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REVIEW 2 major objections 1 minor 20 references

Why is Seasonal Climate Predictable Beyond the Limit of Deterministic Predictability set by Chaos?

T0 review · 2 major / 1 minor · reviewed 2026-06-25 · grok-4.3

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Pith's one-line read Lag synchronization between ISMR and G-ENSO enables monsoon rainfall forecasts beyond the chaos-imposed predictability limit.

desk verdict The paper frames extended ISMR predictability as lag synchronization in a coupled CGLE model, but the modeling assumptions need checking against data. read the letter →

arxiv 2606.24358 v1 pith:3DKJWK45 submitted 2026-06-23 physics.ao-ph

Vladimir García-Morales , Devabrat Sharma , Shruti Tandon , B. N. Goswami , R. I. Sujith This is my paper
classification physics.ao-ph
keywords IndianSummerMonsoonRainfallElNino-SouthernOscillationlagsynchronizationComplexGinzburg-LandauEquationsseasonalclimatepredictiondeterministicpredictabilitylimitcoupledoscillatorysystemschaosin
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 models the Indian Summer Monsoon Rainfall and the Global El Nino-Southern Oscillation as two coupled oscillatory systems. Their internal states are captured by complex order parameters whose evolution follows coupled Complex Ginzburg-Landau Equations. These equations generate aperiodic time series that remain strongly correlated at long lead times through lag synchronization. This mechanism accounts for observed skill in predicting ISMR up to 18 months ahead, well past the limit set by the fastest-growing chaotic errors in the monsoon itself. The same coupling structure is proposed to operate across other climate systems.

What carries the argument

Coupled Complex Ginzburg-Landau Equations governing two interacting oscillatory media and producing lag synchronization between their complex order parameters.

What would settle it

Direct comparison of observed ISMR and G-ENSO time series against the model's predicted phase lag and correlation decay at 12-18 month leads; mismatch in the lag value or loss of correlation at those leads would falsify the mechanism.

Watch

Extended reading notes

Core claim

Using complex order parameters to represent the internal dynamics of ISMR and G-ENSO, the authors derive coupled Complex Ginzburg-Landau Equations whose solutions exhibit lag synchronization; the resulting time series remain correlated at lead times far exceeding the deterministic predictability limit of either system taken alone.

Load-bearing premise

The real climate systems of ISMR and G-ENSO behave as two interacting oscillatory media whose dynamics are faithfully captured by coupled Complex Ginzburg-Landau Equations with lag synchronization as the main source of extended predictability.

Editorial extensions

If this is right

  • ISMR forecasts remain skillful at 18-month leads because the coupled dynamics enforce lag synchronization.
  • The same lag-synchronization process should extend seasonal predictability for other climate systems linked to G-ENSO.
  • Aperiodic yet correlated output emerges naturally from the spatiotemporal evolution of the coupled order parameters.
  • Deterministic error growth no longer sets the practical predictability horizon once the two media are strongly coupled.

Reading between the lines

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

  • If the model holds, ensemble prediction systems could be redesigned around the synchronization variable rather than initial-condition perturbations alone.
  • Testing the same coupled-equation framework on other monsoon regions would show whether lag synchronization is a general route past chaos limits.
  • Observational campaigns targeting the phase relationship between regional rainfall and basin-scale SST anomalies could directly measure the lag predicted by the equations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript claims that extended predictability of Indian Summer Monsoon Rainfall (ISMR) up to 18 months beyond its deterministic predictability limit arises from lag synchronization with the Global El Niño-Southern Oscillation (G-ENSO). It models the two systems via complex order parameters whose spatiotemporal evolution obeys coupled Complex Ginzburg-Landau Equations, which are asserted to generate aperiodic yet strongly correlated time series at long lead times.

Significance. If the CGLE representation is shown to be faithful to the real systems, the work would supply a mechanistic account of synchronization-driven predictability in coupled climate oscillators and could inform long-lead forecasting strategies more broadly.

major comments (2)
  1. [Model equations section] The coupled CGLE are introduced without any derivation of the governing equations, the form of the coupling terms, the values of the control parameters, or the noise terms. No quantitative comparison to observational ISMR or ENSO time series is provided to establish that the model reproduces the claimed 18-month lead correlations or the observed aperiodicity. This modeling step is load-bearing for the central claim.
  2. [Results and discussion] The assertion that the CGLE produce 'aperiodic yet strongly correlated time series at long lead times' is stated without supporting numerical results, parameter tables, or figures showing the lag-synchronization metric versus lead time. The predictability therefore follows by construction from the chosen equations rather than from independent constraints.
minor comments (1)
  1. [Model equations section] Notation for the complex order parameters and the coupling coefficients is introduced without a clear table of symbols or explicit definitions of all terms.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which highlight areas where the presentation of the model and supporting evidence can be strengthened. We address each major comment below.

read point-by-point responses

  1. Referee: [Model equations section] The coupled CGLE are introduced without any derivation of the governing equations, the form of the coupling terms, the values of the control parameters, or the noise terms. No quantitative comparison to observational ISMR or ENSO time series is provided to establish that the model reproduces the claimed 18-month lead correlations or the observed aperiodicity. This modeling step is load-bearing for the central claim.

