclusters of synoptic-scale organization (SYSO) with km, and mesoscale organization (MESO) with km.

The dimensionless precipitation strength for a precipitation efficiency parameter [see in Majda and Shefter (2001b)] and a planetary-scale precipitation with is then estimated as follows:
e31
Furthermore, for heating regions associated with the MJO . In addition, for synoptic-scale heating we have . These numbers agree quite well with other estimates (Majda and Shefter 2001b,a; Yano et al. 1995).

6) Mass flux strength

The mass flux is given by the balance between E and P and its strength is determined by
e32
For the hierarchies of clouds and precipitation discussed above, we have
e33
Considering the whole budget in (33) implies that in the tropical region precipitation is larger than evaporation []. However, in the planetary ITCZ-like organization, as is positive, the atmosphere has a net gain of mass (evaporation stronger than precipitation). In contrast, for smaller organization systems, such as in the planetary MJO-like or in the SYSO-like structures, it follows that precipitation is stronger than evaporation, resulting in a negative mass source and a net mass loss. Therefore, it appears that the scale of moisture convergence can be used as a bifurcation parameter.

4. Multiscale SInEN model

Let us assume now that each component of the system has a solution composed of leading-order and higher-order perturbations, with indicating the atmosphere and the ocean, respectively. We should remember that in our scaled model the ocean evolves in the slowest two time scales (), whereas the atmosphere evolves in the fastest two time scales (). Then the following ansatz is assumed:
e34a
e34b
e34c
e34d
with , and 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,
e35a
e35b
Physically, (35) means that even after a long period the perturbations cannot overcome the leading-order ones. In addition, consistently with the strength estimates the source terms can be expanded as
e36a
e36b
with the oceanic momentum flux given by the wind stress parameterization
e37a
e37b
and the atmospheric heat flux given by
e37c
e37d
e37e
Once the leading-order atmospheric mass flux is specified (recall that it is a free parameter of the model), it drives the perturbations, and, in turn, a combination of the resulting perturbations constitutes either the atmospheric mass forcing for higher-order disturbances or the momentum forcing for leading-order perturbations in the ocean . The evaporation contributes with both linear and nonlinear terms, and precipitation was parameterized in terms of linear moisture convergence. In the ocean, the inhomogeneous term for is due to nonlinear combination of perturbations, whereas the forcing of presents a mixed term. Mixed terms also appear in the atmosphere, but only as a forcing for higher-order terms that are not included in the present study. In Fig. 1 a schematic diagram of the multiscale atmosphere–ocean interactions in the SInEn regime is depicted. The physics introduces nonlinear terms related to the mass and momentum fluxes, and the nonlinear mass flux is related to the evaporative heat flux. Finally, inserting the ansatz [(34)] into the SInEN model equations yields three time-scale models for the SInEN regime:
e38a
e38b
e38c
e38d
e38e
e38f
e38g
e38h
e38i
e38j
e38k
e38l
Fig. 1.
Fig. 1.

Schematic diagram for the SInEN regime. The terms A() and O() represent the slowly variant atmospheric and ocean energies, respectively. Similarly, a() and o() are the leading-order atmospheric, and ocean perturbations and a() and o() are higher-order atmospheric and ocean perturbations. The term adv is the advective nonlinearity, EP is the mass flux, and Qsynoptic is the external synoptic forcing. The scale separation in both the speed and time is given by .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

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 and perturbations must be nonresonant with the linear operator. Since the linear operator describes the anisotropic Rossby and Kelvin waves, the elimination of resonances is achieved by projecting the first-order equations onto the Kelvin and Rossby wave eigenvectors, in their respective traveling reference frames [see, e.g., Boyd (1980) and Majda and Biello (2003)]. This projection results in the slow evolution equations of the amplitudes of Kelvin and Rossby wave packets of both the atmosphere and ocean. These Rossby and Kelvin wave packets undergo their own self-mode interactions as a result of the intrinsic advective nonlinearity of each subsystem [cf. Gill (1980) and Gill and Phlips (1986)]. In addition, the parameterized mass and momentum fluxes coupling the atmosphere and ocean can yield interactions between atmospheric and oceanic wave packets through resonant triads of specific Fourier modes. This latter feature of the multiscale SInEN model is illustrated in the next section.

