REVIEW 4 major objections 4 minor 13 references
Current State of Atmospheric Turbulence Cascades
T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read Simulated turbulence dissipation shows the same box-counting fractal dimension along every direction, so anisotropy must come from experiments.
desk verdict The paper's central isotropy claim is undermined by a dimensionally wrong dissipation surrogate and circular dataset selection; not ready for review. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The argument is carried by the box-counting fractal dimension $D=-\lim_{s\to0} \log N/\log s$ (Eq. 9), applied to the normalized dissipation surrogate $\varepsilon'/\langle\varepsilon\rangle$, where $\varepsilon'=\partial u_1/\partial t$ (Eq. 6). A $32\times32\times32$ sampling mesh with $0.01$ spacing supplies the point set, and the multiplicative-process multifractal equations (3)-(5) frame the expected scaling behavior. The central comparison is the value of $D$ along the $x$, $y$, and $z$ directions: identical values mean the dissipation field is declared isotropic.
What would settle it
Compute the box-counting dimension using the full dissipation tensor $\varepsilon=2\nu S_{ij}S_{ij}$ instead of $\partial u_1/\partial t$ on the same DNS field; if the direction-independence disappears, the isotropy is an artifact of the surrogate. Also repeat with $64^3$ and $128^3$ meshes: if the dimension shifts with resolution, the scaling range was not resolved.
Extended reading notes
Core claim
The central discovery is that, in every dataset examined, the box-counting fractal dimension of the energy-dissipation field is the same along all three spatial directions. The authors estimate dissipation from the time derivative of one velocity component, normalize it, and apply box-counting to DNS and LES fields; both types of simulation return isotropic fractal signatures. The paper reads this as agreement with the earlier multifractal dissipation work and as a limitation of isotropic numerical datasets: anisotropic multifractal behavior must be sought in experimental data.
Load-bearing premise
The result stands or falls on treating the time derivative of one velocity component as a faithful measure of the true energy dissipation rate, and on a $32\times32\times32$ grid with $0.01$ spacing resolving the scales that box-counting needs; if either fails, the measured fractal dimensions could be artifacts of the method rather than properties of the turbulence.
Editorial extensions
If this is right
- Current DNS and LES dissipation fields, as sampled here, behave isotropically at the fractal level, so they cannot by themselves validate direction-dependent multifractal cascade models.
- The box-counting dimension of the dissipation surrogate can serve as a quick diagnostic for isotropy in numerical turbulence datasets.
- Future multifractal turbulence simulations will need to incorporate experimental data to capture anisotropic dissipation, as the paper explicitly concludes.
- Atmospheric remote-sensing metrics built on isotropic dissipation assumptions remain consistent with these numerical datasets over the sampled scales.
Reading between the lines
- If the single-component time-derivative surrogate is trusted, the observed isotropy may still be a grid-symmetry effect of the $32^3$ Cartesian mesh; sampling on a rotated or non-axis-aligned mesh would test whether direction-independence is intrinsic.
- A natural extension is to apply the same box-counting pipeline to experimental hot-wire or particle-image-velocimetry dissipation fields; the paper's framing implies those would show direction-dependent dimensions that the numerical datasets lack.
- The coarse $0.01$ spacing likely probes only a narrow band of the inertial range, so the reported isotropic dimension could change at higher Reynolds numbers; finer DNS data would show whether isotropy persists across the full cascade.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript attempts to test whether the box-counting fractal dimension of the energy dissipation field is isotropic in simulated turbulence, following Meneveau and Sreenivasan's multifractal analyses. The authors use velocity data from the JHU 'isotropic1024' DNS database and from in-house OpenFOAM LES snapshots, define a dissipation surrogate in Eq. (6), compute box-counting dimensions via Eq. (9), and conclude in Section 5 that all simulated datasets exhibit isotropic characteristics with direction-independent box-counting fractal dimensions, in agreement with Meneveau's findings.
