the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 May 2026
Memory-driven cascading tipping dynamics in the Earth system. A regime-switching Volterra framework calibrated with CMIP6 ensembles
Abstract. We show that finite-memory effects fundamentally reshape cascade risk in coupled climate tipping systems by decoupling ensemble stability from pathwise instability. Applying a regime-switching Volterra model with tempered fractional kernels to CMIP6 multi-model ensembles (n = 10 for the Atlantic Meridional Overturning Circulation, AMOC; n = 37 for the Amazon and Greenland), we demonstrate that three tipping elements operate under structurally distinct memory regimes linked to different physical processes. AMOC lower-tail occupancy triples under SSP5-8.5 while ensemble-mean weakening reaches only ≈ 0.5σ; a per-model-consistent memory amplification index M^≈ 2.7–6.0 confirms that persistence, not mean shift, is the primary driver. The Amazon presents a mechanistically contrasting picture (M^< 1): its tail amplification is forcing- dominated, making ensemble-mean drying projections adequate for risk assessment. Greenland internal surface-mass-balance (SMB) variability is strongly long-range dependent (H = 0.89; 89 % of models), anchoring it as a persistent upstream driver. Cascade simulations show that quenched (99th-percentile) pathwise Amazon damage exceeds annealed (median) projections by a factor of > 2 under weak forcing – a divergence invisible to ensemble summaries and absent in memory-free dynamics. These results demonstrate that neglecting long-range dependence systematically understates upper-tail cascade risk, and that AMOC, the Amazon, and Greenland require mechanistically differentiated treatment in climate-risk assessment.
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Status: final response (author comments only)
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RC1: 'Comment on egusphere-2026-2260', Anonymous Referee #1, 29 Jul 2026
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AC1: 'Reply on RC1', Mauricio Herrera-Marín, 07 Aug 2026
We thank the reviewer for the careful and technically demanding assessment of our manuscript. In particular, the comments concerning the definition of variables and states, the distinction between empirical inputs and model parameters, the mapping from CMIP6 diagnostics to the reduced model, and the mathematical status and interpretation of the theoretical results led us to undertake a substantially broader validation than would normally be required for a revision.
That re-examination confirmed that some of the reviewer’s concerns were not matters of exposition alone. Most importantly, we found that the original argument linking positive temporal dependence to an increase in the expected rolling occupancy was not mathematically justified. Dependence can alter the variance, clustering, run lengths, maxima, and upper quantiles of occupancy, but it does not increase its expectation in the manner asserted in the submitted theorem. The theorem must therefore be withdrawn rather than restated.
We also found that the proposed memory-amplification diagnostic did not provide a sufficiently clean decomposition between forcing effects and temporal dependence. In addition, stronger reference-model comparisons showed that temporal history carries reproducible predictive information, particularly in the Amazon component, but the available finite records do not identify a unique memory kernel, support a specifically fractional law, or require a nonlinear memory mechanism. This distinction is important: projection can generate a Volterra memory representation, while the functional form of the corresponding kernel may remain empirically non-identifiable from finite records.
These findings change the scientific interpretation too substantially for us to regard the required changes as an ordinary revision. Rather than defend a formulation that our expanded validation no longer supports, we have decided to withdraw the manuscript in its present form.
Several empirical results remain scientifically meaningful, especially the predictive value of temporal history in parts of the Greenland–AMOC–Amazon system. What does not survive is the stronger attribution of those results to a uniquely fractional or nonlinear memory mechanism.
We appreciate the reviewer’s insistence on precise definitions and on a transparent connection between the data, the mathematical objects, and their physical interpretation. In retrospect, those comments helped expose a distinction that is central to the future development of this work: reproducibility of a computational result is necessary, but it is not sufficient to establish identification of the mechanism used to interpret it.
We therefore regard the reviewer’s report as a constructive contribution to the scientific development of the project, even though it has led us to the more substantial decision of withdrawing rather than revising the submitted manuscript.
