The Power Spectrum of Compressible Turbulence: A Story in Symmetries

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The paper, The power spectrum of compressible turbulence: A story in
symmetries is carried out to simplify a complex problem through a rigorous, symmetry-based approach, leading to a robust theoretical framework that can potentially resolve long-standing controversies in the field of compressible turbulence. My comments on this paper are as follows:
- Use a pronoun as the author in the paper; some sentences use I, our, we, etc.
-Lines 24 to 26 use citation instead of directly citing the paper title
-How does the result compare to the recent study? Mostly, the references are out of date
-The paper uses the theoretical approach, how about the effect in the numerical approach? Such as computational fluid dynamics (CFD)
-Future work needs to add how the implementation of real cases and applications in fluid mechanics
-Please explain why such a model would be necessary to clarify the nature of the transition that occurs between the low Mach number (subsonic) and high Mach number (supersonic) regimes.?
-There is no mention of its applicability to non-Newtonian fluids, turbulence in other media (like plasmas or quantum fluids), or other physical systems that might exhibit turbulent-like behavior. Please elaborate on this in your paper.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The comments are attached in PDF file. 

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript addresses a fascinating and fundamental topic the spectral scaling properties of compressible turbulence analyzed through the lens of symmetries and renormalization‑group (RG) transformations. However, to enhance the overall scientific and structural quality of the manuscript and to make it suitable for publication, several revisions and clarifications are required. The following comments aim to support the author in strengthening the rigor, clarity, and impact of the work.

1- The paper would benefit from a more explicit statement of aims and research questions in the Introduction. The motivation connecting physical intuition (understanding energy spectra in compressible turbulence) with the proposed symmetry‑based framework should be clearly articulated.
2- The literature review is descriptive rather than analytical. It should include a concise comparative summary of major scaling theories (e.g., Kolmogorov‑41, Burgers, KPZ, and RG‑based approaches) and highlight what theoretical gap the present work addresses.
3- The decomposition of the microscopic action into longitudinal and transverse parts (Section 2) lacks mathematical transparency. Each symbol and transformation should be explicitly defined, and the physical interpretation of the terms clarified to ensure reproducibility by readers outside the immediate subfield.
4- The derivation connecting fundamental symmetries to spectral power‑law behavior should be unfolded step‑by‑step. The transition from symmetry constraints to RG fixed‑point behavior currently feels abrupt and requires smoother logical progression.
5- While the manuscript is theoretical, it would significantly benefit from a brief quantitative comparison with existing DNS or LES data at different Mach numbers to illustrate practical consistency with observed spectra.
6- The β‑coefficients and flow functions introduced in Section 4 appear without prior definition or reference. Clarifying whether these are analytical results or approximations extracted from prior studies (e.g., Ref. [26]) is essential.
7- The analysis would gain credibility by discussing possible uncertainty in the estimated scaling exponents (e.g., ±Δβ) and their effect on the dominance of longitudinal versus transverse modes.
8- The conclusion should succinctly address three key points: (a) the exact theoretical novelty compared with the incompressible case; (b) how Mach number and compressibility influence the spectrum; and (c) suggested future directions or possible extensions toward numerical verification.
9- Several sentences are overly long and densely technical, which impacts readability. The manuscript would greatly benefit from careful linguistic polishing and condensation of the most technical passages in Sections 2-3 for clarity.
10- The in‑text citations (e.g., [5], [6]) occasionally lack correspondence or explanatory context. Cross‑checking all references to ensure alignment with the bibliography is necessary before resubmission.

 

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The paper can be accepted for publication.

Reviewer 2 Report

Comments and Suggestions for Authors

The author has thoroughly and constructively addressed the reviewers’ comments. Key aspects, including sector decoupling, pressure formulation, boundary assumptions, and notation clarity, have been satisfactorily clarified. The revised version shows improved coherence and readability. I therefore recommend acceptance of the manuscript in its present form.

Reviewer 3 Report

Comments and Suggestions for Authors

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