The 1988 paper that introduced principal oscillation pattern (POP) analysis - the normal modes of a linear first-order autoregressive model fitted to multivariate geophysical time series - applied it to the tropical 30-60-day oscillation (now called the Madden-Julian oscillation, MJO) in a 10-year simulation with the T21 general circulation model (GCM) of the European Centre for Medium-Range Weather Forecasts (ECMWF), finding a dominant eastward mode of zonal wave number 1 (a single planetary-scale wave encircling the equator) with a period of about 24 days and a damping time of about 10 days. Its associated patterns display striking empirical regularities: the eastward phase speed varies from roughly 6 m/s over Indonesia to more than 30 m/s over the eastern Pacific, with slow propagation accompanying large amplitude and fast propagation accompanying small amplitude; rainfall and upper-air velocity potential are in phase in high-amplitude regions but uncorrelated elsewhere; the surface wind response resembles the classical Gill solution for flow forced by an equatorial heat source; and the annual-mean analysis shows no systematic meridional surface wind. The paper left the interpretation of these regularities open; its conclusions explicitly asked for a theoretical clarification of why the propagation velocity and the wave amplitude vary simultaneously and whether a causal relationship exists between the amplification and deceleration of the wave and the in-phase relation of upper-air velocity potential and rainfall, and, conversely, between the acceleration and damping of the wave and the loss of that phase correlation.

Background

The dominant subseasonal signal of the tropical atmosphere is the 30-60-day oscillation, now called the Madden-Julian oscillation (MJO): a planetary-scale coupling between deep convection and circulation that migrates eastward from the Indian Ocean across the Maritime Continent into the western Pacific at about 5 m/s, with the convective signal weakening over the cooler eastern Pacific while upper-tropospheric circulation anomalies continue around the globe. In the 1980s, general circulation models (GCMs, the spectral atmosphere models used for climate simulation) represented this intraseasonal oscillation only crudely, and diagnosing simulated and observed intraseasonal variability objectively was an open methodological problem.

The source paper (von Storch, Bruns, Fischer-Bruns and Hasselmann, 1988, Journal of Geophysical Research 93(D9), 11022-11036) introduced a diagnostic for this purpose. Empirical orthogonal functions (EOFs) are the decorrelated spatial patterns obtained from the covariance matrix of a multivariate record and are the standard tool for reducing the dimensionality of geophysical fields. Principal oscillation pattern (POP) analysis goes further: it fits a linear first-order autoregressive (Markov) model

x(t+1)=B x(t)+r(t)x(t+1)=B\,x(t)+r(t)

to the EOF-filtered state vector x(t)x(t) of a chosen primary field, with residual noise r(t)r(t), and defines the POPs as the eigenvectors of the system matrix B=C1C0−1B=C_1 C_0^{-1}, estimated from lag-zero and lag-one covariances. A complex-conjugate eigenvalue pair describes an oscillating, generally propagating mode with period T=2π/arg⁡(λ)T=2\pi/\arg(\lambda) and e-folding damping time τ=−1/ln⁡∣λ∣\tau=-1/\ln|\lambda|. "Associated correlation patterns" are regressions of secondary fields (rainfall, sea-level pressure, surface wind) onto the POP coefficient time series, while "composite patterns" average secondary fields over episodes selected by phase class and by an amplitude threshold L>⟨L⟩+d σLL>\langle L\rangle+d\,\sigma_L, where LL is the modal amplitude, σL\sigma_L its standard deviation, and dd a chosen coefficient. Hasselmann's companion paper in the same issue (Hasselmann, 1988) placed POPs in a wider program of reducing complex nonlinear systems to low-dimensional dynamics, of which principal interaction patterns (PIPs) - in which the subspace basis and the dynamical model are fitted jointly, rather than the basis being prescribed in advance - are the more general but harder-to-estimate member.

