• Castanheira, J., and C. Marques, 2015: Convectively coupled equatorial-wave diagnosis using three-dimensional normal modes. Quart. J. Roy. Meteor. Soc., 141, 27762792, https://doi.org/10.1002/qj.2563.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chang, C., and H. Lim, 1988: Kelvin wave-CISK: A possible mechanism for the 30–50 day oscillations. J. Atmos. Sci., 45, 17091720, https://doi.org/10.1175/1520-0469(1988)045<1709:KWCAPM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, S., R. Houze Jr., and B. Mapes, 1996: Multiscale variability of deep convection in relation to large-scale circulation in TOGA COARE. J. Atmos. Sci., 53, 13801409, https://doi.org/10.1175/1520-0469(1996)053<1380:MVODCI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, S., A. Majda, and S. Stechmann, 2016: Tropical–extratropical interactions with the MJO skeleton and climatological mean flow. J. Atmos. Sci., 73, 41014116, https://doi.org/10.1175/JAS-D-16-0041.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Crueger, T., B. Stevens, and R. Brokopf, 2013: The Madden–Julian oscillation in ECHAM6 and the introduction of an objective MJO metric. J. Climate, 26, 32413257, https://doi.org/10.1175/JCLI-D-12-00413.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • DeMott, C., N. Klingaman, and S. Woolnough, 2015: Atmosphere-ocean coupled processes in the Madden-Julian oscillation. Rev. Geophys., 53, 10991154, https://doi.org/10.1002/2014RG000478.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., 2002: Genesis of intraseasonal oscillations and equatorial waves. J. Atmos. Sci., 59, 27612781, https://doi.org/10.1175/1520-0469(2002)059<2761:GOIOAE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., and C. Frederiksen, 1993: Monsoon disturbances, intraseasonal oscillations, teleconnection patterns, blocking, and storm tracks of the global atmosphere during January 1979: Linear theory. J. Atmos. Sci., 50, 13491372, https://doi.org/10.1175/1520-0469(1993)050<1349:MDIOTP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., and H. Lin, 2013: Tropical–extratropical interactions of intraseasonal oscillations. J. Atmos. Sci., 70, 31803197, https://doi.org/10.1175/JAS-D-12-0302.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fuchs, Z., and D. J. Raymond, 2005: Large-scale modes in a rotating atmosphere with radiative–convective instability and WISHE. J. Atmos. Sci., 62, 40844094, https://doi.org/10.1175/JAS3582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendon, H., and M. Salby, 1994: The life cycle of the Madden–Julian oscillation. J. Atmos. Sci., 51, 22252237, https://doi.org/10.1175/1520-0469(1994)051<2225:TLCOTM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kasahara, A., and K. Puri, 1981: Spectral representation of three-dimensional global data by expansion in normal mode functions. Mon. Wea. Rev., 109, 3751, https://doi.org/10.1175/1520-0493(1981)109<0037:SROTDG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kitsios, V., 2010: Recovery of fluid mechanical modes in unsteady separated flows. Ph.D. thesis, University of Melbourne and Université de Poitiers, 259 pp., http://repository.unimelb.edu.au/10187/9088.

  • Kobayashi, A., and Coauthors, 2015: The JRA-55 reanalysis: General specifications and basic characteristics. J. Meteor. Soc. Japan, 93, 548, https://doi.org/10.2151/jmsj.2015-001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lau, K., and L. Peng, 1987: Origin of low-frequency (intraseasonal) oscillations in the tropical atmosphere. Part I: Basic theory. J. Atmos. Sci., 44, 950972, https://doi.org/10.1175/1520-0469(1987)044<0950:OOLFOI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liess, S., and L. Bengtsson, 2004: The intraseasonal oscillation in ECHAM4 part II: Sensitivity studies. Climate Dyn., 22, 671688, https://doi.org/10.1007/s00382-004-0407-z.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lindzen, R., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31, 156179, https://doi.org/10.1175/1520-0469(1974)031<0156:WCITT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Madden, R. A., and P. R. Julian, 1971: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. J. Atmos. Sci., 28, 702708, https://doi.org/10.1175/1520-0469(1971)028<0702:DOADOI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Madden, R. A., and P. R. Julian, 1972: Description of global-scale circulation cells in the tropics with a 40–50 day period. J. Atmos. Sci., 29, 11091123, https://doi.org/10.1175/1520-0469(1972)029<1109:DOGSCC>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, https://doi.org/10.1073/pnas.0903367106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteor. Soc. Japan, 66, 823836, https://doi.org/10.2151/jmsj1965.66.6_823.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J., and I. Held, 1987: Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev., 115, 312, https://doi.org/10.1175/1520-0493(1987)115<0003:MTCBOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J., and J.-Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part I: Analytical theory. J. Atmos. Sci., 51, 18761984, https://doi.org/10.1175/1520-0469(1994)051<1876:MOTVUC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Noack, B., K. Afanasiev, M. Morzynski, G. Tadmor, and F. Thiele, 2003: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497, 335363, https://doi.org/10.1017/S0022112003006694.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., and Ž. Fuchs, 2009: Moisture modes and the Madden–Julian oscillation. J. Atmos. Sci., 22, 30313046, https://doi.org/10.1175/2008JCLI2739.1.

    • Search Google Scholar
    • Export Citation
  • Rui, H., and B. Wang, 1990: Development characteristics and dynamic structure of tropical intraseasonal convection anomalies. J. Atmos. Sci., 47, 357379, https://doi.org/10.1175/1520-0469(1990)047<0357:DCADSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sobel, A., and E. Maloney, 2012: Moisture modes and the eastward propagation of the MJO. J. Atmos. Sci., 69, 16911705, https://doi.org/10.1175/JAS-D-11-0118.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, B., and H. Rui, 1990: Dynamics of the coupled moist Kelvin–Rossby wave on an equatorial β-plane. J. Atmos. Sci., 47, 397413, https://doi.org/10.1175/1520-0469(1990)047<0397:DOTCMK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, B., and T. Li, 1994: Convective interaction with boundary-layer dynamics in the development of the tropical intraseasonal system. J. Atmos. Sci., 51, 13861400, https://doi.org/10.1175/1520-0469(1994)051<1386:CIWBLD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and G. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain. J. Atmos. Sci., 56, 374399, https://doi.org/10.1175/1520-0469(1999)056<0374:CCEWAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and H. Hendon, 2004: An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction. Mon. Wea. Rev., 132, 19171932, https://doi.org/10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., and J. Tribbia, 2017: Tropical atmospheric Madden–Julian oscillation: A strongly nonlinear free solitary Rossby wave. J. Atmos. Sci., 74, 34733489, https://doi.org/10.1175/JAS-D-16-0319.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., and C. Franzke, 2015: Systematic decomposition of the Madden-Julian oscillation into balanced and inertio-gravity components. Geophys. Res. Lett., 42, 68296835, https://doi.org/10.1002/2015GL065130.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., J. Tribbia, J. Anderson, and K. Raeder, 2009: Uncertainties of estimates of inertia–gravity energy in the atmosphere. Part I: Intercomparison of four analysis systems. Mon. Wea. Rev., 137, 38372477, https://doi.org/10.1175/2009MWR2815.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., A. Kashahara, K. Terasaki, J. Tribbia, and H. Tanaka, 2015: Normal-mode function representation of global 3-D data sets: Open-access software for the atmospheric research community. Geosci. Model Dev., 8, 11691195, https://doi.org/10.5194/gmd-8-1169-2015.

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

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    Vertical structure function eigensolution: (a) static stability profile vs an indicative pressure σ×p0, where p0 = 1013 hPa; (b) equivalent height Dm, or eigenvalues of the VSF EVP; and (c) a subset of vertical modes, or eigenvectors of the VSF EVP.

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    Horizontal structure function eigensolutions: (a) time scales 1/νknmq of the HSF for vertical mode of equivalent height = 7 m for balanced (BAL; large dots: n=1; small dots: n>1), easterly propagating inertial gravity waves (EIG; large dots: n=1; small dots: n>1), westerly propagating inertial gravity waves (WIG; large dots: n=1; small dots for all other n), mixed Rossby–gravity (MRG; equivalent to BAL: n=0) and Kelvin waves (KW; equivalent to EIG: n=0); (b) time scales 1/νknmq of the HSF for EIG with (k,n)=(1,0), WIG with (k,n)=(1,0), and BAL with (k,n)=(1,1) vs the equivalent height of each vertical mode; and (c) the longitudinal velocity component Uknmq(ϕ) of the horizontal structure functions for EIG modes with (k,n,Dm)=(1,0,7m) and (1,0,324m), and BAL modes with (k,n,Dm)=(1,1,92m) and (1,1,324m).

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    Time-averaged spectral contribution from each of the NMFs: (a) zonal wavenumber spectra of total energy (i.e., potential and kinetic) summed over n and m decomposed into EIG, WIG, and BAL components, with the dashed black line representing the k3 decay rate; (b) vertical mode spectra of total nonzonal energy summed over k and n as a function of equivalent height; (c) as in (b), but for only the (k,n)=(1,1) horizontal mode pair; and (d) as in (b), but for only the (k,n)=(1,0) horizontal mode pair. Legend and vertical scale in (a) is applicable to all plots.

