Propagation of the Madden-Julian oscillation as a deterministic chaotic phenomenon

Daisuke Takasuka; Hiroaki Miura; Masuo Nakano; Tamaki Suematsu

REVIEW 5 major objections 5 minor 65 references

Propagation of the Madden-Julian oscillation as a deterministic chaotic phenomenon

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Madden-Julian oscillation can take two distinct propagation pathways under identical background conditions, making its eastward timing a deterministic chaotic phenomenon.

desk verdict Big-ensemble NICAM study claiming MJO propagation splits into two timing regimes set by climatological SST asymmetry; the clean SST-swap tests make the core result credible, but the regime detection pipeline is the load-bearing risk and the 'chaos' label outruns the evidence. read the letter →

arxiv 2506.23195 v1 pith:TNRRSAFP submitted 2025-06-29 physics.ao-ph

Daisuke Takasuka , Tamaki Suematsu , Hiroaki Miura , Masuo Nakano This is my paper · ORCID

classification physics.ao-ph

keywords Madden-JulianoscillationMJOpropagationdeterministicchaosregimebifurcationequatorialasymmetryofSSTRossbywavestropical-extratropicalinteractioncloud-system-resolvingensemblesimulation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the eastward progression of the Madden-Julian oscillation (MJO), the dominant tropical weather system, is not a linear process whose behavior can be captured by refining existing linearized theories. Instead, the authors claim, MJO propagation is a deterministic chaotic phenomenon: under one fixed atmosphere-ocean background, the same event can take multiple distinct propagation pathways. Using 4,000 ensemble members from global cloud-system-resolving simulations of two real 2018 MJO events, they find that the November event has a single propagation regime, while the December event bifurcates into two regimes with different arrival times over the western Pacific. The bifurcation is controlled by the equatorial asymmetry of climatological sea surface temperature, and which regime is realized depends probabilistically on whether tropical-extratropical interactions sustain moistening by westward-moving tropical wave disturbances. If correct, this means the timing of MJO propagation is intrinsically probabilistic and that observations capture only one realization of a multi-stable system.

What carries the argument

The machinery has two linked parts. First, the experimental design: 1,000-member ensembles for two real 2018 MJO events, run for 45 days with a 14-km global cloud-system-resolving model and identical boundary conditions, plus two further 1,000-member sensitivity experiments that exchange the seasonal SST fields. Second, the physical mechanism that selects between regimes: an upper-level extratropical trough intruding into the tropical western Pacific, whose refraction is controlled by the strength of a cyclonic upper-level circulation generated by the MJO convection's own circulation response. That trough, when strong, sustains westward-moving tropical waves that moisten the troposphere east of the convection and enable the observed December 22 arrival; when weak and displaced eastward, those waves decay and a slower December 30 pathway emerges through equatorially asymmetric off-equatorial convection. Regime membership is measured by projecting each member's simulated outgoing longwave radiation onto observation-derived patterns and applying four thresholds to the resulting MJO index.

What would settle it

Run the same December 2018 ensemble protocol but identify propagation directly from the model's own outgoing longwave radiation—for instance, the date the equatorial minimum crosses 150°E—with no thresholds or appended observations; if the two-peaked distribution vanishes or shifts with the detection choice, the bifurcation claim is not a property of the simulated atmosphere.

Watch

Extended reading notes

Core claim

The authors' central discovery is that MJO propagation is a multi-stable, deterministic chaotic phenomenon rather than the straightforward consequence of linear moisture dynamics. In their 4,000-member ensemble simulations, the December 2018 MJO splits into two regimes under identical boundary conditions: Dec-1, in which convection reaches the western Pacific around December 22 as observed, and Dec-2, in which it reorganizes about a week later through a different moistening pathway. The existence of these regimes is tied to the seasonal SST field: when December simulations are run with November SSTs, the two-regime structure disappears, and when November simulations are run with December SSTs, propagation becomes more variable. Selection between the two December regimes is governed by the strength of tropical-extratropical interaction before the event: an extratropical trough intruding into the tropical western Pacific, refracted by a cyclonic upper-level circulation generated by the MJO's own convection, determines whether westward-moving tropical waves moisten the western Pacific and enable the observed pathway. The observed December 2018 event is therefore interpreted as one probabilistic realization of a system that could equally have produced a delayed, alternative propagation.

