Nonlinear Fourier Analysis: Rogue Waves in Numerical Modeling and Data Analysis
Round 1
Reviewer 1 Report
In the paper „Nonlinear Fourier Analysis: Rogue Waves in Numerical Modelling and Data Analysis“, the author investigates the significant role of the nonlinear Fourier analysis in the description of ocean rogue waves in directional sea states. The methodology is also applied to real ocean data. Surprisingly, it is also shown that the corresponding nonlinear wave hydrodynamics at play can be discussed by means of linear theory, based on the fundamental superposition of waves.
The paper is well-written while the author is reporting excellent results that will immediately attract the attention of the marine engineering community.
I only have few comments that the author should take into account during the revision:
- There are some typos in displaying operators and mathematical expressions that should be fixed. I presume these are just WORD compatibility issues.
- I do believe that it would be better to cite the original work by Saffman, P. G., and Henry C. Yuen. "Stability of a plane soliton to infinitesimal two‐dimensional perturbations." The Physics of Fluids8 (1978): 1450-1451, when discussing the oblique solutions of the NLSE. There is even an earlier work referring to this magic threshold angle of 36.26°: Longuet-Higgins, Michael Selwyn. "On the nonlinear transfer of energy in the peak of a gravity-wave spectrum: a simplified model." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences347.1650 (1976): 311-328. Such oblique directional structures have been indeed observed recently: Chabchoub, Amin, et al. "Directional soliton and breather beams." Proceedings of the National Academy of Sciences 116.20 (2019): 9759-9763.
- It would be interesting to discuss the effect of dissipation for such oblique breathers / super-breathers. For the uni-directional case, the dynamics, which involves FPU-recurrence, can become quite complex, see for instance: Kimmoun, Olivier, et al. "Nonconservative higher-order hydrodynamic modulation instability." Physical Review E2 (2017): 022219.
- The lifetime of such directional breathers are also worth a note. Clearly when the oblique angle is of 35.26°, things can become quite dramatic as dispersion vanishes.
- I am still puzzled about the fact that linear superposition of linear waves can describe the nonlinear wave evolution. I understand that at a specific moment or location a signal can be transformed into its fundamental Fourier modes. However, each single mode will in reality not evolve following a linear evolution equation. Especially, when considering coherent structures, known to be a result of balance between dispersion and nonlinearity. This needs clarification.
The paper can be published in Journal of Marine Science and Engineering, once the comments above have been addressed.
Author Response
Please see attached Word file.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
Dear Author,
Thank you for this hudge amount of work. The article is interresting and demonstrate a deep knowledge of
both the mathematics and wave modelling.
To be honnest, it is dificult to estimate how relevant is this article for the community :
On one hand, it is interresting and promote innovative and promising method for wave analysis.
On the other hand, the article is too long, adresses too many topics and is sometimes written like a book, sometimes like an oral presentation. This is potentially due the fact it is a special journal issue...
It is also referring too much to the author itself.
From an engineering perspective, I beleve it is difficult to extract methods from the article in its current form.
I acknowledge some examples are given in part 7, but it is far from realistic applications.
From the scientific point of view, the method described would benefit from comparisons to field data or to other numerical models.
It would demonstrate the accuracy of the method, the perfomance one can expect from it and provide concrete results to the reader.
In addition, it would help defining the limits of applicability of the present model.
Last, the paper is really long and difficult to read. Its organization is not optimal, too many topics are adressed and there is
a risk of being lost when reading this paper...
In my opinion, too much pages are devoted to simple concepts and not enough to more complex concepts that would help
the reader to understand more in depth the objective of the paper.
Thus, I believe the article should be modified to :
- provide a more specific contribution on a dedicated point chosen by the author,
- more applications, to be potentially transfered to engineering domain in the Future
- propose validation or verification on concrete cases, comparison with other method or field experiments.
You will find attached some comments in the text.
with my best regards
Comments for author File: Comments.pdf
Author Response
Please see attached Word file.
Author Response File: Author Response.pdf
Reviewer 3 Report
This paper developed a method (NLFA) to simulate the nonlinear waves motions. The problem considered in this paper is definitely of interest. It is of great current interest in the research community of coastal engineering. The paper is well written in this present form. I strongly recommend this manuscript to be published in the Journal of Marine Science and Engineering.
Author Response
Please see attached Word file.
Author Response File: Author Response.pdf
Reviewer 4 Report
The manuscript "Nonlinear Fourier Analysis: Rogue Waves in Numerical Modelling and Data Analysis" is a significant and important contribution to the field of rogue wave and wave-modelling and should be published, however it needs extensive revision before it can be accepted. It contains a range of errors in numbering of equation, in referring of Figures and appears at time as independent sections cohered to one another inconsistently. I will give a series of examples below, however this should be regarded as only the first round of revision. Another referre will have to continue the second revision, as it is a long manuscript and has over 200 equations.
This paper contains interesting and important data, and should therefore be written in a style avoiding exclamation points, and the extensive use of "I". The paper becomes automatically more pleasant to read.
Abstract:
Line: 20. Replace exclamation point with ", which is rather surprising." Or any similar term.
Page 2, line 83. I cannot find the reference Khintchine [1934] in the list. Please add it. Also, the name Rice is not introduced elsewhere, is it related to reference from 1934? Please cite accordingly.
Page 3.:
Line 88: The fields..."conceptualize" not "think".
Line 108: Replace "I have" with "In recent papers, it was.."
Line 111: "Here I construct", replace with " Here, the asymptotic...(2+1 NLS) are constructed, which describes.."