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    Spectral coherence properties of the NMFs: (a) power spectral density in the zonal wavenumber (k)–frequency (f) plane of EIG(k,0,324m), with the white arrow indicating the fast Kelvin wave component; (b) as in (a), but for EIG(k,0,7m); (c) coherence (squared correlation in spectral space) in the (k,f) plane between EIG(k,0,324m) and EIG(1,0,7m); (d) coherence output (correlated variance in spectral space) in the (k,f) plane of EIG(k,0,324m) with respect to EIG(1,0,7m); (e) power spectral density of BAL(k,1,324m); (f) coherence output of BAL(k,1,8) with respect to EIG(1,0,7m); (g) power spectral density of BAL(k,1,92m); (h) coherence output of BAL(k,1,92m) with respect to EIG(1,0,7m); (i) coherence output of EIG(1,0,m) with respect to EIG(1,0,7m) for all vertical modes represented by their equivalent height; and (j) coherence output of BAL(1,1,m) with respect to EIG(1,0,7m) for all vertical modes. Vertical axes in (a), (e), and (i) are applicable to all plots. The horizontal dashed vertical line in all plots indicates a time scale of 30 days.

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    Anomalies of phase-averaged OLR and velocity potential fields, calculated using the EIG (k,n,Dm)=(1,0,7m) mode of phases angles: (a) phase 1 from −180° to −135°, (b) phase 2 from −135° to −90°, (c) phase 3 from −90° to −45°, (d) phase 4 from −45° to 0°, (e) phase 5 from 0° to 45°, (f) phase 6 from 45° to 90°, (g) phase 7 from 90° to 135°, and (h) phase 8 from 135° to 180°. OLR contours range from −4 (blue) to 4 W m−2 (yellow). Velocity potential isosurfaces are shaded for two standard deviations below the mean (light gray) and two standard deviations above the mean (dark gray). The vertical axis, σ×p0, is an indicative pressure where p0 = 1013 hPa.

  • View in gallery

    Composite-averaged normal mode function complex expansion coefficients χknmq: (a) magnitude of the EIG mode triplet (k,n,Dm)=(1,0,7m) relative to day 0, illustrating each individual event (faint lines), the ensemble mean (thick solid line) and plus and minus one ensemble standard deviation (thick dashed lines); and (b) as in (a), but for the phase angle.

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    Composite-averaged velocity potential field at 200 hPa based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −2.5 × 106 (blue) to 2.5 × 106 m2 s−1 (yellow). Minimum (min) and maximum values (max) over the illustrated domain for each plot are listed in their respective titles.

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    Composite-averaged outgoing longwave radiation field based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −8 (blue) to 8 W m−2 (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

  • View in gallery

    Composite-averaged precipitation field based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −2 (blue) to 2 mm day−1 (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

  • View in gallery

    MJO time series: (a) square root of the 91-day centered moving averages (MA) of the RMM index (RMM12 + RMM22), the NMF energy Eknmq of unfiltered modes BAL(1,1,324m) and EIG(1,0,7m), and coherence-filtered modes BAL(1,1,324m), EIG(1,0,324m), and BAL(1,1,92m), with the yellow boxes highlighting the time periods illustrated in the remaining plots; (b) comparison of the phase angles of RMM and the coherence-filtered BAL(1,1,324m) mode for an MJO event in November 2002; (c) as in (b), for January 2006; (d) as in (b), for February 2008; (e) as in (b), for March 2012; (f) RMM in phase space for the March 2012 event; and (g) coherence-filtered BAL(1,1,324m) mode in phase space for the March 2012 event. In (e)–(g), the circle and square markers indicate the start and end of the period of interest, respectively. The legend in (a) is applicable to all plots.

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A Reduced-Order Representation of the Madden–Julian Oscillation Based on Reanalyzed Normal Mode Coherences

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  • 1 Oceans and Atmosphere, CSIRO, Hobart, Tasmania, Australia
  • | 2 Meteorologisches Institute, Universität Hamburg, Hamburg, Germany
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Abstract

The Madden–Julian oscillation (MJO) is presented as a series of interacting Rossby and inertial gravity waves of varying vertical scales and meridional extents. These components are isolated by decomposing reanalysis fields into a set of normal mode functions (NMF), which are orthogonal eigenvectors of the linearized primitive equations on a sphere. The NMFs that demonstrate spatial properties compatible with the MJO are inertial gravity waves of zonal wavenumber k = 1 and the lowest meridional index n = 0, and Rossby waves with (k, n) = (1, 1). For these horizontal scales, there are multiple small vertical-scale baroclinic modes that have temporal properties indicative of the MJO. On the basis of one such eastward-propagating inertial gravity wave (i.e., a Kelvin wave), composite averages of the Japanese 55-year Reanalysis demonstrate an eastward propagation of the velocity potential, and oscillation of outgoing longwave radiation and precipitation fields over the Maritime Continent, with an MJO-appropriate temporal period. A cross-spectral analysis indicates that only the MJO time scale is coherent between this Kelvin wave and the more energetic modes. Four mode clusters are identified: Kelvin waves of correct phase period and direction, Rossby waves of correct phase period, energetic Kelvin waves of larger vertical scales and meridional extents extending into the extratropics, and energetic Rossby waves of spatial scales similar to that of the energetic Kelvin waves. We propose that within this normal mode framework, nonlinear interactions between the aforementioned mode groups are required to produce an energetic MJO propagating eastward with an intraseasonal phase period. By virtue of the selected mode groups, this theory encompasses both multiscale and tropical–extratropical interactions.

© 2019 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: Vassili Kitsios, vassili.kitsios@csiro.au

Abstract

The Madden–Julian oscillation (MJO) is presented as a series of interacting Rossby and inertial gravity waves of varying vertical scales and meridional extents. These components are isolated by decomposing reanalysis fields into a set of normal mode functions (NMF), which are orthogonal eigenvectors of the linearized primitive equations on a sphere. The NMFs that demonstrate spatial properties compatible with the MJO are inertial gravity waves of zonal wavenumber k = 1 and the lowest meridional index n = 0, and Rossby waves with (k, n) = (1, 1). For these horizontal scales, there are multiple small vertical-scale baroclinic modes that have temporal properties indicative of the MJO. On the basis of one such eastward-propagating inertial gravity wave (i.e., a Kelvin wave), composite averages of the Japanese 55-year Reanalysis demonstrate an eastward propagation of the velocity potential, and oscillation of outgoing longwave radiation and precipitation fields over the Maritime Continent, with an MJO-appropriate temporal period. A cross-spectral analysis indicates that only the MJO time scale is coherent between this Kelvin wave and the more energetic modes. Four mode clusters are identified: Kelvin waves of correct phase period and direction, Rossby waves of correct phase period, energetic Kelvin waves of larger vertical scales and meridional extents extending into the extratropics, and energetic Rossby waves of spatial scales similar to that of the energetic Kelvin waves. We propose that within this normal mode framework, nonlinear interactions between the aforementioned mode groups are required to produce an energetic MJO propagating eastward with an intraseasonal phase period. By virtue of the selected mode groups, this theory encompasses both multiscale and tropical–extratropical interactions.

© 2019 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: Vassili Kitsios, vassili.kitsios@csiro.au

1. Introduction

The Madden–Julian oscillation (MJO) is an eastward-propagating intraseasonal mode of variability, with its variance concentrated in the 30–90-day time-scale band. It is a physical phenomenon first characterized by Madden and Julian (1971), involving interactions between tropical deep convection focused within the Maritime Continent, moisture, and atmospheric dynamics (Zhang 2005). Since this initial study there has been a vast array of research into this physical phenomenon, with recent comprehensive reviews in Zhang (2005) and DeMott et al. (2015). In the current study we decompose atmospheric reanalysis into a series of scales using normal mode functions (NMF), and on the basis of these results propose a reduced-order theory of the MJO. The following literature review hence focuses on the spatiotemporal-scale properties of the MJO and the historical theories explaining its dynamics.

The life cycle of the MJO is typically characterized as being initiated by a strong large-scale deep convective event and associated precipitation over the Maritime Continent. This strong deep convection is linked to surrounding regions of weaker convection via the circulation in the vertical–longitudinal plane extending throughout the troposphere (Madden and Julian 1972; Rui and Wang 1990). In the lower troposphere, at an altitude of approximately 850 hPa, the zonal winds converge toward the convective center. In the upper troposphere, at around 200 hPa, the zonal winds diverge and travel away from the source of convection. This large-scale convective system propagates eastward and decays in the western Pacific. This first-order view of the MJO suggests a simple baroclinic structure, since the anomalous zonal velocity at the upper and lower levels are of opposite sign. In reality, however, this perceived large-scale eastward propagation is due to a hierarchy of smaller spatiotemporal-scale convective systems that on average are initiated eastward of the earlier systems (Nakazawa 1988; Hendon and Salby 1994; Chen et al. 1996). At present this finer vertical-scale structure is less well understood. One of the goals of the current study is to infer a hierarchy of important vertical scales from the NMFs. In doing so we aim to clarify the nonlinear processes that couple westward-propagating Rossby waves, fast eastward inertial gravity waves, and extratropical influences, which together give rise to the intraseasonal eastward propagation of the MJO.