Load-bearing premise

The central claim rests on how 'propagation into the western Pacific' is measured: each member is classified by projecting simulated cloud radiation onto observed patterns, appending ten days of observations, and applying four numerical thresholds, so if that detection recipe rather than the simulated atmosphere produces the two timing peaks, the bifurcation result inherits the artifact.

Editorial extensions

If this is right

Observed MJO propagation is a single realization: the same December 2018 MJO could have taken a week-later pathway under identical boundary conditions, so hindcasts should be evaluated probabilistically rather than by matching one observed timing.
Existing linear MJO theories, which assume a time-invariant background and no interaction with higher-frequency waves, are insufficient; the regime structure implies nonlinear cross-scale interactions are essential.
The seasonal dependence of MJO activity, with more frequent propagation into the western Pacific in December-February than in November, can be interpreted as the emergence of additional propagation regimes as climatological SSTs become equatorially asymmetric.
Prediction of December-February MJO propagation timing is inherently uncertain even with a perfect model, and machine-learning systems trained only on observed data cannot cover the full probability space of possible pathways.
Training prediction systems on huge-ensemble storm-resolving simulations that capture these bifurcating solutions may yield better subseasonal-to-seasonal forecasts of MJO onset and propagation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same bifurcation logic suggests that interannual SST variations, such as those associated with El Niño and La Niña, could shift the number or timing of propagation regimes and therefore change how predictable the MJO is from year to year.
A testable extension is to run the same ensemble protocol for additional December MJO events and check whether the two-regime structure and its dependence on SST asymmetry reproduce, or whether the bifurcation is specific to the 2018 event.
The mechanism implies that variations in midlatitude wave activity, not just tropical SST, should alter the probabilities of the two regimes; conditioning the ensemble on different extratropical initial states would test this directly.
Because the regime definition relies on projected radiation fields and hand-set thresholds, an independent verification using direct cloud-tracking metrics would confirm that the bimodality is a property of the simulated atmosphere rather than of the detection recipe.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a serious, large-compute paper with a genuinely new claim. Using 4,000 NICAM members across two 2018 MJO events, it finds that December MJO propagation timing bifurcates into two peaks (Dec-1 and Dec-2) under identical boundary conditions, and that swapping November and December SST fields removes or creates the second peak (Fig. 3). That SST-swap experiment is clean and parameter-free, which makes the central attribution harder to dismiss as pure curve-fitting. The authors also deserve credit for openly reporting that the bimodality disappears at 100–500 ensemble members, and for flagging the filter differences between observations and simulations.

The soft spots are real, though. The biggest one is the regime detection pipeline. Every member's propagation time comes from projecting simulated OLR onto observational EEOFs, appending 10 observed days, and applying four hand-set thresholds (amplitude > 0.4, phase skip and recession limits, Phase 1 within day 20). The Dec-1/Dec-2 split is read off the histogram of that variable (Fig. 1D). The SST-swap experiments show the pipeline responds to the simulated fields, but they do not show the two histogram peaks correspond to dynamically distinct states rather than two projection regimes of a single evolution. That distinction is central to the multi-stability interpretation, and the paper does not test it. A simple robustness check—varying the thresholds, using a different MJO index, or tracking convection directly—would settle it.

The second weakness is the word "chaos." What is shown is multi-stability and ensemble spread, which are necessary but not sufficient for deterministic chaos. There are no quantitative chaos metrics (Lyapunov exponents, Kolmogorov-Sinai entropy, unstable periodic orbits). The paper's own conclusion hedges by saying nonlinearity is a necessary condition, which is honest but undercuts the headline label. Third, generalizability rests on two events, and the Dec-2 regime is never observed in nature, so the claim that it represents a real alternative pathway is inferential.

Citation practice looks fine; the linear-theory papers they contrast with (refs 22–25) are the standard ones. The Dryad statement lacks a DOI, which is a minor reproducibility gap given the Fugaku-scale cost of the runs. Who this is for: MJO theorists, S2S prediction people, and anyone building ML forecasters on synthetic ensembles. It deserves a serious referee, but the referee should push hard on the detection-pipeline robustness before the bifurcation claim is taken as established.