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

The source terms for the perturbations in both the atmosphere [(38d)(38f)] and ocean [(38j)(38l)] have the form of a forced Burger equation, which is a model for the nonlinear interaction of nondispersive wave packets (e.g., Boyd 1980; Menzaque et al. 2001). However, as our focus is on interactions involving waves of different media, we neglect the advective terms and restrict our analysis to the discrete wave mode interactions produced by the physical parameterizations. The motivation for this simplification comes from Fig. 2, where a representative example of the atmosphere–ocean resonant triad is depicted. The triad is composed of atmospheric Kelvin and Rossby waves along with an oceanic Kelvin wave represented by , and (), respectively. The resonance condition for this discrete triad interaction is given by
e39a
e39b
Therefore, in order to study the dynamics of this resonant triad, we consider the following ansatz for the leading-order solution of SInEN model:
e40a
e40b
In (40), the meridional structure functions () are given by
e41a
e41b
where and are the Hermite functions. The weaker meridional confinement near the equator of the atmospheric waves is represented by in the argument of the Hermite functions. Therefore, substituting the ansatz [(40)] in (38) it follows that the leading-order equations are satisfied automatically, since the ansatz is a combination of three linearly independent solutions of the problem. Consequently, requiring orthogonality between the inhomogeneous terms in the equations and the linear operator to eliminate the secular terms yields
e42a
e42b
e42c
Fig. 2.
Fig. 2.

Dispersion wavenumber–frequency diagram depicting the possibility of a resonant triad involving an atmospheric dry Kelvin wave, an atmospheric dry Rossby wave, and an oceanic Kelvin wave. The term “dry” refers to the eigenfrequency of the equatorial wave mode not modified by coupling with deep convection. The wavenumbers/frequencies are nondimensional; however, the dimensional phase speeds (m s−1) of the corresponding waves are also displayed. To determine the resonant triad, the origin of the dispersion curve of the atmospheric Rossby wave is symmetrically displaced to a point over the oceanic Kelvin wave dispersion curve. Thus, the intersection between the dispersion curves of the atmospheric Kelvin and Rossby waves determines a set of three wave modes satisfying the resonance conditions [(39)].

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

where the linear coefficients , , and are given by
e43a
e43b
e43c
and the nonlinear interaction coefficients are
e44a
e44b
e44c
The multiplicative factor 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 is defined by
e45
where the dagger operator defines the transpose conjugated operations. The time and zonal dependency can be evaluated using
e46
By considering the transformation , and omitting the hats for simplicity, (42) can be rewritten to focus on the nonlinear terms
e47a
e47b
e47c
Furthermore, evaluating the nonlinear coupling coefficients, we have
e48a
e48b
e48c
where
e49a
e49b
and
e50a
e50b
Equations (44), (48), and (49) show that the nonlinear interaction coefficients are purely imaginary numbers and are explicit functions of the frequency and wavenumber of the atmospheric Rossby wave (, ). In addition, as the triad interaction considered is due to the parameterized mass and momentum forcings, its dynamics should differ from the resonant triads arising from advective nonlinearity. However, in the specific parameter regime where the model has stable solutions, the triad displays certain properties similar to those of the conservative resonant interactions associated with advective nonlinearity.

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

To further understand the interactions, we first analyze the limiting case where the interaction coefficient of the oceanic Kelvin mode is zero. This is equivalent to considering either a linear parameterization for the wind stress () or the limiting case of . In this case, the oceanic Kelvin mode acts as a catalyst mode; that is, it allows the nonlinear interaction between the two atmospheric waves, but its amplitude is unaffected by the two atmospheric wave modes. This type of resonant interaction is known as “parametric interaction,” as the magnitude of the energy exchange between the other triad members depends on the initial amplitude of the catalyst mode. Under the parametric interaction described above, (47) now reads
e51a
e51b
e51c
An interpretation for (51) is that the slow oceanic wave amplitude evolution is even slower compared with the atmospheric wave amplitude evolution. In other words, there is a widescale separation between the evolution of the atmosphere and the ocean.
Furthermore, from (51) one can obtain an equation for each of the atmospheric wave amplitudes. Thus, for the atmospheric Kelvin wave, we have
e52
where
e53