The large-scale horizontal features of the MJO convection were characterized in Wheeler and Kiladis (1999) via the spectral properties of the outgoing longwave radiation (OLR). Here, the meridionally symmetric and antisymmetric components of OLR were spectrally decomposed into longitudinal wavenumber (k) and temporal-frequency (f) components. The MJO was identified as an eastward-propagating symmetric component of OLR. In the (k, f) plane it has significant variance for wavenumbers k = 1–4 for time scales longer than 30 days, with k = 1 having the largest contribution. Faster eastward-propagating Kelvin waves were also identified, which were aligned with solutions of the shallow-water equations of equivalent heights (or depths) between 12 and 50 m. The fast propagation speed (or equivalently short phase period) of Kelvin waves is commonly cited as a reason for why they are not representative of the MJO. Following on from this physical understanding of the MJO, Wheeler and Hendon (2004) devised a real-time multivariate MJO (RMM) index involving the singular value decomposition of OLR, and the longitudinal velocity at 200 and 850 hPa, all meridionally averaged within the region 15°S–15°N. The first two principal components (referred to as RMM1 and RMM2) are shown to have dominant variance within the intraseasonal time-scale band.

Theories of the MJO typically involve prescribed relationships between convection, moisture, and atmospheric dynamics. Initial models were developed on the basis of wave–conditional instability of the second kind (CISK) theory, in which Kelvin waves become unstable because of low-level convective heating (Lindzen 1974; Lau and Peng 1987; Chang and Lim 1988). These models, however, did not predict the appropriate longitudinal scale of the MJO. The wind-induced surface heat exchange (WISHE) theory of Emanuel (1987) proposed eastward-propagating tropical waves were perturbed via enhanced evaporation. The WISHE model produces modes of appropriate longitudinal scale, and propagation speed, but with inconsistent perturbation structure during the Northern Hemisphere winter. In the frictional wave-CISK theory equatorial Rossby waves are also included, the interaction of which produces MJO-like properties of appropriate length and time scales (Wang and Rui 1990; Wang and Li 1994).

Another prevailing theory of the MJO is that it is driven by a moisture mode instability. This type of instability depends on the evolution of the humidity field, with its growth governed by feedbacks that further moisten the atmosphere (Neelin and Yu 1994; Fuchs and Raymond 2005). Raymond and Fuchs (2009) identify that in general this moistening process occurs in the presence of negative gross moist stabilities (GMSs). GMS is a vertically integrated measure of the moist static stability in the atmosphere (Neelin and Held 1987). Sobel and Maloney (2012) also find in their idealized model of the MJO that eastward-propagating waves can only be growing if the effective GMS is less than zero.

Majda and Stechmann (2009) developed a minimal dynamical model of the MJO capturing the interactions between lower-tropospheric moisture and convectively coupled wave dynamics of prescribed meridional and baroclinic structure. Within this framework they identify a mechanism for fast Kelvin waves and westerly propagating equatorial Rossby waves to couple to each other and become dispersive. This model produced an MJO with a small group velocity, intraseasonal phase speed, and a horizontal quadrapole vortex structure. Chen et al. (2016) extended on this work to model both the onset and decay of the MJO via interactions with extratropical and barotropic modes.

In Frederiksen and Lin (2013) the MJO is characterized by a coupled tropical–extratropical mode energized by moist baroclinic–barotropic instability within a three-dimensional basic state. The linear primitive-equation instability model of Frederiksen (2002) is used to simulate eternal-January basic states. Within this framework the MJO is a single mode of dominant zonal wavenumber k = 1, first internal baroclinic structure in the tropics, equivalent barotropic structure in the extratropics, and a phase period of 34.4 days (Frederiksen and Lin 2013). Other intraseasonal modes have also been identified using this approach with phase periods between 28 and 60 days (Frederiksen and Frederiksen 1993; Frederiksen 2002; Frederiksen and Lin 2013).

There are multiple theories that arrive at length and time scales consistent with the MJO, each of which adopts different approaches and hence has an alternate explanation of the nature of the MJO. At present there is no consensus on the appropriate framework, nor as to which minimal set of scales are required to reproduce the MJO dynamics. Yano and Tribbia (2017) also make the point that the results of certain successful theories are sensitive to parameter selection. Here we divorce ourselves from any parameter selection issues by processing three-dimensional global reanalysis data to identify which atmospheric scales have the appropriate spatiotemporal MJO properties, and the observed relationships between them.

Žagar and Franzke (2015) undertook a scale decomposition of the MJO present in ERA-Interim on the basis of NMFs. NMFs are the eigenvectors of the linearized primitive equations on a sphere, simultaneously capturing the variability in the horizontal velocity field and geopotential height, with the eigenvalues being the temporal period of the mode. The NMFs are characterized by a zonal wavenumber k, meridional wavenumber n, and vertical mode number m, which are in turn decomposed into Rossby waves and westerly and easterly traveling inertial gravity waves. Žagar and Franzke (2015) calculated the MJO-associated spectrum of NMF variance using with the RMM index. They found the largest percentage of the global variance is in the n = 1 Rossby mode of baroclinic structure with vertical modes m > 3. Using a similar approach Castanheira and Marques (2015) isolated the MJO from ERA-Interim via a cross-spectral analysis between the NMF decomposed k = 1 and n = 1 Rossby wave, and the OLR field. However, in isolation (i.e., not interacting with any other modes nor itself), Rossby waves travel westward. In addition, the eigenvalues associated with this mode indicates that the phase periods are too short to be that of the MJO. While the dominant projection of the MJO is onto this particular mode, the dynamics cannot be described by it alone. However, we propose that the pertinent dynamics can be described via its interaction with other NMFs.

In the present paper we extend the previous NMF applications and show that there are smaller-scale vertical modes that do have the appropriate phase periods, and other physical properties representative of the MJO. Unlike the Žagar and Franzke (2015) and Castanheira and Marques (2015) studies, the approach presented here does not require the RMM index nor the OLR field to isolate the MJO. We do, however, compare our results to the RMM index and also generate composites of the OLR field as means of validation. The NMF decomposition presented within is not intended to be a real-time index of the MJO as in Wheeler and Hendon (2004). Instead the intention is to use the NMFs to further understand the three-dimensional nature of the MJO, and its multiscale interactions.

The manuscript is organized as follows. In section 2 we outline the NMF theory, with the NMFs themselves calculated in section 3. Candidate NMFs representative of the MJO are determined on the basis of their temporal and spatial properties. The NMFs are then projected onto the daily fields of the Japanese 55-year Reanalysis (JRA-55; Kobayashi et al. 2015) with the spectral decomposition of the variance presented in section 4. A cross-spectral analysis is undertaken in section 5 to infer the potential for interactions between the pertinent modes. In section 6, one of the MJO-like NMF modes is used to produce three-dimensional phase averages. This same mode is then used to composite large and persistent MJO events in section 7. This analysis demonstrates the propagation of the velocity potential, and oscillation of the OLR and precipitation fields with the appropriate intraseasonal period. The temporal evolution of a combination of NMF modes are shown to be highly correlated with the RMM index in section 8. Finally, concluding remarks are made in section 9.