Significance. If the central claim were correct, the paper would offer a useful numerical check of multifractal dissipation isotropy in DNS and LES and would connect simulation practice to Meneveau-Sreenivasan experimental results. The manuscript also identifies an important problem: simulating anisotropic, realistic atmospheric cascades requires going beyond idealized isotropic datasets. However, the current presentation does not deliver on this promise. No quantitative box-counting dimensions or uncertainty estimates are reported, no code or data are made available, and the central dissipation proxy is dimensionally inconsistent with the energy dissipation rate, so the conclusion is not supported. The paper's positive contribution is limited to a statement of intent and a qualitative observation that all studied datasets are isotropic, which is largely by construction.
major comments (4)
- [Section 3.3, Eq. (6)] The quantity defined as ε′ = ∂u1/∂t is not an energy dissipation rate. Its units are L/T^2, whereas the true dissipation rate ε = 2νSijSij has units L^2/T^3. No derivation, empirical calibration, or scaling argument is provided to show that the multifractal geometry of ∂u1/∂t matches that of the dissipation field. Since Meneveau and Sreenivasan's results concern the dissipation field, the conclusion that the box-counting dimension is direction-independent for these datasets is a statement about an acceleration component, not about turbulent energy dissipation. This is load-bearing because all subsequent box-counting analyses in Sections 3.5 and 4 use this surrogate.
- [Section 5 and Results (Section 4)] The central claim that 'the box-counting fractal dimension remains consistent across all directions' is never backed by any reported numerical value, fitted slope, regression statistic, or error bar. The 32×32×32 sampling mesh described in Section 3.3 has a dynamic range of only about five box sizes, and no convergence or resolution study is presented to show that Eq. (9) yields a well-defined limit. Without reporting the actual dimensions and their uncertainties, the direction-independence conclusion is quantitatively unsupported.
- [Section 3.4 and Section 5] The conclusion that all simulated datasets are isotropic is partially forced by the dataset selection. The JHU dataset is explicitly the 'isotropic1024' dataset, and Section 3.4 states that the OpenFOAM simulation is a cube 'where the turbulence dissipates isotropically.' Running an isotropic dataset through a box-counting analysis and then 'discovering' isotropy is circular. The paper does not compare the DNS and LES results against any anisotropic dataset, so it cannot test Meneveau's anisotropic multifractal framework, which is the stated motivation of the work.
- [Section 3.2, Eqs. (3)-(5)] Equations (3)-(5) are presented as 'Meneveau's multiplicative process,' but they are not the equations from Meneveau and Sreenivasan (1991). The notation is undefined (e.g., C(q), φ(q,x), χ(i,q), ψ(i)), the sums and products are not connected to the cascade construction, and these equations are never used in the subsequent analysis. This mischaracterizes the methodological foundation of the paper and should be corrected or removed.
minor comments (4)
- [Abstract and Introduction] The prose is often unclear; for example, 'the one we have used applies to a more exact representation of turbulence where people use the multifractal representation' and 'vertices' instead of 'vortices' should be corrected. A careful language edit is needed throughout.
- [Section 3.4, Eqs. (7)-(8)] The LES equations are written without any filtering notation and without the subgrid-scale stress term, so they are formally identical to the incompressible Navier-Stokes equations. If LES is actually used, the filtered equations and the SGS model should be stated explicitly.
- [Figures] Figure captions are incomplete or refer to placeholder labels such as 'Fig. 5' and 'Fig. 6', and Figures 7-11 are not described quantitatively in the text. The reader cannot infer from the captions what is being plotted or what conclusion to draw.
- [Results and Conclusions] The statement in Section 4 that 'we see fractal agreement and consistency at 0.01s' and 'at 0.40s' is not supported by any definition of 'fractal agreement' or by any error metric; please specify what quantity is being compared and how the agreement is quantified.
Circularity Check
Isotropy result is the input restated as a discovery: JHU 'isotropic1024' DNS and an OpenFOAM cube 'where the turbulence dissipates isotropically' are used, and Section 5 then reports that all simulated datasets are isotropic, with self-citation [12] invoked for the LES fractal-dimension finding.
-
self definitional
[Section 3.4 and Section 5 Conclusions]
"In this study, we have used OpenFOAM to simulate atmospheric turbulence using the Large Eddy Simulation (LES) model, where we take a cube where the turbulence dissipates isotropically. Knowing that turbulence dissipates isotropically, we wonder if, using the methods described by Meneveau, the fractal dimension would be the same in all directions. We took snapshots at two different times to compare unsteadiness and coherence in turbulence, and we found that all directions have the same structure. We realized that the fractal dimension is the same [12]."
The simulation is deliberately constructed with isotropic dissipation, and the DNS is the JHU 'isotropic1024' dataset (Section 3.3). The paper's central conclusion in Section 5 that 'all simulated datasets exhibit isotropic characteristics' and that the box-counting dimension is direction-independent is therefore a restatement of the construction choices; there is no non-isotropic or experimental dataset against which the claim could fail. In addition, Eq. (6) defines the measured field as epsilon-prime = du1/dt, so the 'dissipation' whose multifractal dimension is analyzed is not Meneveau's epsilon but a surrogate labeled by definition as energy dissipation.