Citation: https://doi.org/10.5194/egusphere-2026-2260-AC1
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AC1: 'Reply on RC1', Mauricio Herrera-Marín, 07 Aug 2026
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RC2: 'Comment on egusphere-2026-2260', Anonymous Referee #2, 05 Aug 2026
Although I find the idea and methodological approach of this manuscript very interesting, I struggled to review it in its current form. In my opinion, the manuscript requires serious revision and reworking to make it easier to follow and understand.
My two main stylistic objections are that both the textual and mathematical derivations are quite unconventional in places, and that the mathematics sometimes lacks proper referencing (for example, throughout Section 2).
Methodologically, as an expert in DFA approaches, I found it a significant omission that the method is not introduced even at a basic level; moreover, it is written out in full only very late in the text. Finally, the choice of different variables for the tipping point elements should be explained in more detail. As far as I understand, this difference drives the scaling behavior that forms the basis of the paper's central hypothesis, so it must be made clear — including how a different choice of variables would alter the results.
I am not an expert in Volterra operators, and I therefore believe that a reviewer with that expertise would be far more useful in the next round of reviews. I look forward to seeing the revised manuscript.
Finally, if AI tools were used in preparing this manuscript, please declare where and for what purposes. This would greatly assist the review process. I used Claude AI to polish this text.
Citation: https://doi.org/10.5194/egusphere-2026-2260-RC2 -
AC2: 'Reply on RC2', Mauricio Herrera-Marín, 07 Aug 2026
We thank the reviewer for the thoughtful assessment of the manuscript and, in particular, for questioning the mathematical presentation, the interpretation of detrended fluctuation analysis, the choice of climate observables, and the sensitivity of the conclusions to those choices.
These comments motivated us to test more directly whether the persistence diagnostics used in the manuscript actually identified the type of memory attributed to them. The resulting analysis showed that an elevated detrended-fluctuation exponent is evidence of persistence, but is not sufficient, over records of the length available here, to distinguish fractional long-range dependence from plausible short-memory alternatives. Consequently, the mapping from an estimated scaling exponent to a specifically fractional kernel was stronger than the data justified.
The reviewer’s concern about observable choice also proved particularly important for the Amazon component. We therefore reconstructed Amazon precipitation at its native monthly resolution and tested the annual dry-month clustering against a monthly surrogate null that preserves calendar-month seasonality, slow evolution, the exact number of dry months within the prescribed time blocks, and the principal monthly dependence structure. We generated 500 surrogate realizations per trajectory.
At the prespecified primary extremal threshold q = 0.90, positive excess clustering was found in 6 of 8 climate models, but the aggregate model-level test was not significant (p = 0.1417). Only one model was significant before multiple-testing correction, and no model remained significant after false-discovery-rate correction. Sensitivity analyses at q = 0.85 and q = 0.95 were also non-significant.
The comparison between annual and native-resolution nulls is itself informative. Under the annual surrogate null matched in marginal distribution and spectrum, the median extremal index was approximately 0.99; under the native-monthly null it was 0.722, compared with an observed value of 0.637. Thus, approximately three quarters of the apparent extremal-index deficit relative to the annual null is absorbed when the null is instead constructed at the native monthly resolution while preserving seasonality, slow evolution, dry-month frequency, and monthly dependence. The remaining departure, Delta theta = 0.085, is not statistically significant at the model level. This shows that inference about nonlinear temporal organization can depend strongly on the resolution at which the null model is constructed. Establishing how generally this phenomenon arises in threshold-based climate indices requires a separate study.
We therefore can no longer retain the submitted claim that nonlinear memory is empirically required at the Amazon node. This does not establish that the underlying climate dynamics are linear; rather, it shows that the present evidence does not require an additional nonlinear memory mechanism beyond the structure represented by the native-resolution null.
Regarding the declaration requested by the reviewer, large-language-model assistants were used for language editing, examination of argument structure, and assistance in checking code and LaTeX implementations. All scientific decisions, analyses, numerical verification, interpretation, and responsibility for the manuscript remain with the author. This would have been stated explicitly in any revised version.
Taken together, the new validation changes the scientific interpretation too substantially for us to regard the required changes as an ordinary revision. We have therefore chosen to withdraw the manuscript rather than replace its central claims with substantially different conclusions within the same public discussion.