2. Normal mode theory

NMFs decompose the flow by vertical scale via vertical structure functions (VSFs), and horizontal scale and mode type via horizontal structure functions (HSFs). The linearized equations of motion from which the NMFs are derived are described below, followed by the two eigenvalue problems (EVPs) required to calculate the VSFs and HSFs. As presented in detail in Kasahara and Puri (1981) and Žagar et al. (2015), the NMFs are derived from the linearized hydrostatic baroclinic primitive equations of the atmosphere on a sphere. The equations are given by
ut2Ωsin(ϕ)υ=gacosϕhλ,
υt+2Ωsin(ϕ)u=gahϕ,and
t{σ[gσRΓ0(σ)hσ]}=V,
where g = 9.81 m s−2 is gravity, t is time, λ is longitude, ϕ is latitude, and the vertical coordinate σ=p/ps, where p is the pressure and ps the surface pressure. These equations govern the linear evolution of the state vector [u,υ,h] in a zero-flow climate, where u and υ are the longitudinal and meridional velocity components, respectively. The modified geopotential height h=z+RT0(σ)ln(ps)/g, where z is the standard geopotential height, T0(σ) is the horizontally averaged temperature at a given σ level, and R = 287 J kg−1 K−1 is the gas constant in air. Note, z is a function of both temperature and specific humidity. The static stability profile Γ0(σ) is given by
Γ0(σ)=RT0(σ)cpσdT0(σ)dσ,
where cp = 1004.6 J kg−1 K−1 is the specific heat of air at constant pressure. Equations (1)(3) are augmented with the surface boundary condition of
hσ+Γ0T0h=0,
at σ=1, and the top boundary condition of
σhσ=0,
at σ=0.
The vertical and horizontal variations of the state vector are assumed separable according to
[u,υ,h]T(λ,ϕ,σ,t)=[u,υ,h]T(λ,ϕ,t)×G(σ),
where G(σ) is a function of only the vertical coordinate. Substituting (7) into (1)(3) returns two EVPs. The first is that of the VSF given by
ddσ[σSdG(σ)dσ]=HDG(σ),
where S(σ)=RΓ0(σ)/(gH), and H=8km is a scaling factor. We use the notation Gm(σ) to refer to the mth VSF (i.e., eigenvector), and Dm the associated equivalent height (i.e., eigenvalue). Using the decomposition in (7), the surface boundary condition in (5) applied at σ=1 becomes
dG(σ)dσ+Γ0T0G(σ)=0.
The top boundary condition becomes
σdG(σ)dσ=0,
which is applied at some small value of σ=σT. This eigenvalue problem has the property that all eigenvalues (i.e., equivalent heights) are positive. Note that this boundary condition is applied at σT as opposed to precisely zero, as a means of avoiding the issue of the VSF EVP becoming singular. This means that we are not solving the exact analytical form of the equations presented in (8)(10). However, Žagar et al. (2009) demonstrates that small differences in the location of the model top for the given vertical discretization and static stability profile have no significant impact on the VSFs.
The second EVP is that of the HSF that require Dm as an input. It is given by
t˜Wm+LmWm=0,
where Wm(λ,ϕ,t˜)=(u˜m,υ˜m,h˜m)T, with nondimensional quantities u˜m=um/gDm, υ˜m=υm/gDm, h˜m=hm/Dm, and t˜=2Ωt. The linear operator Lm is given by
Lm=γm[0sin(ϕ)/γm1cosϕλsin(ϕ)/γm0ϕ1cosϕλtanϕ+ϕ0],
where γm=gDm/(2aΩ). The solutions to (11) take the form
Wm(λ,ϕ,t˜)=Hknmq(λ,ϕ)eiνknmqt˜,
where Hknmq(λ,ϕ) represent the HSFs (i.e., eigenvectors) with zonal wavenumber k, meridional index n, vertical index m, and mode type q. The associated dimensionless frequency (i.e., eigenvalue) is νknmq. Separation of variables about the periodic longitudinal direction produces
Hknmq(λ,ϕ)=Θknmq(ϕ)eikλ,
where Θknmq(ϕ)=[Uknmq(ϕ),iVknmq(ϕ),Zknmq(ϕ)], with Uknmq(ϕ), Vknmq(ϕ), and Zknmq(ϕ) meridionally dependent profiles of the zonal velocity, meridional velocity, and modified geopotential heights, respectively.

Since there are three components in the state vector, there are three eigensolutions (or mode types, q) per scale triplet (k,n,m). One of the solutions is a low-frequency Rossby wave–like balanced (BAL) mode. For even n, Zknmq and Uknmq are antisymmetric about the equator, while Vknmq is symmetric. For odd n, the symmetry properties are swapped. The remaining two solutions are high-frequency inertial gravity (IG) waves, with symmetry properties swapped in comparison to the BAL modes. One solution is an eastward-propagating IG wave (EIG), and the other a westward-propagating IG wave (WIG). We use the index q of values (1, 2, 3), to refer to the mode types (EIG, WIG, BAL). Throughout the manuscript we use the subscript (k,n,Dm) to refer to the horizontal and vertical structure of a given mode class. For example the EIG mode of indices (k,n,m)=(1,0,28), is denoted by EIG(k,n,Dm)=EIG(1,0,D28)=EIG(1,0,7m).

3. Normal mode functions of the Japanese 55-year Reanalysis

The VSF and HSF modes derived in the previous section are calculated here using the MODES code developed and described in Žagar et al. (2015). Upon defining the vertical discretization on the terrain-following coordinate levels σ, the only input data required for the calculation of the VSF modes are the static stability profiles Γ0(σ). The σ grid is defined such that for a nominal surface pressure of 1013 hPa, there are 7 levels below 800 hPa, 6 levels between 800 and 500 hPa, 13 levels from 500 to 100 hPa, 9 levels between 100 and 10 hPa, and 8 levels from 10 to 0.5 hPa, for a total of 43 levels.

Daily temperature, wind and moisture fields of JRA-55 from 1 January 1958 to 31 December 2016 are interpolated to the above-defined σ levels. JRA-55 temperature fields are first formed using 37 isobaric levels defined from 1 to 1000 hPa, and augmented by the surface temperature. These fields are then interpolated onto the above-defined σ grid, and averaged across each σ level to produce T0(σ,t). The daily static stability profiles Γ0(σ,t) are then calculated according to (4). The Γ0(σ,t) profiles are then time averaged to produce Γ0(σ), which is illustrated in Fig. 1a. This profile is everywhere positive and hence stable.

Fig. 1.
Fig. 1.

Vertical structure function eigensolution: (a) static stability profile vs an indicative pressure σ×p0, where p0 = 1013 hPa; (b) equivalent height Dm, or eigenvalues of the VSF EVP; and (c) a subset of vertical modes, or eigenvectors of the VSF EVP.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The eigenvalues of the VSF EVP in (8) are the equivalent heights Dm and illustrated in Fig. 1b. The equivalent heights decrease monotonically with vertical mode index m. The mth VSF has m1 zero crossings. The m=1 mode has no zero crossings and is hence referred to as the barotropic mode, here with an equivalent height of D1 = 10 km. The m=2 mode has one crossing and consequently referred to as the first baroclinic mode, of equivalent height D2 = 7 km. These modes are illustrated in Fig. 1c, along with the m=8, m=15, and m=28 VSFs, with respective equivalent heights of D8 = 324 m, D15 = 92 m, and D28 = 7 m. As the equivalent height decreases (or equivalently m increases) the structures of the VSFs are also concentrated closer to the surface.

The HSF EVP requires Dm as an input, and hence each vertical mode m has a unique set of HSFs and associated eigenvalues νknmq. HSF EVP solutions are calculated for Dm > 1 m. The periods (1/νknmq) calculated from the HSF EVP are illustrated in Fig. 2a for the vertical mode with an equivalent height of 7 m. We select this equivalent height since the EIG mode with k=1 and n=0 has a time scale central to the intraseasonal band of 56 days. The EIG modes (blue dots) are the only easterly propagating modes, having positive k. Both the WIG (red dots) and BAL (black dots) modes have a negative k and hence propagate westerly. The bold dots represent the n=1 meridional index for each of these mode types. The separation in time scales between the IG and BAL modes also decreases as the equivalent heights decrease, or equivalently as m increases (Žagar et al. 2015). As n increases, the time scale of the IG modes decrease, while that of the BAL modes increase. Two other subclasses are also explicitly illustrated. The mixed Rossby–gravity (MRG) modes are by definition the BAL modes with n=0 (for all k and m). Likewise the Kelvin wave are the EIG modes with n=0. Kelvin waves have the particular property that their phase speed (νknmq/k) is equivalent to their group velocity (νkmnq/k).

Fig. 2.
Fig. 2.

Horizontal structure function eigensolutions: (a) time scales 1/νknmq of the HSF for vertical mode of equivalent height = 7 m for balanced (BAL; large dots: n=1; small dots: n>1), easterly propagating inertial gravity waves (EIG; large dots: n=1; small dots: n>1), westerly propagating inertial gravity waves (WIG; large dots: n=1; small dots for all other n), mixed Rossby–gravity (MRG; equivalent to BAL: n=0) and Kelvin waves (KW; equivalent to EIG: n=0); (b) time scales 1/νknmq of the HSF for EIG with (k,n)=(1,0), WIG with (k,n)=(1,0), and BAL with (k,n)=(1,1) vs the equivalent height of each vertical mode; and (c) the longitudinal velocity component Uknmq(ϕ) of the horizontal structure functions for EIG modes with (k,n,Dm)=(1,0,7m) and (1,0,324m), and BAL modes with (k,n,Dm)=(1,1,92m) and (1,1,324m).

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

At this stage we look to identify a set of modes representative of the MJO. To restate, the MJO is an eastward-propagating tropics centric intraseasonal oscillation with horizontal velocity components of dominant zonal wavenumber k=1, and whose zonal velocity is also predominantly symmetric about the equator. For the inertial gravity waves, the n=0 modes are tropics centric with their magnitude decreasing monotonically with distance from the equator, and also have the appropriate symmetry properties. Hence the horizontal-scale pair of (k,n)=(1,0) is an appropriate starting point for the EIG and WIG modes. For the Rossby waves, the n=1 modes have the appropriate symmetry properties, hence the (k,n)=(1,1) is the appropriate horizontal-scale pair. The time scales of the associated Kelvin wave (EIG with n=0), WIG and BAL modes are illustrated as a function Dm in Fig. 2b. This figure demonstrates power-law relationships between the time scales and equivalent depths. Both the Kelvin wave and BAL modes contains temporal scales within the intraseasonal time-scale band. Specifically the EIG(1,0,Dm) modes, or Kelvin waves, with Dm of order 10 m have time scales between 30 and 90 days. As mentioned above, the eigenvalue associated with EIG(1,0,7m) has a time scale of 56 days and is central to this intraseasonal range. The balanced modes, BAL(1,1,Dm), have phase periods within intraseasonal time-scale band for Dm of order 100 m, with BAL(1,1,92m) having a time scale of 46 days. The linear dispersion relationships are such that the Kelvin waves propagate eastward, while the BAL modes propagate westward. In order for the BAL modes to propagate eastward like the MJO (as later observed in the reanalysis), we hypothesize that some form of interaction with these modes is required to modify their effective dispersion relations.