Referee Report

5 major / 5 minor

Summary. The manuscript uses 4,000-member ensembles of 14-km NICAM hindcast simulations for two MJO events in November and December 2018 to argue that MJO propagation into the western Pacific is a deterministic chaotic phenomenon. For the December event, the distribution of simulated propagation timing is reported to be bimodal (Dec-1 near December 22 and Dec-2 near December 30), whereas the November event is unimodal. SST-swap experiments show that replacing December SSTs with November SSTs removes the second mode, and replacing November SSTs with December SSTs produces a later-timing mode. Composites attribute the regime selection to tropical-extratropical interaction and to moistening by westward-propagating equatorial Rossby waves over the western Pacific.

Significance. If correct, the paper would be an important demonstration with a global storm-resolving model that MJO propagation under fixed boundary conditions is intrinsically multi-stable, with direct implications for subseasonal predictability and for training machine-learning prediction systems. The SST-swap experiments are a genuine out-of-sample test of the SST attribution, and the authors honestly report that the bimodality is not visible with 100 or 500 ensemble members. These strengths are substantial. However, the central claim currently rests on a detection-pipeline definition of propagation timing whose robustness is not established, so the significance can only be fully assessed after that premise is secured.

major comments (5)

[Methods: Identifying the MJO propagation and its composite] The entire regime structure is defined by a detection pipeline: 45-day simulated OLR is projected onto observationally derived DJF EEOFs with 10 days of observed OLR appended, a 5-day running mean is applied to the PCs, and four hand-set criteria are imposed (A > 0.4 with at most 4 low-amplitude days, at most one phase skip, at most three phase recessions, and Phase 1 within the first 20 days). Because the Dec-1/Dec-2 split is read off the histogram of this variable, the central bifurcation claim requires demonstration that the bimodality is robust to each of these choices. Please provide sensitivity tests, for example by varying the EEOF base period and lag structure, the appended-segment length, the amplitude threshold, and the phase-tracking limits, and by confirming the two peaks with a direct cloud-tracking measure such as the longitude-time evolution of OLR minima or column water vapor. If the pipeline itself generates the two clusters, the SST attribution and the mechanism analysis inherit that artifact.
[Results: Multiple regimes of the MJO propagation (Fig. 1D)] The claim of two December regimes is based on a histogram with colored vertical lines, and the paper states that the bimodality is not obtained with 100 or 500 members. This raises the question of statistical significance and mode separation. Please report the number of members assigned to each regime, apply a formal unimodality test (for example a Hartigan dip test or a two-component Gaussian mixture with a model-selection criterion), and show sensitivity to bin width and to the placement of the separation boundary. The 100/500-member subsampling result should be interpreted explicitly: if 1000 members are required for the two modes to appear, the separation may be marginal, and the reader needs a quantitative measure of how distinct the Dec-1 and Dec-2 states are.
[Results: Processes supporting the bifurcated regimes (Figs. 2 and 4)] Because members are grouped into regimes by the arrival-timing variable itself, composite differences between Dec-1 and Dec-2 may partly be mechanical consequences of the grouping criterion. The paper should show that the observed differences in convection location, vertical motion, and tropical-extratropical interaction are not simply lead-lag artifacts of different propagation dates. One way is to recompute the composites relative to each regime's own propagation day, or to condition on a narrow propagation-timing window and compare members across the SST sensitivity experiments. The current presentation does not rule out the possibility that the composites are describing the detection algorithm's phase definition rather than the atmosphere's dynamics.
[Results: Climatological SSTs as a key (Fig. 3) and Methods: Model and simulation setups] The SST-swap experiments exchange the entire seasonally evolving SST field between November and December, not just its equatorial asymmetry. The conclusion that 'the bifurcation originates from the equatorial asymmetry' is therefore stronger than what the experiments test. Please quantify the change in the meridional SST gradient and, ideally, perform a test in which the equatorially asymmetric component of SST is removed while the zonal-mean and total-warming components are retained (or vice versa). This would clarify whether the specific SST asymmetry, rather than the overall seasonal warming or the calendar shift of the boundary condition, controls the existence of the second regime.
[Discussion and title] The phrase 'deterministic chaotic phenomenon' is stronger than what the results currently establish. Multiple stable regimes under an external parameter (multi-stability) is a nonlinear-dynamical phenomenon but is not synonymous with chaos, which requires sensitive dependence on initial conditions within an attractor. The manuscript should either define the precise sense in which 'chaotic' is being used or temper the wording to 'nonlinear multi-stable regime behavior' unless quantitative evidence of sensitive dependence (for example, divergence estimates among nearby ensemble members) is provided.