As and are polynomials in ε; both of them in combination with and determine the character of the slow wave modulation. Figure 3 displays as a function of the oceanic to atmospheric meridional decay ratio ε for and () taken from the equatorial Rossby wave depicted in Fig. 2. Thus, we note that for typical values of the ratio between oceanic and atmospheric equatorial wave trappings it follows that and, consequently, the atmospheric wave amplitudes undergo periodic modulation with frequency . In contrast, for smaller values of ε (e.g., ) it results that , and the atmospheric wave amplitudes undergo an exponential growth, indicating an unstable character. In fact, Fig. 4 illustrates the integration of (51) for the unstable regime. A very fast exponential growth of the solutions at intraseasonal time scales can be noted. Furthermore, as , it is suggested that the meridional trapping of the interacting equatorial waves is an important parameter to define the stability of the atmosphere–ocean resonant interactions.

Fig. 3.
Fig. 3.

Shown is as a function of the ocean to atmosphere meridional decay ratio for .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

Fig. 4.
Fig. 4.

Energy modulation of the resonant triad interaction illustrated in Fig. 2 for the unstable regime depicted in Fig. 3. The thick line is the total energy, and the thin line is the energy of the oceanic wave, which remains bound. The dotted line is the energy of the atmospheric Rossby wave, and the long dashed line is the atmospheric Kelvin wave energy. The log10 of the energy scale indicates the exponential growth at an intraseasonal time scale.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

In the parametric oscillatory regime of (51) (i.e., ), the total triad energy is bound, and the oceanic wave amplitude is not modulated at the slow time scale. However, the oceanic wave is essential to allow the energy exchange between the atmospheric waves. The long period of energy exchange depends on the initial energy with which the triad was established. Thus, whenever then , and the modulation period becomes infinite. On the other hand, large values of result in large values of and, consequently, short periods for the energy exchange.

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: and , respectively. In these numerical experiments , and, therefore, the solution is stable (see Fig. 3). The selected initial energy distribution deposits more energy into the oceanic mode and less energy into the atmospheric Kelvin mode (). The atmospheric Rossby mode is initiated with an intermediate energy value (). From the time integration it can be seen that most of the energy goes to the atmospheric Kelvin mode, whereas the atmospheric Rossby wave is modulated with the exact opposite phase. Thus, when the Rossby wave is at its maximum energy level, the atmospheric Kelvin wave is at its minimum energy level and vice versa. Furthermore, as decreases, the frequency modulation decreases, and the period of energy exchange increases from around 60 days to around 75 days. In both experiments, the energy of the oceanic Kelvin wave remains constant for the whole period.

Fig. 5.
Fig. 5.

Time evolution of the mode energies for the resonant triad interaction illustrated in Fig. 2, but for (stable regime). The triad is composed of an atmospheric Kelvin wave (mode 1—long dashed line), an atmospheric Rossby wave (mode 2—short dashed line), and an oceanic Kelvin wave (mode 3—thin continuous line). The total energy of the triad is also displayed (thick continuous line). The initial amplitudes of the modes are set to . Other parameters are set to

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

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 () or similarly relaxing the limiting case of by considering ε small but finite. Although the coupling coefficient of the oceanic Kelvin mode must still be much smaller than those of the atmospheric members of the triad, now the amplitude of the oceanic mode is allowed to vary in time. Consequently, all the triad members undergo nonlinear amplitude modulation.