Applied to a ten-year simulation with the T21 GCM of the European Centre for Medium-Range Weather Forecasts (ECMWF), where T21 denotes triangular spectral truncation at total wave number 21, using equatorial 200-hPa divergence (the divergent outflow at the upper troposphere that marks deep convection; its horizontal flow is described by the velocity potential, the potential of the irrotational part of the wind) as the primary field, the method isolated one dominant POP pair of zonal wave number 1 (a single planetary-scale wave encircling the equator) with a 24-day period and 10-day e-folding time - the model counterpart of the observed 30-60-day wave. The associated patterns reproduced observed hallmarks, including a Gill-type surface response (the linear shallow-water response to an equatorial heat source, with an equatorial trough to the east and two off-equatorial cyclones to the west of the heating; Gill, 1980). They also revealed two regularities that demanded dynamical explanation. First, the eastward phase speed varies with longitude, from about 6 m/s in the Indonesian region to more than 30 m/s over the eastern Pacific, and slow propagation coincides with large modal amplitude while fast propagation coincides with small amplitude. Second, rainfall anomalies and upper-air velocity potential are in phase in the high-amplitude regions but essentially uncorrelated in the low-amplitude regions. The authors conjectured a two-stage lifecycle: over the main precipitation region the disturbance behaves as a slowly migrating, strongly forced state with phase-locked convection, and away from it the disturbance propagates almost freely, accelerating and decaying, with the convection-circulation phase relation lost. The annual-mean analysis showed no regular meridional surface-wind structure, which the authors attributed to the disregard of seasonal asymmetries.

The question that the source paper's own conclusions explicitly posed for future work is whether the covariation of propagation velocity and wave amplitude, and the accompanying convection-circulation phase relation, can be given a causal dynamical explanation. The paper also recorded interpretive caveats about its own construction - that the missing regular meridional surface-wind structure may be due to the lack of seasonal stratification, that the EOF subspace was prescribed rather than jointly optimized with the dynamical model (the more general principal-interaction-pattern formulation of Hasselmann's companion paper), and that the composite amplitude-threshold coefficient was chosen arbitrarily - but it did not pose these caveats as open targets. Three decades of MJO research (reviews by Madden and Julian, 1994, and Zhang, 2005) have documented the observed oscillation and its seasonal variation far more completely, and theoretical frameworks such as moisture-mode theory have been proposed, but no generally accepted dynamical theory of MJO propagation, amplitude, and convection-circulation coupling exists; the empirical regularities that the POP analysis surfaced remain ingredients that any such theory must explain.

Problem Statement

This problem asks for a causal dynamical explanation of the empirical regularities of the dominant eastward-propagating intraseasonal mode identified by the principal oscillation pattern (POP) analysis of von Storch, Bruns, Fischer-Bruns and Hasselmann (1988), replacing the purely empirical description based on POPs (the normal modes of the linear first-order autoregressive model fitted to the data). Following the question posed in the paper's own conclusions, the target is a theoretical/dynamical clarification of:

Why the eastward propagation velocity and the wave amplitude of the dominant intraseasonal mode covary - the empirical fact that the mode is slow and amplified (with rainfall in phase with upper-level divergence, described by the upper-air velocity potential, the potential of the divergent part of the upper-tropospheric wind) over the main precipitation regions, and fast and damped (with rainfall and velocity potential decorrelated) elsewhere. Two parts of this target are jointly required: (i) determine whether this covariation reflects a genuine dynamical mechanism rather than coincidence and, if it does, identify and quantify that mechanism, specifying how diabatic (convective) forcing and the background flow combine to produce the covariation and the accompanying convection-circulation phase relation; and (ii) assess the source paper's own conjectured mechanism - a two-stage lifecycle in which the wave is essentially a Kelvin wave that occurs either as a forced wave - slow, large-amplitude, and phase-locked to longitudinally stationary rainfall anomalies over the main forcing region - or as an almost freely propagating wave that, after propagating out of that region, accelerates and decays with the convection-circulation phase relation lost - on its merits, against alternative mechanisms, rather than assuming it.

If part (i) is answered affirmatively, its mechanism must be stated quantitatively - specifying how the mode's phase speed, damping rate and amplitude depend on the convective forcing and the background flow - and validated against observed and/or simulated intraseasonal data by reproducing the observed longitude dependence of phase speed (roughly 6 m/s over Indonesia to more than 30 m/s over the eastern Pacific) and its covariation with amplitude; if part (i) is answered negatively, it must be supported by evidence that the covariation is not a genuine dynamical mechanism - for example, that it is an artifact of the POP construction or of the data. Part (ii) is posed independently of the answer to part (i).

