the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Oct 2025
A deterministic cascade model to infer intermittency stochastics of Navier-Stokes
Abstract. The ubiquity of the fundamental characteristic of turbulence, intermittency, is increasingly recognized in many fields. The multifractal analysis of various turbulence data, particularly from lab experiments and atmospheric sensed data, has rather constantly yielded a multifractality index of α ≈ 1.5 and a mean codimension of C1 ≈ 0.25, but with a given uncertainty. To reduce this uncertainty and understand the dynamical origin of these estimates, the multifractality of turbulence is investigated with the help of the deterministic Scaling Gyroscope Cascade (SGC) model. In this study, the forced SGC model is run with cascade levels of up to 14 and a duration of 2.5 × 104 large eddy turnover times. These simulations exhibit extreme spatial-temporal intermittency. Multifractal analysis confirms the empirical values α ≈ 1.5, C1 ≈ 0.25, showing almost independence on the forcing. It raises doubts about the Log-normal model, at least for hydrodynamic turbulence. In addition, the remaining uncertainty in multifractality resulting from the discrete numerical simulation method is investigated.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Nonlinear Processes in Geophysics.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: open (until 23 Feb 2026)
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RC1: 'Comment on egusphere-2025-4895', Anonymous Referee #1, 14 Dec 2025
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AC1: 'Reply on RC1', Xin Li, 28 Jan 2026
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The authors greatly appreciate the constructive comment from RC1, which does help us improve the quality of the manuscript.
1. We thank the reviewer for pointing out the issue of the poor quality of the figures. Upon careful examination, we found that the problem was caused by excessive downscaling of the figures during layout preparation, which resulted in reduced font sizes and poor readability. All figures will be regenerated and reformatted with appropriately scaled labels and annotations. The corrected figures will be included in the revised manuscript.
2. We appreciate the reviewer’s careful attention to detail. All listed typographical errors and suggested minor edits will be fully addressed. In addition, the manuscript will undergo thorough proofreading and language revision to improve clarity, consistency, and overall fluency. We believe that these changes will significantly enhance the readability of the paper.
3. We thank the reviewer for raising this important point. We agree that physical conclusions should not depend on numerical methods. The original wording may have caused ambiguity. The hierarchical structure of the SGC model causes its computational cost to rise exponentially with more cascade steps. For Case 2 with n=14, the simulation requires one week on the sever. Due to these computational costs, the use of higher-order or significantly more advanced time-integration schemes was not feasible within the scope of the present study. Importantly, our intention was not to imply that the main physical conclusions (the multifractality index α≈1.5 and the mean intermittency codimension C1≈0.25) depend on the numerical method. Rather, the results indicate that the values of multifractality parameters exhibit only a limited sensitivity to the numerical scheme, and therefore the qualitative behavior and the principal conclusions are robust. To avoid any misunderstanding, we will carefully revised the text associated with Fig. 3 and the corresponding discussion to clearly emphasize this distinction. For instance, we will clarify that Euler’s explicit scheme strictly respects the detailed conservation of energy but is only conditionally stable, while the semi-implicit scheme is unconditionally stable, but only approximatively respects the detailed conservation of energy.
4. Following our previous comment that SGC is more than an ad-hoc toy model, and in order to preserve “the manuscript [to be] brief, and relatively easy to read” we will revise the appendix A a much less abrupt beginning and to provide more pedagogical insights on the derivation of SGC from Navier-Stokes. A concise summary will be provided in the main text. The latter will focus the physical issue of intermittency. It will highlight the fact that the SGC structure, whose number of eddies increases like the inverse of scale, easily deals with the high space heterogeneity that is an indispensable component of the space-time intermittency. This is absent from shell models that have only scales. We will highlight this property, in particular with the help of a figure presenting the structure of both models. Moreover, our detailed multifractal analysis of the SGC simulations shows that their multifractality index α≈1.5 and their mean intermittency codimension C1≈0.25 are consistent with those observed in empirical data, resulting therefore in quantitative agreement. This was the main goal of our paper and it should be understood both ways, e.g., the SGC results, whose estimate uncertainties can be reduced at a rather arbitrary level, comfort the empirical results whose uncertainties are more difficult to reduce.
This also raises interesting theoretical questions for future works. Indeed, the phenomenology of turbulence is usually based on local interactions, i.e., between components of rather similar scale, whereas SGC is based on semi-local interactions in the Fourier space. Therefore, it is rather remarkable that SGC is so successful at simulating intermittency.
Citation: https://doi.org/10.5194/egusphere-2025-4895-AC1
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AC1: 'Reply on RC1', Xin Li, 28 Jan 2026
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- 1
This manuscript considers a numerical “toy” model on intermittency. The manuscript is brief, and relatively easy to read. The three figures, however, are of very poor quality and it was difficult to gauge the scientific value of the work with such poor quality figures. I suspect just making the figures “proper sized” would help, but I am a bit puzzled how one submits something for peer review in which a reader literally cannot make out the figures.
I provide a substantial list of typos and small changes which I trust the authors would readily implement. Some of these indicate that the manuscript could be proofread better prior to resubmission, in particular for language flow.
There is a more serious issue relating to the numerical methods. I start my own numerical explorations with Euler methods, too, but the reality is that most of the time something higher order, and more current is adopted for an eventual set of production runs prior to submission. In the manuscript, Figure 3 and the attendant discussion both seem to suggest that results are numerical method dependent. This is either a very poor choice of communication, or a fatal flaw. Valid scientific results are never numerical method dependent and this manuscript cannot be published unless this is resolved.
In a broader sense, the connection to science is tenuous. By shuffling the derivation to Appendix A, and not really taking much care to make sure this section is broad audience appropriate, the authors substantially decrease the legibility of the manuscript and its relevance to a broad audience (and I write this as an applied mathematician, who is familiar with the various bits of notation used). It is important to point out what aspects of the physics the toy model captures, and what it misses. I am a big fan of toy models, but the presentation here greatly limits the audience.
Detailed comments:
45 “of the Bernoulli form” not “of Bernoulli form”
45 “has only scales but no space” does not make sense
55 I think the expression for energy should be a numbered equation
60 Is there a primary reference for the Mellin transform that can be added?
65 What does “a rather deterministic framework” mean?
90 “it is strongly unstable” I presume the “it” is the solution in equation (7)? It might be better to say that explicitly.
100 I think the energy transfer expression should be a numbered equation
105 “structure of the SGC model” currently the “the” is missing.
145 I think the standard terminology is “classical” not “classic”. The issue also occurs in the Conclusions and Appendix B
The figures are all tiny to the point of being illegible.
Figure 3 caption: Do the authors really mean differences due to numerical methods? That would be very bad if this was the case.
Acknowledgements: there is a capitalization issue in the name of the institute, and note that latex allows for the accent on “Ecole”.
175 “the gyroscope equation”
180 I think “\nabla” and “\nabla \times” would look better than grad and curl written out in words. If you do want to use words please use \hbox{grad} and similar to avoid the math environment italicization. It is bizarre that later in the same paragraph the proper latexed nabla is used.
180 This paragraph should be rewritten for clarity. I think a number of articles are missing or out of place, and it’s really hard to make out what’s going on.