As in the parametric case, the full triad system [(47)] is integrable, and its solutions are described in terms of Jacobi elliptic functions (Abramowitz and Stegun 1972; Arfken et al. 1995; Lynch 2003; Craik 1985). In this sense, to somewhat simplify the solution, we set the mode with the highest energy modulation (mode 1—the atmospheric Kelvin mode) to have zero initial amplitude. In this case, the solution of the system in (47) is similar to that used by Domaracki and Loesch (1977) and Raupp et al. (2008):
e54a
e54b
e54c
where sn, cn, and dn are the Jacobi elliptic functions, with argument (corresponding to the rescaled time) and parameter given by
e55a
e55b
The analytic solution [(54)] may exhibit different behaviors depending on the initial energy partition among the triad members. This is evidenced by the dependence of on the ratio of the initial amplitudes of modes 2 and 3. For example, when , the triad essentially undergoes the parametric regime discussed above, with mode 3 (the oceanic Kelvin mode) acting as a catalyst mode for the energy exchanges between the atmospheric waves. Moreover, when , the elliptic functions become trigonometric functions, with , , and , resulting that is a constant. On the other hand, as the parameter tends to 1, the elliptic functions describe a parabola, and instability of the highest-frequency mode (mode 1) might occur. In addition, for intermediate values of the parameter , the triad undergoes considerable energy exchanges, with all the wave amplitudes being significantly modulated in time. As the coupling coefficient of the oceanic mode is one order of magnitude smaller than those of the atmospheric waves, a sufficiently small initial amplitude of the oceanic Kelvin mode, in comparison with the atmospheric Rossby mode, is required in this regime.

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 regime, and other parameters were set to yield . As can be noted, all the triad members undergo significant interannual energy modulation. The dimensional natural oscillation periods associated with the triad members are days, days, and days. Therefore, the energy modulation of the triad members is much slower than their natural oscillation periods. Thus, the nonlinear triad interaction analyzed here allows for a multi-time-scale interaction, yielding an interannual energy modulation through nonlinear interactions involving waves with synoptic and intraseasonal time scales.

Fig. 7.
Fig. 7.

Similar to Fig. 5, but for the general case where the wave amplitudes are governed by (47). The triad is made of an atmospheric Kelvin mode (mode 1—long dashed line), an atmospheric Rossby mode (mode 2—short dashed line), and an oceanic Kelvin mode (mode 3—continuous line). Total energy of the triad is also displayed (thick continuous line). The initial amplitudes are set as and . The model parameter values considered in this integration are and .

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

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 equatorial Rossby wave (mode 2)—are displayed. The Kelvin wave is of planetary scale and produces strong westerly winds throughout the Pacific Ocean (peaking over the central Pacific) and easterly winds outside the basin. In addition, the equatorial Rossby (mode 2) produces a symmetric pattern about the equator, with strong westerlies both to the west and east of the Pacific basin and strong easterlies over the central Pacific. The spatial scale of the Rossby wave is compatible with the pattern of twin cyclones around the eastern Indian Ocean, the Maritime Continent, and the western Pacific Ocean that is associated to the MJO (e.g., Ferreira et al. 1996).

Fig. 8.
Fig. 8.

Horizontal structure of the low-level horizontal wind and height fields for (a) the atmospheric first baroclinic equatorial Kelvin wave with the dimensionless zonal wavenumber and frequency and (b) the atmospheric first baroclinic equatorial Rossby wave, with dimensionless zonal wavenumber and frequency . Both of these waves constitute the resonant triad illustrated in Figs. 57.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

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.

Fig. 9.
Fig. 9.

(a) Zonal wind stress produced by the interaction of the atmospheric Kelvin wave and the Rossby wave (Figs. 8a,b). Over the Pacific Ocean, the wind stress is associated with westerlies. The intense westerlies over the western Pacific Ocean may trigger the oceanic Kelvin wave propagation, which is important for onset of El Niño. Westerly winds over the western Pacific and Indian Oceans are sometimes associated with the MJO. Over the eastern Pacific, low-level westerlies may lead to the weakening of the trade winds and relax the pressure gradient that maintains warm waters and deep convection to the west of the Pacific Ocean; (b) spatial pattern of the evaporation field produced by the coupling between the atmospheric Kelvin–Rossby waves and the oceanic Kelvin mode. The spatial modulation over the western Pacific is about 60° (≈6000 km), and an internal structure of synoptic or meso-γ scale (≈2000 km) is noted. Over the western Pacific and Indian Oceans the internal structure may lead to the propagation of synoptic-scale convection to the east along with the MJO signal. This configuration also reinforces the atmospheric configuration displayed in (a) and Fig. 8.