References

  1. von Storch, H., T. Bruns, I. Fischer-Bruns, and K. Hasselmann (1988), "Principal oscillation pattern analysis of the 30- to 60-day oscillation in general circulation model equatorial troposphere", Journal of Geophysical Research: Atmospheres 93(D9), 11022-11036, doi:10.1029/JD093iD09p11022, https://doi.org/10.1029/JD093iD09p11022
  2. Hasselmann, K. (1988), "PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns", Journal of Geophysical Research: Atmospheres 93(D9), 11015-11021, doi:10.1029/JD093iD09p11015, https://doi.org/10.1029/JD093iD09p11015
  3. Gill, A. E. (1980), "Some simple solutions for heat-induced tropical circulation", Quarterly Journal of the Royal Meteorological Society 106(449), 447-462, doi:10.1002/qj.49710644905, https://doi.org/10.1002/qj.49710644905
  4. Gutzler, D. S., and R. A. Madden (1989), "Seasonal variations in the spatial structure of intraseasonal tropical wind fluctuations", Journal of the Atmospheric Sciences 46(5), 641-660, doi:10.1175/1520-0469(1989)046<0641:SVITSS>2.0.CO;2, https://doi.org/10.1175/1520-0469(1989)046<0641:SVITSS>2.0.CO;2
  5. von Storch, H., and Jinsong Xu (1990), "Principal oscillation pattern analysis of the 30- to 60-day oscillation in the tropical troposphere", Climate Dynamics 4(3), 175-190, doi:10.1007/BF00209520, https://doi.org/10.1007/BF00209520
  6. Slingo, J. M., and R. A. Madden (1991), "Characteristics of the tropical intraseasonal oscillation in the NCAR community climate model", Quarterly Journal of the Royal Meteorological Society 117(502), 1129-1169, doi:10.1002/qj.49711750203, https://doi.org/10.1002/qj.49711750203
  7. von Storch, H., and D. P. Baumhefner (1991), "Principal oscillation pattern analysis of the tropical 30- to 60-day oscillation", Climate Dynamics 6(1), 1-12, doi:10.1007/BF00210577, https://doi.org/10.1007/BF00210577
  8. Bürger, G. (1993), "Complex principal oscillation pattern analysis", Journal of Climate 6(10), 1972-1986, doi:10.1175/1520-0442(1993)006<1972:CPOPA>2.0.CO;2, https://doi.org/10.1175/1520-0442(1993)006<1972:CPOPA>2.0.CO;2
  9. Schnur, R., G. Schmitz, N. Grieger, and H. von Storch (1993), "Normal modes of the atmosphere as estimated by principal oscillation patterns and derived from quasigeostrophic theory", Journal of the Atmospheric Sciences 50(15), 2386-2400, doi:10.1175/1520-0469(1993)050<2386:NMOTAA>2.0.CO;2, https://doi.org/10.1175/1520-0469(1993)050<2386:NMOTAA>2.0.CO;2
  10. Hendon, H. H., and M. L. Salby (1994), "The life cycle of the Madden-Julian oscillation", Journal of the Atmospheric Sciences 51(15), 2225-2237, doi:10.1175/1520-0469(1994)051<2225:TLCOTM>2.0.CO;2, https://doi.org/10.1175/1520-0469(1994)051<2225:TLCOTM>2.0.CO;2
  11. Madden, R. A., and P. R. Julian (1994), "Observations of the 40-50-day tropical oscillation - A review", Monthly Weather Review 122(5), 814-837, doi:10.1175/1520-0493(1994)122<0814:OOTDTO>2.0.CO;2, https://doi.org/10.1175/1520-0493(1994)122<0814:OOTDTO>2.0.CO;2
  12. Kwasniok, F. (1996), "The reduction of complex dynamical systems using principal interaction patterns", Physica D: Nonlinear Phenomena 92(1-2), 28-60, doi:10.1016/0167-2789(95)00280-4, https://doi.org/10.1016/0167-2789(95)00280-4