-
self citation load bearing
[Section 3.4, reference [12]]
"We realized that the fractal dimension is the same [12]."
The bibliographic support offered for the key LES result (direction-independent fractal dimension) is [12], Dudu et al. 2021, whose author list overlaps with the present authors (Rodriguez and Kumar). The manuscript does not report the fitted dimensions, standard errors, box sizes, or scaling ranges that would make the LES finding independently checkable here, so the load-bearing support for the claim is the authors' own earlier conference paper rather than an external, machine-checked, or parameter-free result.
full rationale
The paper's central claim is that the simulated DNS and LES dissipation fields are isotropic in the sense that the box-counting fractal dimension is direction-independent. That conclusion is forced by the construction of the inputs: the JHU dataset is explicitly called 'isotropic1024', and the OpenFOAM simulation is set up 'where the turbulence dissipates isotropically'. Section 5 then reports that 'all simulated datasets exhibit isotropic characteristics', which is the setup restated as an empirical discovery. This is a self-definitional circularity rather than an independently tested prediction. A separate, non-circular correctness risk is that Eq. (6) labels the time derivative of a single velocity component as 'energy dissipation'; this disconnect from the physical dissipation rate should be weighed in review, but it is not what drives the circularity score. The self-citation to [12] adds a second, milder circular thread because that prior work by the same group is the only reference given for the LES fractal-dimension finding and no quantitative dimensions or uncertainties are provided in the present manuscript. Overall, the derivation chain reduces to the input definition, so a high circularity score is warranted.
Assumptions & free parameters
free parameters (3)
- sampling mesh interval =
0.01
- time step dt =
0.002 s
- snapshot times for LES =
0.01 s and 0.40 s
assumptions (4)
- standard math Box-counting dimension, as defined in Eq. (9), is a valid measure of the multifractal scaling of turbulent energy dissipation.
- domain assumption The JHTDB isotropic1024 dataset and the OpenFOAM LES run are representative of fully developed turbulence.
- ad hoc to paper The time derivative of streamwise velocity, ∂u1/∂t, represents the energy dissipation rate.
- ad hoc to paper A 32x32x32 sampling mesh with 0.01 spacing resolves the scaling range needed for box counting.
Cite this review
Pith. "Pith review of Current State of Atmospheric Turbulence Cascades." pith.science (2026). https://pith.science/paper/O6Q4OSKQ
@misc{pith2026241219953,
author = {Pith},
title = {Pith review of: Current State of Atmospheric Turbulence Cascades},
year = {2026},
howpublished = {\url{https://pith.science/paper/O6Q4OSKQ}},
note = {Machine review of arXiv:2412.19953}
}
read the original abstract
Turbulence cascade has been modeled using various methods; the one we have used applies to a more exact representation of turbulence where people use the multifractal representation. The nature of the energy dissipation is usually governed by partial differential equations that have been described, such as Navier-Stokes Equations, although usually in climate modeling, the Kolmogorov turbulence cascading approximation leads towards an isotropic representation. In recent years, Meneveau et al. have proposed to go away from Kolmogorov assumptions and propose multifractal models where we can account for a new anisotropic representation. Our research aims to use Direct Numerical Simulations (DNS) from the JHU Turbulence Database and Large Eddy Simulations (LES) we simulated using OpenFOAM to predict how accurate these simulations are in replicating Meneveau experimental procedures with numerical simulations using the same rigorous mathematical approaches. Modeling turbulence cascading using higher fidelity data will advance the field and produce faster and better remote sensing metrics. We have written computer code to analyze DNS and LES data and study the multifractal nature of energy dissipation. The box-counting method is used to identify the multifractal dimension spectrum of the DNS and LES data in every direction to follow Meneveau work to represent turbulence-cascading effects in the atmosphere better.