Citation: Journal of the Atmospheric Sciences 74, 8; 10.1175/JAS-D-15-0325.1

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

In this paper, we have developed a novel nonlinear multiscale model to study synoptic–intraseasonal–interannual–El Niño (SInEN) interactions in a coupled atmosphere–ocean system. For this purpose, we have considered a simple setup—that is, two coupled equatorial 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,
e56
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 is constant, but in the real atmosphere–ocean system the exchanges of evaporation and precipitation along with the salinity also affect the density structure and play an important role in thermocline fluctuations. Furthermore, the ocean dynamics–thermodynamics coupling and the geometry of the Pacific Ocean are crucial for the formation of the warm (cold) tongue during El Niño (La Niña) events, and thus it would be important to include these effects in our model along with the seasonal cycle.

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 Rossby wave, with the modes interacting resonantly through the parameterized atmosphere–ocean heat and momentum fluxes. The analytic solution of the triad equations shows that the oceanic wave may act as a catalyst mode for the energy exchanges between the atmospheric waves for linearized momentum flux. The oceanic Kelvin wave can also undergo significant energy modulations for a small but nonzero interaction coefficient, provided that this mode has a sufficiently smaller initial amplitude than the atmospheric waves. The results also show that the atmospheric Kelvin mode always supplies/receives energy to/from the remaining two triad components. In this situation, the wave amplitude modulations occur at interannual time scales, while the phase propagation periods of the wave fields are of synoptic and intraseasonal time scales.

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 Rossby wave activity. The phases displayed for the atmospheric waves are in agreement with what is required by the amplitude modulation. Over the Pacific Ocean, strong westerlies are found over both the western and eastern sides of the basin, whereas moderate winds are in the central Pacific. Associated to the wind patterns, planetary-scale wind stress patches are found (~5000 km; see Fig. 9), and their tropical nature, magnitude, and spatial scale suggest that they can be associated to the MJO. In addition, the nonlinear coupling to the ocean produces a synoptic-scale structure for the evaporation field (~2000 km) that is modulated at planetary scales (~6000 km; see Fig. 9). The up and down synoptic-scale pattern of the evaporation field may stimulate farther eastward propagation of the intraseasonal activity and trigger oceanic Kelvin waves. Over the eastern Pacific Ocean, a relatively weak wind stress patch is found, which is associated with westerlies and thus tends to weaken the climatological trade winds and to reduce the east–west pressure gradient that maintains warm water to the west over the Pacific Ocean.

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

The evaluation of the product of modes requires the evaluation of integrals involving three or more Hermite functions. For the product of modes of the ocean and atmosphere, the meridional decays are different. The ocean is meridionally more confined than the atmosphere. In the ocean the Hermite functions are given by , whereas in the atmosphere they are given by , where ε is a measure of the meridional confinement. The Hermite functions are related to the Hermite polynomials by
ea1
where . The meridional integration of the product of three Hermite functions with , and p nodes and different decays can be expressed by . A practical rule to evaluate three or more Hermite functions is to reduce the functions in pairs until just one Hermite function remains and then use the parity condition
ea2

a. Reduction in pairs for same meridional decays

In the simplest case of the same meridional decays for the Hermite functions, the reduction in pairs is given according to Lord (1948) and Busbridge (1948):
ea3

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

The general case of different meridional decays can also be performed by reduction in pairs. However, the decaying parameter enters in the product, and the results depend on the decaying parameter. Let and represent two Hermite polynomials with different meridional decays:
ea4
where and represent the lowest nearest integer number, also known as the floor of the division and , respectively. The order of the resulting polynomial is ; however, their coefficients depend on the decaying parameter.
Thus, for , , , and we have
ea5
For , , , and ,
ea6
Both results contrast with the simplest case of , , and :
ea7
Thus, the evaluation of the meridional integrals will depend on both the latitudinal structure of the involved modes and on their 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, 28892919, doi:10.1175/1520-0469(1988)045<2889:DATOAW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Biello, J. A., and A. J. Majda, 2005: A new multiscale model for the Madden–Julian oscillation. J. Atmos. Sci., 62, 16941721, doi:10.1175/JAS3455.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bony, S., and et al. , 2015: Clouds, circulation and climate sensitivity. Nat. Geosci., 8, 261268, doi:10.1038/ngeo2398.