We are grateful for the reviewer’s comments. They contributed directly to a more rigorous distinction among persistence, predictive value of temporal history, and evidence for a specific fractional or nonlinear memory mechanism.
Citation: https://doi.org/10.5194/egusphere-2026-2260-AC2
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AC2: 'Reply on RC2', Mauricio Herrera-Marín, 07 Aug 2026
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EC1: 'Comment on egusphere-2026-2260', Norbert Marwan, 07 Aug 2026
I agree with the concerns of the other two reviewers. The manuscript is difficult to digest. Much more fundamental explanations on all aspects are required.
Moreover, please provide the computational code and (if possible) the data for reproducing the results already for the review stage.
Citation: https://doi.org/10.5194/egusphere-2026-2260-EC1 -
AC3: 'Reply on EC1', Mauricio Herrera-Marín, 07 Aug 2026
We thank the Editor and the referees for the careful evaluation of our manuscript and for the opportunity provided by the open-discussion process to examine its assumptions in greater depth.
Following the referee reports, we undertook an extensive post-submission validation of the theoretical arguments, the data-to-model mapping, the memory diagnostics, the reference models, and the temporal resolution at which the principal claims were tested. This validation went considerably beyond the specific revisions requested by the referees.
The audit produced five conclusions that materially affect the submitted manuscript.
1. THE ORIGINAL ROLLING-OCCUPANCY ARGUMENT IS NOT VALID AS STATED.
The submitted theoretical argument linked positive temporal dependence to an increase in expected rolling occupancy. On re-examination, this is incorrect: dependence affects the variance, clustering, run lengths, maxima, and upper quantiles of the rolling occupancy, but not its expectation in the manner asserted. The corresponding theorem must therefore be withdrawn rather than locally corrected.
2. THE PROPOSED MEMORY-AMPLIFICATION STATISTIC DOES NOT CLEANLY ISOLATE MEMORY.
The statistic compared an empirical rolling-tail response with a marginal reference based primarily on changes in mean and scale. Stronger analysis showed that forcing, distributional changes, heterogeneity, and temporal dependence are not separated sufficiently by this construction to support the causal memory attribution made in the manuscript.
3. THE DATA DO NOT IDENTIFY A UNIQUE MEMORY KERNEL OR FRACTIONAL LAW.
After introducing stronger reference models, temporal history remained predictively informative in parts of the system. However, conventional finite-history autoregressive representations and distributed-history representations performed comparably, and broad ranges of fractional-kernel parameters were empirically indistinguishable. The appropriate conclusion is therefore that history contains predictive information, while the available finite records do not determine the functional form of the corresponding memory representation. This does not invalidate Volterra reduction as a mathematical description of projected dynamics; it invalidates the stronger claim that the submitted data identify a particular fractional or nonlinear kernel.
4. THE AMAZON NONLINEAR-MEMORY CLAIM DOES NOT SURVIVE A NATIVE-RESOLUTION CONTROL.
We reconstructed Amazon precipitation at monthly resolution and repeated the extremal-clustering analysis using a constrained monthly surrogate null that preserves seasonality, slow evolution, the exact number of dry months within the prescribed blocks, and the principal monthly dependence structure. With 500 surrogate realizations per trajectory, the prespecified q = 0.90 test produced positive excess clustering in 6 of 8 models but an aggregate model-level p-value of 0.1417, with no model remaining significant after multiple-testing correction. Sensitivity analyses at q = 0.85 and q = 0.95 were also non-significant.
Under the annual surrogate null, the median extremal index was approximately 0.99; under the native-monthly null it was 0.722, compared with an observed value of 0.637. Most of the apparent departure relative to the annual surrogate null is therefore absorbed when the null is reconstructed at the native monthly resolution; the remaining departure is not statistically significant. We can no longer maintain the submitted claim that nonlinear memory is empirically required at the Amazon node.
5. THE SUBMITTED AMOC COHORT CONTAINED A MATERIAL BASIN-SELECTION PROBLEM.
During the same audit, we identified that part of the files treated in the submitted ten-model AMOC cohort as Atlantic overturning diagnostics corresponded to a different basin designation. We subsequently reconstructed an independently audited Atlantic–Arctic cohort, but the submitted ten-model cohort cannot be treated as the homogeneous Atlantic overturning ensemble described in the manuscript, and the quantitative AMOC results derived from that cohort cannot be maintained as submitted.