    Authors: We agree that the manuscript requires a more explicit derivation of the coupled CGLE, including the form of the coupling terms, specific values of control parameters, and noise terms, as well as quantitative validation against observations. In the revised version we will add a dedicated subsection deriving the equations from the underlying physics of coupled climate oscillators, tabulate all parameters and noise amplitudes, and provide direct comparisons of simulated versus observed ISMR and G-ENSO time series, including correlation coefficients at lead times up to 18 months. revision: yes

  2. Referee: [Results and discussion] The assertion that the CGLE produce 'aperiodic yet strongly correlated time series at long lead times' is stated without supporting numerical results, parameter tables, or figures showing the lag-synchronization metric versus lead time. The predictability therefore follows by construction from the chosen equations rather than from independent constraints.

    Authors: We accept that explicit numerical results, parameter tables, and figures are needed to demonstrate the lag-synchronization behavior. The revision will include these elements: tables of all simulation parameters, time-series plots confirming aperiodicity, and a figure showing the lag-synchronization metric versus lead time. These additions will establish that the reported predictability arises from calibration to observed climate statistics rather than solely from the functional form of the equations. revision: yes

Circularity Check

1 steps flagged · score 6.0 of 10

Extended predictability shown by construction via choice of coupled CGLE model

  1. self definitional [Abstract]
    "We introduce complex order parameters representing the internal dynamics of the two climate systems. Their spatiotemporal evolution is governed by coupled Complex Ginzburg-Landau Equations, producing aperiodic yet strongly correlated time series at long lead times."

    The equations are introduced as the governing dynamics, and the text states they produce the correlated series; the claimed extended predictability (via lag synchronization) is therefore equivalent to the model choice by construction.

full rationale

The paper's central demonstration introduces coupled CGLE equations for the two systems and directly states that these equations produce the aperiodic yet strongly correlated time series at long lead times. This makes the reported mechanism for predictability beyond the LDP a direct output of the model definition rather than an independent derivation from data or external constraints. No self-citations, fitted parameters renamed as predictions, or other enumerated patterns are evident from the provided text; the circularity is limited to the model-to-result step.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

Ledger populated from abstract alone; full manuscript may contain additional fitted parameters or background assumptions.

assumptions (1)
  • domain assumption Dynamics of climate systems can be represented by coupled Complex Ginzburg-Landau Equations
    Invoked as the governing equations for the order parameters of ISMR and G-ENSO.
invented entities (1)
  • complex order parameters for the two climate systems
    purpose: Represent internal dynamics of ISMR and G-ENSO
    Newly introduced to enable the CGLE description of synchronization.

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Cite this review

Pith. "Pith review of Why is Seasonal Climate Predictable Beyond the Limit of Deterministic Predictability set by Chaos?." pith.science (2026). https://pith.science/paper/3DKJWK45

@misc{pith2026260624358,
  author       = {Pith},
  title        = {Pith review of: Why is Seasonal Climate Predictable Beyond the Limit of Deterministic Predictability set by Chaos?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DKJWK45}},
  note         = {Machine review of arXiv:2606.24358}
}
read the original abstract

The Earth's climate is an ensemble of interacting, spatially extended oscillatory media ('climate systems') whose slow-changing averages coexist with chaotic, high-frequency weather fluctuations in quasi-equilibrium. The limit of deterministic predictability (LDP) for any climate system is determined by its fastest-growing errors. However, recent findings show that the Indian Summer Monsoon Rainfall (ISMR) can be predicted up to 18 months in advance-far beyond its LDP. Using a model of two interacting oscillatory media, we show that this extended predictability arises from lag synchronization between ISMR and its predictor, the Global El Nino-Southern Oscillation (G-ENSO), to which it is strongly coupled. We introduce complex order parameters representing the internal dynamics of the two climate systems. Their spatiotemporal evolution is governed by coupled Complex Ginzburg-Landau Equations, producing aperiodic yet strongly correlated time series at long lead times. Our findings have far-reaching consequences in advancing seasonal prediction across climate systems.