  13. Gualdi, S., A. Navarra, and H. von Storch (1997), "Tropical intraseasonal oscillation appearing in operational analyses and in a family of general circulation models", Journal of the Atmospheric Sciences 54(9), 1185-1202, doi:10.1175/1520-0469(1997)054<1185:TIOAIO>2.0.CO;2, https://doi.org/10.1175/1520-0469(1997)054<1185:TIOAIO>2.0.CO;2
  14. Wheeler, M. C., and H. H. Hendon (2004), "An all-season real-time multivariate MJO index: Development of an index for monitoring and prediction", Monthly Weather Review 132(8), 1917-1932, doi:10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2, https://doi.org/10.1175/1520-0493(2004)132<1917:AARMMI>2.0.CO;2
  15. Zhang, C. (2005), "Madden-Julian oscillation", Reviews of Geophysics 43(2), RG2003, doi:10.1029/2004RG000158, https://doi.org/10.1029/2004RG000158
  16. Egger, J. (2008), "Sensitivity of principal oscillation patterns (POPs) to data distribution in space", Meteorologische Zeitschrift 17(5), 673-677, doi:10.1127/0941-2948/2008/0328, https://doi.org/10.1127/0941-2948/2008/0328
  17. Kikuchi, K., B. Wang, and Y. Kajikawa (2011), "Bimodal representation of the tropical intraseasonal oscillation", Climate Dynamics 38(9-10), 1989-2000, doi:10.1007/s00382-011-1159-1, https://doi.org/10.1007/s00382-011-1159-1
  18. Sobel, A. H., and E. D. Maloney (2013), "Moisture modes and the eastward propagation of the MJO", Journal of the Atmospheric Sciences 70(1), 187-192, doi:10.1175/JAS-D-12-0189.1, https://doi.org/10.1175/JAS-D-12-0189.1
  19. Adames, Á. F., and J. M. Wallace (2014), "Three-dimensional structure and evolution of the MJO and its relation to the mean flow", Journal of the Atmospheric Sciences 71(6), 2007-2026, doi:10.1175/JAS-D-13-0254.1, https://doi.org/10.1175/JAS-D-13-0254.1
  20. Adames, Á. F., and D. Kim (2016), "The MJO as a dispersive, convectively coupled moisture wave: Theory and observations", Journal of the Atmospheric Sciences 73(3), 913-941, doi:10.1175/JAS-D-15-0170.1, https://doi.org/10.1175/JAS-D-15-0170.1
  21. Yadav, P., and D. M. Straus (2017), "Circulation response to fast and slow MJO episodes", Monthly Weather Review 145(4), 1577-1596, doi:10.1175/MWR-D-16-0352.1, https://doi.org/10.1175/MWR-D-16-0352.1
  22. Kim, G.-U., and K.-H. Seo (2018), "Identifying a key physical factor sensitive to the performance of Madden-Julian oscillation simulation in climate models", Climate Dynamics 50(1-2), 391-401, doi:10.1007/s00382-017-3616-y, https://doi.org/10.1007/s00382-017-3616-y
  23. Chen, G., and B. Wang (2020), "Circulation factors determining the propagation speed of the Madden-Julian oscillation", Journal of Climate 33(8), 3367-3380, doi:10.1175/JCLI-D-19-0661.1, https://doi.org/10.1175/JCLI-D-19-0661.1
  24. Mayta, V. C., and Á. F. Adames (2023), "Is the Madden-Julian oscillation a moisture mode?", Geophysical Research Letters 50(15), e2023GL103002, doi:10.1029/2023GL103002, https://doi.org/10.1029/2023GL103002
  25. Wu, J., H.-L. Ren, F.-F. Jin, X. Xu, and L. Guo (2024), "Distinct propagating and standing modes of tropical intraseasonal variability as represented by the two principal oscillation patterns", Atmospheric Research 309, 107574, doi:10.1016/j.atmosres.2024.107574, https://doi.org/10.1016/j.atmosres.2024.107574
  26. Sekizawa, S., H. Nakamura, and Y. Kosaka (2026), "Australian monsoon modulates the amplitude and eastward propagation of the MJO in late austral summer", Journal of Climate 39(15), 4733-4748, doi:10.1175/JCLI-D-25-0520.1, https://doi.org/10.1175/JCLI-D-25-0520.1