Reference graph
Works this paper leans on
-
[1]
Atmospheric Turbulence Inter- mittency Effects on Remote Sensing Laser Propagation,
Rodriguez, A., Gudimetla, V. S. R., Adansi, R., Terrazas, J., Corral, V., Harris, C., Kumar, V., Baez, R., and Paez, B., 2022, “Atmospheric Turbulence Inter- mittency Effects on Remote Sensing Laser Propagation,” American Society of Mechanical Engineers, Fluids Engineering Division, FEDSM, American Society of Mechanical Engineers, p. V001T03A008
work page 2022
-
[2]
Kumar, V., 2020, Remote Sensing and Imaging Physics: Developing New Metrics for Deep Turbulence Effects on Laser Propagation Through Long Path, TEXAS UNIV AT EL PASO
work page 2020
-
[3]
Rodriguez, A., Cuellar, C. R., Rodriguez, L. F., Garcia, A., Rao Gudimetla, V. S., Krushnarao Kotteda, V. M., Munoz, J. A., and Kumar, V., 2020, “Stochas- tic Analysis of LES Atmospheric Turbulence Solutions with Generative Machine Learning Models,” American Society of Mechanical Engineers, Fluids Engineering Division, FEDSM
work page 2020
-
[4]
Parameter Sensitivity and Statisti- cal Correlation Found in Atmospheric Turbulence Studies,
Rodriguez, L. F., Kumar, V., Rodriguez, A., Krushnarao Kotteda, V. M., Rao Gudimetla, V. S., and Munoz, J. A., 2020, “Parameter Sensitivity and Statisti- cal Correlation Found in Atmospheric Turbulence Studies,” American Society of Mechanical Engineers, Fluids Engineering Division, FEDSM, 2
work page 2020
-
[5]
Performing Fourier Transform on a Velocity Profile from Atmospheric Turbulence Studies,
Adansi, R., Terrazas, J., Rodriguez, A., Krushnarao Kotteda, V. M., Kumar, V., Rubio, A., and Avalos, E., 2021, “Performing Fourier Transform on a Velocity Profile from Atmospheric Turbulence Studies,” American Society of Mechanical Engineers, Fluids Engineering Division, FEDSM, American Society of Mechanical Engineers, p. V001T02A035
work page 2021
-
[6]
Deblur- ring of Optical Images Due to Atmospheric Turbulence Effects Using Image Processing
Rezaa, K., Rezaa, R., Rodriguezb, A., Adansib, R., and Kumarc, V., “Deblur- ring of Optical Images Due to Atmospheric Turbulence Effects Using Image Processing.”
-
[7]
Simulation of Atmospheric Turbulence with Generative Machine Learning Models,
Rodriguez, A., Cuellar, C., Rodriguez, L., Garcia, A., Terrazas, J., Kotteda, V. M., Gudimetla, R., Kumar, V., and Munoz, J., 2020, “Simulation of Atmospheric Turbulence with Generative Machine Learning Models,” Bulletin of the American Physical Society, 65
work page 2020
-
[8]
The Multifractal Nature of Turbulent Energy Dissipation,
Meneveau, C., and Sreenivasan, K. R., 1991, “The Multifractal Nature of Turbulent Energy Dissipation,” J Fluid Mech, 224, pp. 429–484
work page 1991
Show all 13 references
-
[9]
Simple Multifractal Cascade Model for Fully Developed Turbulence,
Meneveau, C., and Sreenivasan, K. R., 1987, “Simple Multifractal Cascade Model for Fully Developed Turbulence,” Phys Rev Lett, 59(13), pp. 1424–1427. 11
1987
-
[10]
Multifractal Measures, Especially for the Geophysi- cist,
Mandelbrot, B. B., 1989, “Multifractal Measures, Especially for the Geophysi- cist,” Pure and Applied Geophysics PAGEOPH, 131(1–2), pp. 5–42
1989
-
[11]
Multifractal Nature of Fully Developed Turbulence and Chaotic Systems.,
Benzi, R., Paladin, G., Parisi, G., and Vulpiani, A., 1984, “Multifractal Nature of Fully Developed Turbulence and Chaotic Systems.,” J Phys A Math Gen, 17(18), p. 3521
1984
-
[12]
Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning,
Dudu, N., Rodriguez, A., Moran, G., Terrazas, J., Adansi, R., Krushnarao Kot- teda, V. M., Harris, C., and Kumar, V., 2021, “Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning,” American Society of Mechanical Engineers, Fluids Engineering...
2021
-
[13]
High-Reynolds-Number Fractal Signature of Nascent Turbulence during Transition,
Wu, Z., Zaki, T. A., and Meneveau, C., 2020, “High-Reynolds-Number Fractal Signature of Nascent Turbulence during Transition,” Proc Natl Acad Sci U S A, 117(7), pp. 3461–3468. 12
2020
Reviewed August 10, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.