minor comments (5)

[Throughout] There are several typographical errors and missing spaces, including 'conductO(103)-member' in the Results opening, 'occurrs' on page 7, and 'a open question' in the Discussion; these should be corrected.
[Methods, Eq. (2)] The displayed EKE budget equation contains LaTeX brace artifacts ('bracehtipupleft', 'bracehtipdownright') that make the equation unreadable in the submitted PDF; please provide a cleanly typeset equation.
[Fig. 1C and 1D] The dark and light gray histograms for the 500- and 100-member subsamples are difficult to distinguish in gray scale; please use different line styles or colors and add a legend.
[Fig. 4 caption] The parenthetical phrase '(Same below)' is informal and unclear; please replace it with a precise statement about the significance convention used in all panels.
[Data and materials availability] The statement that simulation and observational data and scripts are stored at Dryad should include a DOI or accession link so that the reproducibility claims can be verified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central regime/bifurcation claim is supported by out-of-sample SST-swap experiments, and the MJO-index detection pipeline is a pre-existing diagnostic rather than a fitted predictor.

full rationale

The paper's derivation chain is: huge-ensemble NICAM simulations -> MJO events identified by a fixed, published MJO index (Kikuchi et al. 2012) with four pre-specified tracking criteria -> histograms of propagation timing -> two regimes for Dec-MJO -> SST-swap experiments -> composite/QG diagnostics for the mechanism. No step defines the predicted quantity in terms of the input in a way that forces the result. The EEOFs are derived from observed OLR and the amplitude/phase thresholds are fixed before examining the simulated timing distributions; the same criteria are applied to Nov-MJO, Dec-MJO, and both SST sensitivity experiments. The SST-swap experiments are genuine out-of-sample manipulations (December calendar with November SST and vice versa), so the attribution of the bifurcation to climatological SST asymmetry is not forced by construction. The only mild structural feature is that regime composites are conditioned on the outcome variable (arrival timing), so mechanism differences are not fully independent of the regime definition; however, the mechanism variables (QGPV, wave-activity flux, EKE budget terms) are distinct physical fields measured from the simulations, not the same quantity as the classification variable. This is a standard composite-analysis limitation, not circularity. Self-citations (refs 49 and 56) concern model configuration and a supporting observational analysis, and they are not load-bearing for the central chaotic-bifurcation claim. The central result therefore stands on the simulations and the controlled SST experiment rather than on a self-referential construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims rest on model realism (NICAM at 14 km represents MJO and synoptic-scale waves), on the validity of the hybrid EEOF-based MJO index with its hand-set thresholds, and on the independent-member assumption used in all significance tests. There are no invented physical entities; the Nov-1, Dec-1, and Dec-2 names are statistical groupings, not new ontology. Three free parameters of the detection pipeline (tracking thresholds, the histogram-derived regime split, and the observation-vs-simulation filter windows) shape the headline result and are not sensitivity-tested.

free parameters (3)

MJO index tracking thresholds = A > 0.4 (tolerance 4 days); at most 1 phase skip; at most 3 phase recessions; Phase 1 within first 20 days
Hand-chosen criteria that define which ensemble members are judged to have propagated into the WP and when. The bimodal Dec-MJO histogram (Fig. 1D) and hence the Dec-1/Dec-2 partition is produced by this detector; different thresholds would shift the timing distribution.
Dec-1/Dec-2 regime split point = Propagation around Dec 22 vs Dec 30, split at the gap in the Fig. 1D histogram
The two regimes are not defined a priori; they are read post hoc from the bimodal distribution of the very quantity (propagation timing) the paper then explains. The regime membership of each of the 1,000 December members is fixed by this histogram reading.
Composite filter windows = 11-day running mean deviations (simulations); 5-30 day bandpass, zonal wavenumbers -20 to -1 (observations); 15-day…
Different filter definitions for observations vs simulations are adopted because of the short 45-day simulation records; the budget terms and regime differences depend on these choices, though the paper argues both definitions capture equatorial Rossby waves.