These findings are not minor corrections. They alter both the theoretical support and the empirical interpretation of the central claims. Several results remain scientifically meaningful, including evidence that temporal history improves prediction for some components of the Greenland–AMOC–Amazon system. However, the present evidence supports predictive historical dependence more strongly than it supports identification of a unique fractional, Volterra-kernel form, or nonlinear memory mechanism.
For this reason, we believe that the scientifically responsible course is to withdraw the manuscript rather than attempt to preserve its original thesis through incremental revision. We intend to retain the results that survive the audit, subject them to additional independent validation, and reconsider them in new work only after the surviving claims have been reformulated and tested under a frozen validation protocol.
We thank the Editor and the referees for their time and scrutiny. We take full responsibility for the claims made in the submitted manuscript and equally for the decision not to maintain claims that no longer meet the evidentiary standard we consider necessary.
Citation: https://doi.org/10.5194/egusphere-2026-2260-AC3
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AC3: 'Reply on EC1', Mauricio Herrera-Marín, 07 Aug 2026
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- 1
The authors introduce a stochastic model for the tipping cascade between Greenland meltwater discharge, the weakening AMOC, and Amazon precipitation. The model is designed to incorporate memory effects using a fractional-tempered memory kernel which makes extreme events more likely to cluster temporally. The results of the paper are summarized in four claims R1 to R4 which are stated in the beginning of Section 4.
The model and methodology section describes a multitude of model variables, many of which are not formally introduced. This is having a particularly negative effect on the understandability of the paper since it is not clear how the data analysis of Section 3 uses the models described in Section 2. The model and the data analysis of the paper are involved and seem to the thought-through. However, there are a very large number of moving parts and the paper is lacking a rigorous structure which unfortunately makes many of the results near impossible to interpret. A detailed description of these problems is given below. It is for these reasons that I believe that the paper does not meet the standards of Nonlinear Processes in Geophysics.
Detailed analysis of Section 2 and 3:
Sections 2.1 to 2.3 are mostly coherent and rigorous. There is an external forcing term F_ext which drives a two-state Markov chain z(t), which is turn influences the memory kernel of the Volterra-dynamics of the damage variables D_G, D_A, D_R. Some remarks are:
It is not clearly stated which variables are assumed to be given from the outside (like the CO2 concentration), which variables are model constants (like \gamma_SU), and which variables are dynamical output variables. Having too many model constants without giving good reasoning on how to choose them makes the model less reliable and its results more ambiguous.
In line 80, the manuscript reads 'the Greenland DFA confirms this as \overline H = 0.89 implies \tilde \alpha=0.36'. It is not being stated what \overline H, nor what \tilde \alpha are and how they are related.
Regarding Section 2.4:
It is not mentioned what F_hist^-1 or q are. The variable \hat p(t) is introduced as the 'rolling window tail frequency' without defining what that means in terms of the dynamics introduced in the previous sections. The 'definition' of \Delta p_marg involves \Phi, u_m, \mu_fut,m and \sigma_hist,m; all of which are variables that appear nowhere else in Sections 2 and 3 (u does, but it is not clear what the subscript m means).
Theorem 1 is formulated very vaguely without mathematical rigour. Also, the meaning of the theorem and its purpose for the results are not discussed.
Section 2.5 mentions more variable names, non of which are formally introduced. It is not clear whether these variables are observables from the model output, whether they are extracted from data, or whether they are simply model parameters.
Section 3 does not explain how the data is fed into all of the variables and dynamics introduced in Section 2. Optimally, it should clearly be stated what type of data is being fed into what model, which parameters are chosen, and what output is being measured. Instead Section 3 introduces yet more variables that are not rigorously defined.
Lastly, The first results of Section 4, in Figure 1, show the detrended fluctuation analysis F(s) on the y-axis, even though it has not been explained what a detrended fluctuation analysis is, how it is computed how to interpret this value. In particular, the variable F(s) has never been mentioned in this form before in the Sections 2 and 3.