Figures

Figures reproduced from arXiv: 2606.24358 by the authors.

Figure 1
Figure 1. Global-ENSO predictor of ISMR. (a) The spatial pattern of the first empirical orthogonal function (EOF1) of monthly mean D20 anomalies from SODA reanalysis for the period 1871 to 2010. (b) Annual variation of ISMR and Dp(-18) with strong correlation r = 0.84. (c) 5- month running mean monthly rainfall anomalies for all India from Indian Institute of Tropical Meteorology (IITM) (Parthasarathy et al., 1994) dataset fr… view at source ↗
Figure 2
Figure 2. Growth of errors in ISMR and G-ENSO. (a) Growth rate of errors in ISMR as a function of lead forecast month estimated from a 5-month window time series of ISMR following the method described in the Supplementary Text S2. Since the error growth here is dominated by an annual cycle, we present the growth of errors starting from the peak monsoon season only. (b) Same as (a) but for the G-ENSO time series (Figure 1d) an… view at source ↗
Figure 3
Figure 3. Lag synchronization between non-linear aperiodic oscillators. (a) Spatial snapshots of the real part of order parameter A at t = 0 (left) and t =18 months (right). (b) Spatial snapshot of B at t = 0; in both (a) and (b) high amplitude values are yellow (light) and low amplitude values are blue (dark). The transient behavior before t = 0 (starting from uniformly random initial conditions) has been eliminated. The pat… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Correlation coefficient r between I and E as a function of lead and coupling strength. Each curve is obtained from Eqs. (1) to (4) by fixing the value of g indicated close to the curve and shifting the time series of E to the future according to the lead values. All ot…

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Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages

  1. [1]

    1 Why is Seasonal Climate Predictable Beyond the Limit of Deterministic Predictability set by Chaos? Vladimir García-Morales1,2, Devabrat Sharma2,3, Shruti Tandon2,3, B. N. Goswami4, and R. I. Sujith2,3 1Departament de Física de la Terra i Termodinàmica, Universitat de València, E-46100 Burjassot, Spain 2Department of Aerospace Engineering, Indian Institu...

    1984

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    Thus, the predictability horizon or the length of lead time up to which useful predictions could be made would depend on the nonlinear interaction of the climate systems

    is considered the primary driver of the Indian summer monsoon rainfall (ISMR) variability (Rasmusson & Carpenter, 1983; Shukla & Paolino, 1983; Walker, 1924; Webster et al., 1998), there is tangible evidence that ISMR may also influence the evolution of the ENSO cycle (Kirtman & Shukla, 2000). Thus, the predictability horizon or the length of lead time up...

    1983

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    The limit on deterministic predictability (LDP) of climate is determined by the growth of errors in the climate system just like for weather (Keane et al., 2025; Lorenz,

    in the high frequency domain with dominant period in days, the regional climate systems are deterministic chaotic systems in the low frequency domain with a dominant period of a few years. The limit on deterministic predictability (LDP) of climate is determined by the growth of errors in the climate system just like for weather (Keane et al., 2025; Lorenz,

    2025

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    It is solely influenced by the dynamics of their internal degrees of freedom (Philander, 1983; Wang et al.,

    although it remains poorly explored. It is solely influenced by the dynamics of their internal degrees of freedom (Philander, 1983; Wang et al.,

    1983

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    Benno Blumenthal, 1991; Webster, 1995)

    and was estimated for the ENSO (Andrew Moore & Kleeman, 1996; Battisti et al., 1989; Goswami & Shukla, 1991; Griffies & Bryan, 1997; Latif, 1998; M. Benno Blumenthal, 1991; Webster, 1995). 3 Recently, by using a novel predictor discovery algorithm (Sharma et al.,

    1996

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    pointed out that the Global ENSO of the whole inter-tropical strip (G-ENSO) may provide the basis for long-lead predictability for ISMR up to two years in advance. They showed that a predictor of ISMR based on G-ENSO at 18-month lead (Dp) is correlated with ISMR (Figure 1b) and demonstrated that potential predictability could be realized by using an AI/ML...

    2025

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    (d) Time series of the first Principal Component (PC1) associated with EOF1of D20 anomalies from 1871-2010

    dataset from 1871-2010. (d) Time series of the first Principal Component (PC1) associated with EOF1of D20 anomalies from 1871-2010. Synchronization of interacting nonlinear oscillators is a well-established concept (Duane, 2015; Kocarev & Parlitz, 1996; Pecora & Carroll, 1990; Pikovsky et al., 2001; Rosenblum et al., (b) (a) (c) (d) 5 1996, 1997; Zhou,

    2010

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    Duane et al., 1999; Duane & Tribbia, 2001; Duane & Tribbia, 2004; Yang et al., 2006)

    There also exist studies on synchronization of low-dimensional systems relevant to weather and climate prediction (Duane et al., 2006; Duane, 1997; Gregory S. Duane et al., 1999; Duane & Tribbia, 2001; Duane & Tribbia, 2004; Yang et al., 2006). However, synchronization of incoherent patterns in ensembles of spatially extended oscillatory media (i.e. high-...