assumptions (4)

domain assumption NICAM at a 14-km grid with the "MJO run" physics configuration represents the essential dynamics of the MJO and of synoptic-scale tropical waves.
The entire study is a model-based inference about the atmosphere. The configuration is inherited from Takasuka et al. (2024) and is argued to be reliable because simulated propagation patterns in the Nov-1 and Dec-1 regimes match observed OLR, but a 14-km mesh is still not convection-resolving.
domain assumption The hybrid MJO index, projecting simulated OLR onto observed DJF EEOFs with appended observed OLR, is a valid measure of MJO phase and amplitude for each ensemble member.
Adopted because 45 days is too short for computing robust EEOFs; follows the precedent of Shibuya et al. (2021). The propagation timing for every member is defined through this index.
domain assumption Ensemble members are statistically independent for the purposes of composite significance tests.
Stated in Methods verbatim. Members share boundary SST and land conditions and partially overlapping initial information, so the effective sample size for the 99% t-tests is likely smaller than the member count.
domain assumption Beta-plane QG dynamics with a reference latitude of 35 degrees N is adequate for diagnosing wave behavior that reaches into the tropics (10S to 10N).
Used for QGPV, Q-vector, wave activity flux and refractive index (eqs. 3-7). Applying a mid-latitude beta-plane approximation to tropical and cross-equatorial latitudes is a simplification; it is used as a diagnostic rather than as a governing model.

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Cite this review

Pith. "Pith review of Propagation of the Madden-Julian oscillation as a deterministic chaotic phenomenon." pith.science (2026). https://pith.science/paper/TNRRSAFP

@misc{pith2026250623195,
  author       = {Pith},
  title        = {Pith review of: Propagation of the Madden-Julian oscillation as a deterministic chaotic phenomenon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TNRRSAFP}},
  note         = {Machine review of arXiv:2506.23195}
}

read the original abstract

The Madden-Julian oscillation (MJO), a gigantic tropical weather system, is marked by eastward travel of cumulus cloud clusters over the Indo-Pacific region and often causes severe weather and climate events worldwide. The physics and predictability of MJO propagation remain elusive, partly because of little attention to untangling roles of multi-scale processes relevant to the MJO. Here, we reveal the chaotic nature of MJO propagation arising from cross-scale nonlinear interactions, based on 4,000-member ensemble global cloud-system-resolving simulations of two MJO events. Against conventional linearized thinking, multiple regimes with distinct timings of MJO propagation emerge under a single atmosphere-ocean background. The bifurcation emergence depends critically on the equatorial asymmetry of climatological sea surface temperature. Selection of the bifurcated regimes is probabilistic, influenced by whether tropical-extratropical interplay promotes moistening associated with westward-propagating tropical waves over the western Pacific. These aspects help build a comprehensive MJO model and foresee when the MJO propagates.

Figures

Figures reproduced from arXiv: 2506.23195 by the authors.

**Figure 1.** Two MJOs and their regimes revealed by huge-ensemble hindcast simulations. (A and B) Time-longitude diagrams of observed equatorial (10 ◦S–10◦N) outgoing longwave radiation (OLR) at the top-of-atmosphere (shading) and off-equatorial (0◦–10◦N) 850-hPa meridional winds filtered for westward zonal wavenumbers 1–20 and periods of 5–30 days (contours) for Nov-MJO (A) and Dec-MJO (B). Contour interval is 1.6 m s−1 , wit… view at source ↗

**Figure 2.** Time evolution of the MJO propagation regimes in December and background effects on it. (A) OLR (shading) and 850-hPa wind (vectors) anomalies composited for the Dec2 regime on December 8 (top), 17 (middle), and 26 (bottom). Contours indicate OLR anomalies for the Dec-1 regime. Contour interval is 12 W m−2 , with zero values or higher omitted. Stippling indicates statistical significance of color-shaded values at … view at source ↗

**Figure 3.** Impacts of seasonal SST components on the timing of MJO propagation. (A) Number distributions by dates of propagation into the WP for the Dec-MJO simulation under SSTs used in the Nov-MJO simulation (filled) and for the control Dec-MJO simulation (open). (B and C) As in (A), but for the control Nov-MJO simulation (B) and the Nov-MJO simulation under SSTs used in the Dec-MJO simulation (C). Results for each 100 membe… view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Regime selection by the MJO-extratropical interaction. (A) Observed Q-vector divergence in 400-250 hPa (shading), 200-hPa QGPV (black), and column water vapor (CWV) anomalies (> 2.5 kg m−2 ; hatched) averaged over December 5–9, and 850-hPa EKE tendency on December 7 (m…

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