| 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.
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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.