The abovementioned NMFs have a meridional structure appropriate for the MJO as indicated by the zonal velocity component of the HSFs. Four HSFs are illustrated in Fig. 2c: the Kelvin wave with a 56-day phase period, EIG(1,0,7m); the most energetic Kelvin wave, EIG(1,0,324m); the Rossby wave with a 46-day phase period, BAL(1,1,92m); and an energetic Rossby wave, BAL(1,1,324m). The energy content of the NMFs are discussed in the following section. The zonal velocity of the abovementioned modes are tropics centric with their maximum magnitude located at the equator, and are also symmetric about the equator. Note, since these profiles are the eigenvectors of the HSF EVP, the sign of the modes is arbitrary. The EIG(1,0,7m) mode has significant amplitudes between 15°S and 15°N. The EIG(1,0,324m) mode is far broader, and is nonzero within extratropics from 50°S to 50°N. The BAL(1,1,92m) mode changes sign at 10° either side of the equator, with secondary peaks at 15°S and 15°N. The BAL(1,1,324m) mode has the same topological features as BAL(1,1,92m), but is broader and extends farther into the extratropics. In general, the inertial gravity waves with n=0 and BAL modes with n=1 become more equatorially trapped as m increases as well as with increasing k.

4. Scale and mode type decomposition

The contribution of the VSFs and HSFs to the daily JRA-55 fields are calculated using the NMF orthogonality properties presented in Žagar et al. (2015). Physical space fields are decomposed according to
[u(λ,ϕ,σ,t)υ(λ,ϕ,σ,t)h(λ,ϕ,σ,t)]=m=1MSmGm(σ)Xm(λ,ϕ,t).
The total number of vertical modes is M, and
Sm=[gDm000gDm000Dm].
The dimensionless horizontal coefficients vector is given by
Xm(λ,ϕ,t)=k=KKn=0Nq=13χknmq(t)Hknmq(λ,ϕ),
where K is the zonal truncation wavenumber, N is the meridional truncation wavenumber, and q takes on the values (1, 2, 3), referring to mode types (EIG, WIG, BAL). Note, this notation is different from Žagar et al. (2015) with N one-third of the size, because of the introduction of index q to explicitly represent the BAL, EIG, and WIG motions in the summation. The scalar complex expansion coefficients χknmq are then calculated according to
χknmq(t)=12π02π11Xm(λ,ϕ,t)[Hknmq]*dμdλ,
where μ=sinϕ and the asterisk is the complex conjugate operation. The energy of mode type q for a given scale triplet is Eknmq=χknmq(t)χknmq(t)*Dmg/2, where the angle brackets denote time averaging. The energy spectra of mode type q, as a function of a particular index (e.g., k), is given by the sum of Eknmq over the remaining indices (e.g., n and m). The total energy is the sum of energy from all mode classes.

The expansion coefficients are calculated for the daily JRA-55 fields from 1 January 1958 to 31 December 2016. All of the analysis throughout the paper is undertaken over this period. The time-averaged zonal wavenumber spectrum of each mode type and the total energy is illustrated in Fig. 3a. The total energy exhibits a k3 decay consistent with the constant enstrophy flux inertial range present in two-dimensional turbulence. When broken down into its components the BAL spectra is steeper than both EIG and WIG. The BAL spectra is the dominant component in the large scales (small wavenumbers). The IG spectra, given by the sum of EIG and WIG spectra, has more energy from k40 onward.

Fig. 3.
Fig. 3.

Time-averaged spectral contribution from each of the NMFs: (a) zonal wavenumber spectra of total energy (i.e., potential and kinetic) summed over n and m decomposed into EIG, WIG, and BAL components, with the dashed black line representing the k3 decay rate; (b) vertical mode spectra of total nonzonal energy summed over k and n as a function of equivalent height; (c) as in (b), but for only the (k,n)=(1,1) horizontal mode pair; and (d) as in (b), but for only the (k,n)=(1,0) horizontal mode pair. Legend and vertical scale in (a) is applicable to all plots.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The wave energy (i.e., excluding the k=0 component) in each vertical mode is illustrated in Fig. 3b, and indicates that the BAL component is dominant for all equivalent heights. Of these, the barotropic mode (Dm = D1 = 10 km) is the most energetic. The BAL modes are again dominant when considering only the (k,n)=(1,1) horizontal-scale pair, as illustrated in Fig. 3c. The first baroclinic mode (Dm = D2 = 7 km) has the greatest energy contribution for k=1 balanced modes, with a secondary peak centered at Dm = 324 m. The latter is the BAL(1,1,324m) mode discussed in section 3. The BAL(1,1,92m) mode with an eigenvalue-determined time scale central to the intraseasonal band has an order of magnitude less energy. When considering only the (k,n)=(1,0) horizontal-scale pair the EIG component is now dominant for all m, as illustrated in Fig. 3d. The most energy occurs at Dm = 324 m, which is associated with the EIG(1,0,324m) mode discussed in the previous section. The EIG(1,0,7m) mode, with a time scale of 56 days, has an order of magnitude less energy as compared to EIG(1,0,324m).

5. Cross-spectral analysis

From the analysis in section 3, EIG(1,0,7m) is the only mode with an MJO-like horizontal structure, eastward propagation, and an eigenvalue-determined phase period central to the intraseasonal time-scale band. However, as demonstrated in section 4 this particular mode does not have a dominant contribution to the energy. Of the EIG(1,0,Dm) modes, Dm = 324 m is the most energetic, but has an eigenvalue-derived phase period too short to be associated with the MJO. The spectral structure of energy contained within these scales of EIG modes is illustrated via the variance distribution in the (k,f) plane, where f is the temporal frequency. Fourier transforms are calculated for the expansion coefficient [χknmq(t)] anomalies about the yearly cycle, scaled by Dmg/2 to represent the energy content. The power spectral density (PSD) of the EIG modes with n=0, Dm = 324 m, and 1k10 is illustrated in Fig. 4a. Note that only the positive frequencies are visualized. It indicates that there are in fact two main propagation speeds: one fast and one slow. The fast component is associated with the classical view of a Kelvin wave and is indicated by the white arrow overlaid in the contour plot. The slow component is concentrated within time scales longer than 30 days, as indicated by the white horizontal dashed line, and is presumably associated with the MJO. These features are topologically consistent with the OLR spectra in Wheeler and Kiladis (1999). The PSD of the EIG mode with n=0 and Dm = 7 m is illustrated in Fig. 4b, and indicates that the slow component is dominant.

Fig. 4.
Fig. 4.

Spectral coherence properties of the NMFs: (a) power spectral density in the zonal wavenumber (k)–frequency (f) plane of EIG(k,0,324m), with the white arrow indicating the fast Kelvin wave component; (b) as in (a), but for EIG(k,0,7m); (c) coherence (squared correlation in spectral space) in the (k,f) plane between EIG(k,0,324m) and EIG(1,0,7m); (d) coherence output (correlated variance in spectral space) in the (k,f) plane of EIG(k,0,324m) with respect to EIG(1,0,7m); (e) power spectral density of BAL(k,1,324m); (f) coherence output of BAL(k,1,8) with respect to EIG(1,0,7m); (g) power spectral density of BAL(k,1,92m); (h) coherence output of BAL(k,1,92m) with respect to EIG(1,0,7m); (i) coherence output of EIG(1,0,m) with respect to EIG(1,0,7m) for all vertical modes represented by their equivalent height; and (j) coherence output of BAL(1,1,m) with respect to EIG(1,0,7m) for all vertical modes. Vertical axes in (a), (e), and (i) are applicable to all plots. The horizontal dashed vertical line in all plots indicates a time scale of 30 days.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The covariability and potential for interaction between EIG(1,0,7m) with the correct phase period, and the EIG(k,0,324m) high energy modes, is determined via their spectral coherence. The magnitude-squared coherence (referred to simply as coherence from this point onward) is given by the squared magnitude of the cross-spectral density between two complex time series divided by the PSDs of both time series. Figure 4c indicates that only the slow component of these modes are highly coherent. The coherence output in Fig. 4d is defined as the coherence in Fig. 4c multiplied by the PSD in Fig. 4a. The fast components have largely been filtered out. We now apply the same filtering to the energetic balanced mode, BAL(k,1,324m), and also the balanced mode with appropriate phase period BAL(k,1,92m). For BAL(k,1,324m), the PSD is illustrated in Fig. 4e, and coherence output in Fig. 4f. The PSD and coherence output of BAL(k,1,92m) are illustrated in Fig. 4g and Fig. 4h, respectively. In both cases, the coherence output is shown to retain only the slow components, as previously demonstrated for the EIG(k,0,324m) mode. Note, Figs. 4a and 4d–4h all have the same color bar scale.

