N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study analytically examines the dynamic modes of motion of vortex symmetric solitons (hetons) in a two-layer ocean with layers of equal thickness in the quasi-geostrophic approximation. Bifurcation diagrams are constructed and analyzed, examples of vortex trajectories are given, including special modes of absolute and relative choreographics. The article contains new analytical results and their interpretation. They are interesting and relevant. The article is optimally structured and very neatly formatted. Minor technical shortcomings: 1) the notation \rho for the ratio of the scales of N-gons in the upper and lower layers is not very successful, since previously it was used to denote the density of a liquid medium. 2) Figures 5-12, 15-18 do not have labels along the axes and do not have explanations about this in the captions. I recommend that the article was accepted after minor technical corrections.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Manuscript Number: fluids-2938894-v1

Full Title: N-symmetric interaction of N hetons. II: Analysis of the case of arbitrary N

 

I - General Comments

 

The authors have extended the results obtained in [66], hereafter referred to as SKDR, both in the limit of point vortices and for finite-core vortices, to the case of an arbitrary N. The manuscript needs improvement before to be accepted as Fluids ‘paper.

 

II - Specific Comments

 

(i) Please, use “vortical structure(s)” instead of “vortex” or “vortices” when writing the text. Another possibility/suggestion is to include the scientific term “vortical structure” into the “Introduction” aiming to justify the popular use of “vortices” along the manuscript.

(ii) The “Abstract” should describe the objectives, the methodology and the main conclusions of the paper in about 200 words. It should contain neither formulae nor reference to bibliography.

(iii) The “Abstract” needs to be complemented to clarify the effective contribution of the work. Please, include the present art state of the investigated topic to justify the publication in Fluids.

(iv) The “Introduction” must be restructured too. The authors should clarify the motivation to develop the present research. A technical overview of the present art state of the investigated topic is necessary. It also required a description of the exact question and hypothesis in which the paper will be developed in next sections. How is the manuscript contributing to engineering practical problems of interest for readers?   

(v) In “Introduction” there is citation of only one previous work published in Fluids by other researchers, namely the paper so referred as [53]. Why? The authors also need to justify the intended publication into the Fluids’ scope. The cited “References” are older than eight years. The authors can check that 2% of the professional literary sources used have been published in the last five years. Thus, it is recommended to use/add more up to date references and reduce the total number of citations (if possible for less than 60) too.

(vi) In “Mathematical model and problem formulation”, it is suggested to include some citations before introducing each one of equations.

(vii) All the symbols and notation must be defined into the manuscript. It is suggested a Table including Nomenclature.

(viii) In “Analysis of possible motion regimes”, please, introduce a first paragraph explaining the general aspects for that section.

(ix) The trade-off between increased computation time and accuracy of results also needs to be mentioned by authors. It is necessary to comment about the computer specifications and CPU time related to a typical numerical simulation.

(x) In “Discussion and concluding remarks” the authors have commented the following: “The concept of a two-layer quasi-geostrophic heton is attractive, first of all, because it is the simplest model for the horizontal transfer of temperature anomalies.” Another comment given by the authors: “In this paper, we have not addressed the stability of stationary configurations. This will be addressed in a future paper. We expect that the motions of unstable configurations may also be irregular, breaking the initial symmetry of the heton”.

So, it is necessary to include comments with respect to the numerical results behaviour as compared to previous results available in the literature (particularly, some baseline data, if possible). The readers need to be better convinced of the novelties presented by the current manuscript. What are the drawbacks in this numerical study? And what are the ways to resolve them? Please, complete the section with new comments to clarify the main contribution of the present manuscript; which will help to justify the intended publication in Fluids.

(xi) The original manuscript needs improvement in accordance with all points above mentioned before to be considered for publication in Fluids.

(xii) English language and style are fine/minor spell check required.

 

Comments on the Quality of English Language

English language and style are fine/minor spell check required.

Author Response

See file attached

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

I reviewed the manuscript entitled as "N-symmetric interaction of N hetons. II: Analysis of the case of arbitrary N" by Koshel et al.. Authors focus on the cases with N hetons, which are concepts of two-layer heat-advection flows. Families of new stationary solutions are obtained and their properties, including asymptotic ones, are investigated in detail. These results are then expected to help the study in the domain of geophysics.

Overall the paper is well written. Literature are well introduced, and all derivations are clearly described. Results are convincing to support the theoretical analysis.

I have still the following questions.

1) As I understand, the model of two-layer quasi-geostrophic flow, i.e., equation (1), is only an approximation of Navier-Stokes (NS) equations. I am then confused on the significance of searching for exact solutions of such an approximated model. I suggest authors to discuss more realistically on the use of these exact solutions, and the difference by comparing to NS flows. Can the present solutions reveal some topological structure or dynamics of real NS flows? Otherwise, the present work will be quite less significant.

(2) Can we consider the present setup as a simplification of any real physical problem? That may help to link the results to the reality.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Manuscript Number: fluids-2938894-v2

Full Title: N-symmetric interaction of N hetons. II: Analysis of the case of arbitrary N

 

The manuscript was not reviewed in accordance with my review report send to the authors last week. As a reviewer of “Fluids”, I do not recommend the text for publication.

Comments on the Quality of English Language

Moderate editing of English language required.

Author Response

Dear Referee,

We had addressed most of your suggestions in our reply letter.

We have now included a comment on the numerical for the finite-core hetons:

``Simulations are run using Contour Surgery with the method’s standard set-up[14]. Equations are marched in time using a fourth-order Runge-Kutta scheme with a time step set by the vortices PV: 421 ∆t = π/(10q).''

[14] Dritschel, D. G. Contour surgery: A topological reconnection scheme for extended integrations using contour dynamics J. Comput. Phys., 1988, vol. 79. no. 2, pp. 240-266.

 

Reviewer 3 Report

Comments and Suggestions for Authors

I can recommend for the publication of the present manuscript.

Author Response

Thank you for your review on the paper.

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Manuscript Number: fluids-2938894-v3

Full Title: N-symmetric interaction of N hetons. II: Analysis of the case of arbitrary N

 

The manuscript has been sufficiently improved to warrant publication in Fluids.

Comments on the Quality of English Language

English language and style are fine/minor spell check required.