  • Boyd, J., 1980: The nonlinear equatorial Kelvin wave. J. Phys. Oceanogr., 10, 111, doi:10.1175/1520-0485(1980)010<0001:TNEKW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Busbridge, I. W., 1948: Some integrals involving hermite polynomials. J. London Math. Soc., s1-23, 135141, 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, 629665.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Export Citation
  • Domaracki, A., and A. Z. Loesch, 1977: Nonlinear interactions among equatorial waves. J. Atmos. Sci., 34, 486498, doi:10.1175/1520-0469(1977)034<0486:NIAEW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Donguy, J. R., and G. Meyers, 1987: Observed and modelled topography of the 20°C isotherm in the tropical Pacific. Oceanol. Acta, 10, 4148.

    • Search Google Scholar
    • Export Citation
  • Emanuel, K., 1987: An air-sea interaction model of intraseasonal oscillations in the tropics. J. Atmos. Sci., 44, 23242340, doi:10.1175/1520-0469(1987)044<2324:AASIMO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ferreira, R., W. Schubert, and J. Hack, 1996: Dynamical aspects of twin tropical cyclones associated with the Madden–Julian oscillation. J. Atmos. Sci., 53, 929945, doi:10.1175/1520-0469(1996)053<0929:DAOTTC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fritz, S., 1958: Seasonal heat storage in the ocean and heating of the atmosphere. Arch. Meteor. Geophys. Bioklimatol., 10A, 291300, doi:10.1007/BF02247144.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 447462, doi:10.1002/qj.49710644905.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 6991, doi:10.1002/qj.49711247105.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., 1986: Unstable and damped equatorial modes in simple coupled ocean-atmosphere models. J. Atmos. Sci., 43, 606630, doi:10.1175/1520-0469(1986)043<0606:UADEMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hirst, A. C., and K.-M. Lau, 1990: Intraseasonal and interannual oscillations in coupled ocean-atmosphere models. J. Climate, 3, 713725, doi:10.1175/1520-0442(1990)003<0713:IAIOIC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Holm, D. D., and P. Lynch, 2002: Stepwise precession of the resonant swinging spring. SIAM J. Appl. Dyn. Syst., 1, 4464, doi:10.1137/S1111111101388571.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 777793, doi:10.1007/s003820000148.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Johnson, R., T. Rickenbach, S. Rutledge, P. Ciesielski, and W. Schubert, 1999: Trimodal characteristics of tropical convection. J. Climate, 12, 23972418, doi:10.1175/1520-0442(1999)012<2397:TCOTC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kevorkian, J. K., and J. D. Cole, 1996: Multiple Scale and Singular Perturbation Methods. Springer-Verlag, 634 pp.

    • Crossref
    • Export Citation
  • Klein, R., and A. Majda, 2006: Systematic multiscale models for deep convection on mesoscales. Theor. Comput. Fluid Dyn., 20, 525551, doi:10.1007/s00162-006-0027-9.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Krishnamurti, T. N., H. S. Bedi, and V. M. Hardiker, 1998: Physical processes. An Introduction to Global Spectral Modeling, Oxford University Press, 138–182.

    • Crossref
    • Export Citation
  • Liu, F., and B. Wang, 2013: An air–sea coupled skeleton model for the Madden–Julian oscillation. J. Atmos. Sci., 70, 31473156, doi:10.1175/JAS-D-12-0348.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, M. S., A. E. Gill, and K. Kenyon, 1967: Resonant interactions between planetary waves. Proc. Roy. Soc. London, 299A, 120144, doi:10.1098/rspa.1967.0126.

    • Search Google Scholar
    • Export Citation
  • Lord, R. D., 1948: Some integrals involving hermite polynomials. J. London Math. Soc., 24, 101112.

  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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.