The coherence output with respect to EIG(1,0,7m) is now calculated for the EIG(1,0,Dm) modes for all equivalent heights, and illustrated in Fig. 4i. This figure indicates that the slow MJO time scales of the EIG(1,0,Dm) modes are the most coherent with EIG(1,0,7m). For time scales shorter than 30 days, the coherence output decreases with decreasing time scale (or increasing frequency). The coherence output is organized into two clusters of equivalent heights, the first ranging between 1 km and 100 m and the second 100 m and below. The maxima in these two clusters are associated with the EIG(1,0,324m) and EIG(1,0,7m) modes, respectively.

The associated coherence output for BAL(1,1,Dm) is illustrated in Fig. 4j. Again, the intraseasonal time scales are the most coherent, with the coherence output decreasing as the frequency increases. The coherence output is again organized into two clusters of equivalent heights, spanning the same ranges observed in the EIG(1,0,Dm) case. The first cluster ranges from 1 km to 100 m, is aligned with the secondary peak in energy in Fig. 3c, and has a local maximum at BAL(1,1,324m). In Žagar and Franzke (2015) a mode of similar Dm was shown to have components correlated with the RMM index of Wheeler and Hendon (2004) with more variance than any of the other NMFs. The second cluster is below 100 m, and has a maxima at BAL(1,1,92m) modes. This cluster could be further split into two considering there are two local maxima. However, we treat them as one group since they have many consistent properties, including similar horizontal structure, time scales within the intraseasonal band, and similar energy levels. For completeness we have also calculated the coherence output for the WIG(1,0,Dm) modes, and found that they are two orders of magnitude lower than the EIG(1,0,Dm) and BAL(1,0,Dm) modes. This suggests that the WIG modes play a negligible role in the evolution of the MJO.

In summary four pertinent mode clusters are identified as interacting with each other, and having properties compatible with the MJO. The cluster centered on the EIG(1,0,7m) mode has a horizontal structure that is equatorially trapped within 15°S–15°N, and a phase period as determined from the linearized equations of motion to be appropriate for the MJO. This mode has all of the necessary temporal and horizontal properties. In itself, it has low energy, but is coherent with the intraseasonal components of the following more energetic modes. The energetic EIG(1,0,324m) mode cluster has a broader meridional extent, with the HSFs nonzero between approximately 50°S and 50°N. The BAL(1,1,92m) group has an appropriate time scale, but in isolation propagates westward. The BAL(1,1,324m) cluster has a broader meridional extent, and is the most energetic of the aforementioned clusters. However, in isolation this mode propagates too quickly and in the opposite direction as compared to the MJO.

We hypothesize that the nonlinear interaction between each of these mode four groups ensures that a component of the energetic modes propagate eastward with an intraseasonal phase period. A minimal model of the MJO would require the interacting triads formed by at least one mode central to each of the abovementioned clusters. The reduced-order model equations would be a set of ordinary differential equations (ODEs), derived from the projection of these orthogonal modes onto the nonlinear equations of motion, akin to what is done in the reduced-order modeling of canonical fluid flows (Noack et al. 2003; Kitsios 2010). These ODEs would be solved numerically to return the time-varying contribution of each of these modes to the MJO dynamics. To refine the representation of the MJO, the projection could be systematically expanded to include additional equivalent heights radiating out from the aforementioned four modes, and also finer-scale longitudinal waves. This framework incorporates the interactions between multiple vertical scales, and also between the tropics and extratropics via the varying meridional extents of the HSFs. Since EIG(1,0,7m) has all of the necessary temporal properties and horizontal structure, we will use this mode to extract the MJO in the following analysis.

6. Phase averages

To provide further details on the three-dimensional spatial structure of the MJO, phase averages are constructed using EIG(1,0,7m). Akin to the process undertaken in Wheeler and Hendon (2004) we calculate phase angles from the real and imaginary components of the expansion coefficients χknmq(t), in place of their principal components RMM1 and RMM2. The phases are split into eight discrete octants of indices j = 1–8, with phase angle ranges defined between 180°+45°(j1) and 135°+45°(j1). Anomalous daily fields are first calculated about the yearly cycle, and then bandpass filtered to retain the intraseasonal time scales. The anomalous fields associated with the dates in each phase octant are then averaged. Phase average anomalies are then calculated by removing the time-averaged field. While the EIG(1,0,7m) mode is used to identify the dates associated with each phase octant, since we average through the daily fields all NMFs are essentially contributing.

The phase average anomalies are illustrated by three-dimensional isosurfaces of the velocity potential overlaid with contours of OLR in Fig. 5. To account for the change in magnitude of the velocity potential with level, the fields are normalized by the standard deviations on each level. The dark and light gray structures represent isosurfaces two standard deviations above and below the mean, respectively. From phase 1 to phase 8, this figure demonstrates a consistent eastward propagation of the velocity potential structures, and a dipole type behavior of the OLR field dominant over the Maritime Continent. The phases are also in line with those presented in Wheeler and Hendon (2004). In phase 1 convection from the previous MJO event (positive convection anomaly) is centered over the east of the Maritime Continent. Subsequent phases see convection ramping up in magnitude initially over the east of Africa. There is a strong coupling between the OLR and velocity potential fields in phase 5 with the interface between positive and negative anomalies in both fields occurring at approximately 90°E. The imprint of the upper-level velocity potential is also evident outside of the Maritime Continent in the OLR, with anomalies of like sign aligned.

Fig. 5.
Fig. 5.

Anomalies of phase-averaged OLR and velocity potential fields, calculated using the EIG (k,n,Dm)=(1,0,7m) mode of phases angles: (a) phase 1 from −180° to −135°, (b) phase 2 from −135° to −90°, (c) phase 3 from −90° to −45°, (d) phase 4 from −45° to 0°, (e) phase 5 from 0° to 45°, (f) phase 6 from 45° to 90°, (g) phase 7 from 90° to 135°, and (h) phase 8 from 135° to 180°. OLR contours range from −4 (blue) to 4 W m−2 (yellow). Velocity potential isosurfaces are shaded for two standard deviations below the mean (light gray) and two standard deviations above the mean (dark gray). The vertical axis, σ×p0, is an indicative pressure where p0 = 1013 hPa.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

The velocity potential also changes sign in the vertical, which is representative of convection being associated with upper-level divergence (positive velocity potential) and lower-level convergence (negative velocity potential). This is again consistent with the results of Wheeler and Hendon (2004), in which the first two EOFs have their 200-hPa zonal wind component of opposite sign to the 850-hPa zonal wind.

7. Composite fields

By construction phase averages, such as those presented in the previous section, contain only spatial structure and no temporal information. A more stringent test is to determine if the spatial and temporal properties of fields composited on the basis of EIG(1,0,7m) are representative of the MJO. The expansion coefficients χknmq are bandpass filtered to retain only the intraseasonal time scales, from which phase angles are then calculated. The phase angle time series is split into a series of events, demarcated by a discontinuous change in phase angle. A discontinuity is defined as when the phase angle changes by more than 90° in one day. To select the strong and persistent events, we retain only those that propagate for at least 270° and have an average magnitude in the upper quartile. Over the 1958–2016 period, 96 events are identified, with on average 1.6 MJO events identified per year. The alignment of each event is defined using only the phase information, and hence not explicitly dependent upon the magnitude. Day 0 of each event is defined as the day in which a least squares fit to the phase angle passes through 0°. This is by definition associated with the transition from phase 4 to phase 5 as presented in section 6.

The magnitude of the selected events is illustrated as the faint gray lines in Fig. 6a. The ensemble mean and standard deviation is calculated for each day offset from day 0. The solid bold line in this figure is the ensemble-averaged magnitude, with the dashed lines showing the ensemble average plus and minus one ensemble standard deviation. There is a local maximum in magnitude in the vicinity of day 0, which indicates that these events are stronger than the background variability of this mode. The ensemble-averaged phase angles are illustrated in Fig. 6b, and indicate a persistent eastward propagation.

Fig. 6.
Fig. 6.

Composite-averaged normal mode function complex expansion coefficients χknmq: (a) magnitude of the EIG mode triplet (k,n,Dm)=(1,0,7m) relative to day 0, illustrating each individual event (faint lines), the ensemble mean (thick solid line) and plus and minus one ensemble standard deviation (thick dashed lines); and (b) as in (a), but for the phase angle.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

To confirm the spatial structure, ensemble-averaged composites of velocity potential, OLR, and precipitation are calculated for the days leading up to and following day 0. First, anomalous daily fields are calculated to have the yearly cycle removed and also ENSO time scales filtered out. The fields of dates associated with day 0 are averaged together to produce an ensemble-averaged day 0. Likewise, the same process is applied to produce ensemble-averaged composites for the nonzero day shifts. The ensemble-averaged velocity potential at a pressure level of 200 hPa is illustrated in Fig. 7 for a selection of days from 20 days before to 20 days after. There is a clear propagation of the structures over this 41-day period. The ensemble-averaged OLR is illustrated in Fig. 8 over the same relative day range. The OLR fields demonstrate a dipole type oscillation over the Maritime Continent. Composites of precipitation are illustrated in Fig. 9, which demonstrates that this field has structures very similar to those observed in the OLR fields, but of opposite sign. The observed strong anticorrelation between these two fields is consistent with the view that anomalously low (high) OLR leads to high (low) temperatures and increased (decreased) evaporation and water vapor concentration, which finally produces anomalously high (low) rainfall. By design the structure at day 0, of approximate phase angle 0°, is similar to that of phase 5 illustrated in Fig. 5e. Consistent with Fig. 6a, the magnitude of the velocity potential, OLR, and precipitation fields all decrease the further in time they are away from day 0.