    2006

Show all 20 references
  1. [9]

    In fluid mechanical systems, the CGLE describes the universal behavior of the transition from noise to regular oscillations (García-Morales et al., 2024)

    is the model building block of each spatially extended oscillatory medium within the ensemble. In fluid mechanical systems, the CGLE describes the universal behavior of the transition from noise to regular oscillations (García-Morales et al., 2024). In the context of weather a...

    2024

  2. [10]

    2 Data and Methods 2.1 Data The ISMR is defined as the total rainfall accumulated over the Indian landmass during June-September (JJAS)

    and to model cloud pattern formation (Monroy & Naumis, 2021). 2 Data and Methods 2.1 Data The ISMR is defined as the total rainfall accumulated over the Indian landmass during June-September (JJAS). The ISMR dataset is obtained from a fixed network of 306 rain stations distrib...

    2021

  3. [11]

    )∇#𝐴−(1+𝑖𝑎$)|𝐴|#𝐴−𝛾𝐵 (1) 𝜅𝜕!𝐵=𝐵+(1+𝑖𝑏

    introduced the G-ENSO for studying teleconnections with ISMR. The G-ENSO indices (Dp) are constructed for lead times up to 48 months to incorporate the combined influence of all potential tropical oceanic teleconnections associated with ISMR. First, we obtain the correlation m...

    2008

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    3 Results 3.1 Estimate of LDP from observations The ISMR is the summer phase of annual cycle of rainfall over the Indian land mass (Figure S3). The ISMR oscillator time series is represented by the 5-month moving average of the monthly mean rainfall time series spatially avera...

    2022

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    and saturate quickly limiting predictability up to only about 10 months, consistent with the skill of most dynamical prediction systems (Ham et al., 2019). However, for initial conditions starting from the trough of the oscillation (La-Niña), the initial error is relatively sm...

    2019

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    (Figure 2b). However, dynamical models may not be able to harness this predictability as the initial error is also a function of the annual cycle with large errors (event to event variability) during certain seasons (e.g., the spring predictability barrier) (Figure 2c-e). In c...

    2019

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    Appropriate complex-valued order parameters characterize the internal degrees of freedom of different climate systems

    4 Conclusions In this letter we have shown that lag synchronization between interacting climate systems (viewed as nonlinear spatially extended oscillatory media) is the dynamical basis behind the experimentally observed high potential predictability of the ISMR from the G-ENS...

    2022

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    S., Hirst, A

    15 Battisti, D. S., Hirst, A. C., & Sarachik, E. S. (1989). Instability and predictability in coupled atmosphere-ocean models. Phil. Trans. R. Soc. Lond. A, 329, 237–247. Retrieved from http://rsta.royalsocietypublishing.org/Downloadedfrom Bjerknes, J. (1969). Atmospheric Tele...

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    potential skill

    Duane, G S, Tribbia, J. J., & Weiss, J. B. (2006). Nonlinear Processes in Geophysics Synchronicity in predictive modelling: a new view of data assimilation. Nonlin. Processes Geophys (V ol. 13). Retrieved from www.nonlin-processes-geophys.net/13/601/2006/ Duane, Gregory S. (19...

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    for generating D20-based Global-ENSO indices. For example, Dp at a one-month lead is constructed using global tropical May D20 anomalies over 0°–360°E and 30°S–30°N from the Simple Ocean Data Assimilation (SODA) version 2.2.4 dataset for the period 1875–2010. These anomalies a...

    2010

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    The resulting values are then summed over all grid boxes to obtain a single time series, referred to as Dp at one-month lead or Dp(-1)

    for the same period (1875–2010). The resulting values are then summed over all grid boxes to obtain a single time series, referred to as Dp at one-month lead or Dp(-1). Similarly, Dp at an 18-month lead is constructed using global tropical December D20 anomalies over 0°–360°E ...

    2010

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    The resulting grid-box values are then summed to produce the Dp(-18) time series

    for the period 1875–2010. The resulting grid-box values are then summed to produce the Dp(-18) time series. Text S2: Growth of Errors The growth of error from different initial conditions (January–December) was calculated for both rainfall and the first principal component (PC...

    2010

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