    • Crossref
    • Export Citation
  • Majda, A., and M. Shefter, 2001a: Models for stratiform instability and convectively coupled waves. J. Atmos. Sci., 58, 15671584, doi:10.1175/1520-0469(2001)058<1567:MFSIAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and M. Shefter, 2001b: Waves and instabilities for model tropical convective parametrizations. J. Atmos. Sci., 58, 896914, doi:10.1175/1520-0469(2001)058<0896:WAIFMT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and J. Biello, 2003: The nonlinear interaction of barotropic and equatorial baroclinic Rossby waves. J. Atmos. Sci., 60, 18091821, doi:10.1175/1520-0469(2003)060<1809:TNIOBA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and R. Klein, 2003: Systematic multiscale models for the Tropics. J. Atmos. Sci., 60, 393408, doi:10.1175/1520-0469(2003)060<0393:SMMFTT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Majda, A., and S. Stechmann, 2009: The skeleton of tropical intraseasonal oscillations. Proc. Natl. Acad. Sci. USA, 106, 84178422, doi:10.1073/pnas.0903367106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 41184133, doi:10.1175/1520-0469(1999)056<4118:IOLSEW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • McPhaden, M. J., 1999: Genesis and evolution of the 1997-98 El Niño. Science, 283, 950954, 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, 5182, doi:10.1090/conm/283/04714.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteor. Soc. Japan, 66, 823839, doi:10.2151/jmsj1965.66.6_823.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—Formulation. J. Atmos. Sci., 57, 17411766, doi:10.1175/1520-0469(2000)057<1741:AQETCM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed. Springer-Verlag, 710 pp., doi:10.1007/978-1-4612-4650-3.

    • Crossref
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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.

    • Search Google Scholar
    • Export Citation
  • 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, 14461462, doi:10.1175/JAS3412.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Raupp, C. F. M., and P. L. Silva Dias, 2006: Dynamics of resonantly interacting equatorial waves. Tellus, 58A, 263279, doi:10.1111/j.1600-0870.2006.00151.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 31653183, doi:10.1175/2009JAS2899.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 706718, doi:10.1111/j.1600-0870.2010.00463.x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 33983418, doi:10.1175/2008JAS2387.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 11171133, doi:10.1175/1520-0469(1971)028<1117:SAPOSS>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1982: Nonlinear wave–wave interactions in a one-layer reduced-gravity model on the equatorial β plane. J. Phys. Oceanogr., 12, 97111, doi:10.1175/1520-0485(1982)012<0097:NWIIAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1983a: Weak interactions of equatorial waves in a one-layer model. Part I: General properties. J. Phys. Oceanogr., 13, 12081226, doi:10.1175/1520-0485(1983)013<1208:WIOEWI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Ripa, P., 1983b: Weak interactions of equatorial waves in a one-layer model. Part II: Applications. J. Phys. Oceanogr., 13, 12271240, doi:10.1175/1520-0485(1983)013<1227:WIOEWI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Rogers, R. R., 1976: Water vapor and its thermodynamic effects. A Short Course in Cloud Physics, Pergamon Press, 1526.

  • 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.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 32833287, doi:10.1175/1520-0469(1988)045<3283:ADAOFE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Thual, S., and A. J. Majda, 2016: A skeleton model for the MJO with refined vertical structure. Climate Dyn., 46, 27732786, doi:10.1007/s00382-015-2731-x.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Torrence, C., and P. J. Webster, 1999: Interdecadal changes in the ENSO–monsoon system. J. Climate, 12, 26792690, doi:10.1175/1520-0442(1999)012<2679:ICITEM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, C., and R. Weisberg, 1994: Equatorial trapped waves of a coupled ocean–atmosphere system. J. Phys. Oceanogr., 24, 19781998, doi:10.1175/1520-0485(1994)024<1978:ETWOAC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • 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, 17231742, doi:10.1175/1520-0469(1995)052<1723:HTCSIA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zdunkowski, C., and A. Bott, 2004: Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 268 pp.

    • Crossref
    • Export Citation
  • Zebiak, S., 1982: A simple atmospheric model of relevance to El Niño. J. Atmos. Sci., 39, 20172027, doi:10.1175/1520-0469(1982)039<2017:ASAMOR>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zebiak, S., and M. Cane, 1987: A model El Niño–Southern Oscillation. Mon. Wea. Rev., 115, 22622278, doi:10.1175/1520-0493(1987)115<2262:AMENO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Zhang, C., 2005: Madden-Julian Oscillation. Rev. Geophys., 43, RG2003, doi:10.1029/2004RG000158.

Save