Fig. 7.
Fig. 7.

Composite-averaged velocity potential field at 200 hPa based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −2.5 × 106 (blue) to 2.5 × 106 m2 s−1 (yellow). Minimum (min) and maximum values (max) over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Fig. 8.
Fig. 8.

Composite-averaged outgoing longwave radiation field based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −8 (blue) to 8 W m−2 (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

Fig. 9.
Fig. 9.

Composite-averaged precipitation field based upon the EIG (k,n,Dm)=(1,0,7m) mode for days relative to day 0: (a) 20 days before, (b) 15 days before, (c) 10 days before, (d) 5 days before, (e) day 0, (f) 5 days after, (g) 10 days after, (h) 15 days after, and (i) 20 days after. Contour values in all plots range from −2 (blue) to 2 mm day−1 (yellow). Min and max values over the illustrated domain for each plot are listed in their respective titles.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

8. Characterization of individual MJO events

We next contrast the NMF MJO characterization with the RMM index of Wheeler and Hendon (2004). The evolution of the NMF energy Eknmq is illustrated for the modes central to the four clusters identified in section 5. To recap, these clusters in order of decreasing energy content are the energetic Rossby waves, the energetic EIG waves, the Rossby waves of intraseasonal phase period, and the EIG waves of appropriate phase period. The modes central to each of the above groups are BAL(1,1,324m), EIG(1,0,324m), BAL(1,1,92m), and EIG(1,0,7m), respectively.

Akin to what is done in Wheeler and Hendon (2004), 91-day centered moving averages are calculated. The square root of these moving averages are illustrated in Fig. 10a for the period from 2000 to 2016. The BAL(1,1,324m) mode is indicated by the red dotted line. These coefficients do not appear to follow the RMM index particularly well. It has a correlation coefficient with RMM of 0.42 over the entire period from 1958 to 2016. However, the agreement improves significantly when filtered to retain only the temporal scales that have a coherence with EIG(1,0,7m) greater than 0.5. The filtered time series is illustrated by the solid red line, and has a correlation coefficient with RMM of 0.78. In effect, filtering on the basis of EIG(1,0,7m) removes all negative-frequency contributions (i.e., westward-propagating waves), low-frequency time scales, and high-frequency weather. The following NMFs, also illustrated in Fig. 10a, are discussed in order of decreasing energy content and correlation. The EIG(1,0,324m) mode is filtered in the aforementioned manner and has a correlation with RMM of 0.82. Likewise, the filtered BAL(1,1,92m) mode has a correlation of 0.62. Finally the unfiltered EIG(1,0,7m) mode has a correlation with RMM of 0.32. This mode has the least energy contribution, but is required to isolate the MJO component in the more energetic modes.

Fig. 10.
Fig. 10.

MJO time series: (a) square root of the 91-day centered moving averages (MA) of the RMM index (RMM12 + RMM22), the NMF energy Eknmq of unfiltered modes BAL(1,1,324m) and EIG(1,0,7m), and coherence-filtered modes BAL(1,1,324m), EIG(1,0,324m), and BAL(1,1,92m), with the yellow boxes highlighting the time periods illustrated in the remaining plots; (b) comparison of the phase angles of RMM and the coherence-filtered BAL(1,1,324m) mode for an MJO event in November 2002; (c) as in (b), for January 2006; (d) as in (b), for February 2008; (e) as in (b), for March 2012; (f) RMM in phase space for the March 2012 event; and (g) coherence-filtered BAL(1,1,324m) mode in phase space for the March 2012 event. In (e)–(g), the circle and square markers indicate the start and end of the period of interest, respectively. The legend in (a) is applicable to all plots.

Citation: Journal of the Atmospheric Sciences 76, 8; 10.1175/JAS-D-18-0197.1

From the perspective of the vertical structure, the high energy modes, BAL(1,1,324m) and EIG(1,0,324m), have an MJO-appropriate baroclinic structure for altitudes lower than 100 hPa, but an additional six zero crossings within the stratosphere. This is illustrated by the associated VSF in Fig. 1c. The VSF associated with the EIG(1,0,7m) mode, however, is essentially zero above 100 hPa, also illustrated in Fig. 1c. Filtering on the basis of EIG(1,0,7m) removes the higher-altitude, non-MJO-related dynamics.

We have identified four MJO events defined as local maxima of the RMM index, which in both order of increasing magnitude and chronological order are in November 2002, January 2006, February 2008, and March 2012. Note these dates also correspond to local maxima in our BAL(1,1,324m) mode. The event in March 2012 is the largest-magnitude event within the 2000–16 period. Each of these events are highlighted by the yellow box in Fig. 10a. During the November 2002 event, Fig. 10b illustrates that the phase angle for the coherence-filtered BAL(1,1,324m) mode and the RMM index and exhibit very similar phase propagation. This is also true for the phase propagation of the January 2006, February 2008, and March 2012 events illustrated in Figs. 10c, 10d, and 10e, respectively. For the March 2012 event, the phase portrait of the RMM index illustrated in Fig. 10f demonstrates a counterclockwise phase progression of relatively consistent magnitude. The associated phase portrait of the coherence-filtered BAL(1,1,324m) mode is illustrated in Fig. 10g, and demonstrates a similar counterclockwise evolution. In summary both the magnitude and the phase of the coherence-filtered energetic BAL(1,1,324m) mode are very similar to those of the widely accepted RMM index of Wheeler and Hendon (2004).

9. Concluding remarks

The MJO is described here as a combination of interacting balanced (BAL) modes (i.e., Rossby waves) and eastward-propagating inertial gravity (EIG) waves of varying vertical scale and meridional extent. The MJO was isolated from JRA-55 via a normal mode function (NMF) scale and mode class decomposition. The only NMFs that have the appropriate MJO-like horizontal structure are the Rossby wave BAL modes of zonal wavenumber k=1 and meridional index n=1, and the EIG modes of (k,n)=(1,0). EIG modes with n=0 are by definition Kelvin waves. The potential for the nonlinear interactions between these modes was quantified by the coherence output, which identified four distinct interacting clusters of modes spanning various vertical scales. These groups are listed below in order of increasing energy content. The first group consists of Kelvin waves, with a maximum coherence output occurring at an equivalent height of 7 m. The modes in this cluster propagate eastward, have eigenvalue-determined time scales within the intraseasonal band, are equatorially trapped between 15°S and 15°N, but have minimal energy as compared to the following mode groups. This is perhaps dynamically the most important cluster since it is the only one to have all of the MJO-appropriate horizontal and temporal properties. The second group consists of Rossby waves with the coherence output maxima located at an equivalent height equal of 92 m. They also have the appropriate time scales and reasonable energy, but in isolation would propagate westward (as governed by their eigensolution). The third group contains the most energetic Kelvin waves with the equivalent height of 324 m having the highest coherence. These modes also encompass the extratropics with a broader meridional extent from 50°S to 50°N, but have time scales that are too short. The final group is the most energetic and contains Rossby waves with the same equivalent height, and similar meridional extents as the previous group. Their time scales are again too short and also propagate westward.

A cross-spectral analysis indicated that the above mode groups are coherent and presumably interact with one another. The only way that these orthogonal NMFs can interact is through nonlinear triads. We, therefore, hypothesize that the nonlinear interaction between the above mode groups conspire such that on average the energetic modes propagate eastward with an intraseasonal time scale, as observed. In this representation of the MJO a minimal set of interacting triads is formed by the Kelvin waves EIG(k,n,Dm)=EIG(1,0,7m) and EIG(1,0,324m), and the Rossby waves BAL(1,1,92m) and BAL(1,1,324m). A reduced-order model of the MJO could be developed in the future by projecting these modes onto the nonlinear equations of motion, and solving numerically for the time-varying coefficients of these modes. The MJO dynamics could be made more complete by expanding the range of vertical modes included in the projection. Finer-scale longitudinal waves could also be included. This framework captures the interactions within a hierarchy of vertical scales. Interactions between the tropics and extratropics are also included, since the horizontal structure functions extend farther into the extratropics as their equivalent height increases.

On the basis of the EIG(1,0,7m) Kelvin wave, phase averages and composites of large and persistent MJO events were calculated. Both analyses demonstrate the eastward propagation of the velocity potential and the dipole-like oscillation of outgoing longwave radiation and precipitation over the Maritime Continent. The composite analysis also indicates that this reference Kelvin wave isolates the appropriate intraseasonal time scales. The time-scale components of the more energetic modes coherent with EIG(1,0,7m) are also shown to be highly correlated with the accepted MJO index of Wheeler and Hendon (2004).

These findings may in some way explain the influence of vertical resolution on the representation of the MJO in general circulation models (GCMs). Crueger et al. (2013) undertook a meta-analysis of 42 experiments of varying configuration and resolution, including both atmospheric and coupled GCMs. One of their key findings was that eastward propagation was enhanced when vertical and horizontal resolution were increased together, while changing only vertical resolution had a lesser influence. However, as stated in Liess and Bengtsson (2004), the relative lesser importance of the vertical grid could be due to having an inappropriate ratio of vertical to horizontal resolution. Regardless of the quantum of the influence of the vertical grid, we speculate that associated improvements in the representation of the MJO is related to the importance of the EIG(1,0,7m) mode within this NMF framework. This mode has a small equivalent height of 7 m and its vertical structure has oscillations distributed throughout the troposphere—see Fig. 1c. If this mode is not explicitly resolved on the vertical grid, then the mechanism for changing the direction and slowing down the energetic Rossby waves would be broken, and hence the MJO would be poorly represented.

Acknowledgments

The authors were supported by the Australian Commonwealth Scientific and Industrial Research Organisation Decadal Forecasting Project (research.csiro.au/dfp). The authors would also like to acknowledge NCI for providing the computational resources.

REFERENCES

  • Castanheira, J., and C. Marques, 2015: Convectively coupled equatorial-wave diagnosis using three-dimensional normal modes. Quart. J. Roy. Meteor. Soc., 141, 27762792, https://doi.org/10.1002/qj.2563.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chang, C., and H. Lim, 1988: Kelvin wave-CISK: A possible mechanism for the 30–50 day oscillations. J. Atmos. Sci., 45, 17091720, https://doi.org/10.1175/1520-0469(1988)045<1709:KWCAPM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, S., R. Houze Jr., and B. Mapes, 1996: Multiscale variability of deep convection in relation to large-scale circulation in TOGA COARE. J. Atmos. Sci., 53, 13801409, https://doi.org/10.1175/1520-0469(1996)053<1380:MVODCI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Chen, S., A. Majda, and S. Stechmann, 2016: Tropical–extratropical interactions with the MJO skeleton and climatological mean flow. J. Atmos. Sci., 73, 41014116, https://doi.org/10.1175/JAS-D-16-0041.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Crueger, T., B. Stevens, and R. Brokopf, 2013: The Madden–Julian oscillation in ECHAM6 and the introduction of an objective MJO metric. J. Climate, 26, 32413257, https://doi.org/10.1175/JCLI-D-12-00413.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • DeMott, C., N. Klingaman, and S. Woolnough, 2015: Atmosphere-ocean coupled processes in the Madden-Julian oscillation. Rev. Geophys., 53, 10991154, https://doi.org/10.1002/2014RG000478.

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

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., 2002: Genesis of intraseasonal oscillations and equatorial waves. J. Atmos. Sci., 59, 27612781, https://doi.org/10.1175/1520-0469(2002)059<2761:GOIOAE>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., and C. Frederiksen, 1993: Monsoon disturbances, intraseasonal oscillations, teleconnection patterns, blocking, and storm tracks of the global atmosphere during January 1979: Linear theory. J. Atmos. Sci., 50, 13491372, https://doi.org/10.1175/1520-0469(1993)050<1349:MDIOTP>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Frederiksen, J. S., and H. Lin, 2013: Tropical–extratropical interactions of intraseasonal oscillations. J. Atmos. Sci., 70, 31803197, https://doi.org/10.1175/JAS-D-12-0302.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Fuchs, Z., and D. J. Raymond, 2005: Large-scale modes in a rotating atmosphere with radiative–convective instability and WISHE. J. Atmos. Sci., 62, 40844094, https://doi.org/10.1175/JAS3582.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Hendon, H., and M. Salby, 1994: The life cycle of the Madden–Julian oscillation. J. Atmos. Sci., 51, 22252237, https://doi.org/10.1175/1520-0469(1994)051<2225:TLCOTM>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kasahara, A., and K. Puri, 1981: Spectral representation of three-dimensional global data by expansion in normal mode functions. Mon. Wea. Rev., 109, 3751, https://doi.org/10.1175/1520-0493(1981)109<0037:SROTDG>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Kitsios, V., 2010: Recovery of fluid mechanical modes in unsteady separated flows. Ph.D. thesis, University of Melbourne and Université de Poitiers, 259 pp., http://repository.unimelb.edu.au/10187/9088.

  • Kobayashi, A., and Coauthors, 2015: The JRA-55 reanalysis: General specifications and basic characteristics. J. Meteor. Soc. Japan, 93, 548, https://doi.org/10.2151/jmsj.2015-001.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lau, K., and L. Peng, 1987: Origin of low-frequency (intraseasonal) oscillations in the tropical atmosphere. Part I: Basic theory. J. Atmos. Sci., 44, 950972, https://doi.org/10.1175/1520-0469(1987)044<0950:OOLFOI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Liess, S., and L. Bengtsson, 2004: The intraseasonal oscillation in ECHAM4 part II: Sensitivity studies. Climate Dyn., 22, 671688, https://doi.org/10.1007/s00382-004-0407-z.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Lindzen, R., 1974: Wave-CISK in the tropics. J. Atmos. Sci., 31, 156179, https://doi.org/10.1175/1520-0469(1974)031<0156:WCITT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Madden, R. A., and P. R. Julian, 1971: Detection of a 40–50 day oscillation in the zonal wind in the tropical pacific. J. Atmos. Sci., 28, 702708, https://doi.org/10.1175/1520-0469(1971)028<0702:DOADOI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Madden, R. A., and P. R. Julian, 1972: Description of global-scale circulation cells in the tropics with a 40–50 day period. J. Atmos. Sci., 29, 11091123, https://doi.org/10.1175/1520-0469(1972)029<1109:DOGSCC>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, https://doi.org/10.1073/pnas.0903367106.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Nakazawa, T., 1988: Tropical super clusters within intraseasonal variations over the western Pacific. J. Meteor. Soc. Japan, 66, 823836, https://doi.org/10.2151/jmsj1965.66.6_823.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J., and I. Held, 1987: Modeling tropical convergence based on the moist static energy budget. Mon. Wea. Rev., 115, 312, https://doi.org/10.1175/1520-0493(1987)115<0003:MTCBOT>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Neelin, J., and J.-Y. Yu, 1994: Modes of tropical variability under convective adjustment and the Madden–Julian oscillation. Part I: Analytical theory. J. Atmos. Sci., 51, 18761984, https://doi.org/10.1175/1520-0469(1994)051<1876:MOTVUC>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Noack, B., K. Afanasiev, M. Morzynski, G. Tadmor, and F. Thiele, 2003: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech., 497, 335363, https://doi.org/10.1017/S0022112003006694.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., and Ž. Fuchs, 2009: Moisture modes and the Madden–Julian oscillation. J. Atmos. Sci., 22, 30313046, https://doi.org/10.1175/2008JCLI2739.1.

    • Search Google Scholar
    • Export Citation
  • Rui, H., and B. Wang, 1990: Development characteristics and dynamic structure of tropical intraseasonal convection anomalies. J. Atmos. Sci., 47, 357379, https://doi.org/10.1175/1520-0469(1990)047<0357:DCADSO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Sobel, A., and E. Maloney, 2012: Moisture modes and the eastward propagation of the MJO. J. Atmos. Sci., 69, 16911705, https://doi.org/10.1175/JAS-D-11-0118.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, B., and H. Rui, 1990: Dynamics of the coupled moist Kelvin–Rossby wave on an equatorial β-plane. J. Atmos. Sci., 47, 397413, https://doi.org/10.1175/1520-0469(1990)047<0397:DOTCMK>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wang, B., and T. Li, 1994: Convective interaction with boundary-layer dynamics in the development of the tropical intraseasonal system. J. Atmos. Sci., 51, 13861400, https://doi.org/10.1175/1520-0469(1994)051<1386:CIWBLD>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and G. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain. J. Atmos. Sci., 56, 374399, https://doi.org/10.1175/1520-0469(1999)056<0374:CCEWAO>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Wheeler, M., and H. Hendon, 2004: An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction. Mon. Wea. Rev., 132, 19171932, https://doi.org/10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Yano, J.-I., and J. Tribbia, 2017: Tropical atmospheric Madden–Julian oscillation: A strongly nonlinear free solitary Rossby wave. J. Atmos. Sci., 74, 34733489, https://doi.org/10.1175/JAS-D-16-0319.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., and C. Franzke, 2015: Systematic decomposition of the Madden-Julian oscillation into balanced and inertio-gravity components. Geophys. Res. Lett., 42, 68296835, https://doi.org/10.1002/2015GL065130.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., J. Tribbia, J. Anderson, and K. Raeder, 2009: Uncertainties of estimates of inertia–gravity energy in the atmosphere. Part I: Intercomparison of four analysis systems. Mon. Wea. Rev., 137, 38372477, https://doi.org/10.1175/2009MWR2815.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Žagar, N., A. Kashahara, K. Terasaki, J. Tribbia, and H. Tanaka, 2015: Normal-mode function representation of global 3-D data sets: Open-access software for the atmospheric research community. Geosci. Model Dev., 8, 11691195, https://doi.org/10.5194/gmd-8-1169-2015.

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

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