Line 114: Replace "I call this" with "This is here named..."
Line 117: "I show that" , "It is here shown that..:"
Line 126: "I use", "We apply the.."
Page 4:
Line 134: "I have shown" and exclamation point at the end. Replace with: "In recent papers, it is remarkably shown that.." and remove exclamation point.
Line 150: "I also suggest" "In light of this, it is suggests that..."
Line 155: Remove exclamation point at the end of the sentence "In this new notation" and add "remarkably" to "...looks remarkably like...".
Line 156: "This we have a miracle." Replace by an exceptional finding or "extraordinary finding".
Line 158-163. Does this really apply to every case on nonlinear phenomena? What about cases where the spectrum is completely inessential and entirely complex? Will the spectrum affect the numerical results? Please discuss this in this paragraph.
Line 250: Fig 2 shows some quite detailed values (angles in the wavenumber domain). Is the author aware of any empirical observations that sustain at least certain patterns of this figure, in terms of citations? For instance, the author describes in the legend of Fig 2 that Fig 8 described Phillips four-wave interactions which occur for wind-wave interactions. This is an example, but it should stay in the main text, and not in the legends. Furthermore, this very statement on Fig.8 should be supported by a citation.
Line 176: Repetitive use of exclamation points. Please avoid this in the entire manuscript, and replace such comments with a supportive word, such as "namely", "interestingly" or "surprisingly". Keep the text as concise as possible, avoiding grammar excitations that may obscure the logical reasoning, in order for these excitations to have a function, the reader and the author should be in the same room, and experience the same reasoning, which is impossible when the reader is in a different time when he reads it. Thus, make the text neutral so it can be read at any time, with a preserved logical thread. Also, "[refs]" in this line needs references.
Also, avoid "I", and try to use neutral terms, or if approaching a pedagogic sentence, rather use "We", as it includes the reader in a "symbolic" manner.
Page 13, line 427: Change "Are the first ...to last?" to "In order to find out whether the first three expressions are equivalent to last, we try using ..:"
Page 14: Line 429: The author quotes that Fig 2 (which was mentioned here in the previous points) is physically satisfying. Which example can be made to confirm this? Can a reference be given?
Line 448: Please do not give NLS the equation twice, it was given in eqn. 11 already: Refer to "eqn (11)". As it stands, it is given as eqn. nr. 59 and 11 at the same time.
Line 461: Which reference is related to Baker in 1907? In 1907 there were no rogue wave models.
Line: 530, p 16: Avoid using "classical" for the Schrödinger equation, as it can be confused with "for classical" (mechanics), Use "conventional" or "standard".
Page 17:
Equation 76: Please write the terms -2F_xG_x as -2G_xF_x as it then corresponds exactly to eqn 77 (should commutation play a role).
Line 568: "the nonlinear wave equations themselves". Which?
Line 571-573: Rephrase.
Line 79: Add space after equation
Line 583: Why is this equation written as an equation of Xi, and H? The sentence in line 581 states that one can compute H from Xi and F, so it should be given accordingly in the form it is given in eqn 82 instead.
Line 594: "we solve (76) or (78) rephrase "solve 76 by the Hirotha operator in 77"
Page 18:
Eqn 82: Write with fraction 1/a, not the exponential function divided by a.
Eqn 83: In following line, 630 "is a source term of (88). Usually 88 must be given before it is cited. It is logically sound to introduce it. Either remove this sentence or introduce 88.
Page 20:
Line 698: Remove the block line around this paragraph. Start the paragraph with "In summary, we first measure..."
Page 23:
Line 782: Add "natural" in front of logarithm
Line 797: merges with equation 117. Correct.
Page 24:
Line 805: Equation 121 has an error. It should have a minus sign in the exponent, in front of i Omega_j t
Line 807: Add the" last two boxed equations". Is this from another source? The last equations are not boxed! Use equation number in all citations of equations.
Line 809: Same problem here. Use equation number.
Page 27:
Line 904: "...in the simplest of terms! We can celebrate at this point!" Please rephrase.
Page 25:
Line 859: "This means.." Does this mean that a rogue wave on 1000 meter deep ocean cannot be higher than 5 kilometers? Is the condition then in kh>1.36 really unique? Please consider how the statement "This means..." May be interepreted. If it was in shallow water, one could indeed understand that one wave appearing in a say X-crossing pattern, in 20 cm of water, it could never be higher than 1 meter. But this condition may be strange to give for open ocean deep water, unless we consider asteroid impacts. But these are rogue wave models from nonlinear interaction between wavetrains, so this condition seems a little odd, if not superfluous and computationally strange, as it would impose numerical limitations where the solutions are already far from realistic. Would it be better to generate a different condition, that keeps computational prediction within reasonability? I don't see what is reasonable about predicting that a wave at open oceans cannot be higher than a mountain from the Alpes, for isntance.
I leave the remaining of the manuscript to be checked by the author, following this structure. Please check that all equations are coherently cited, numbered correctly, not given twice as one was, and that there are no errors in them.
Author Response
Please see attached Word file.
Author Response File: Author Response.pdf
Round 2
Reviewer 2 Report
Dear Author,
Thank you for the corrections made on the original paper. This is really appreciated.
The paper is still long and difficult to read, I believe it could be better if focused on a specific point of your NLFA, instead of presenting so many details and aspects of the method.
This was already underlined in my previous review, so as the need to remove some trivial parts in order to shorten the paper, which is really too long.
However, the new version is much clearer and sections 8 and 9 are really usefull for the reader to understand what the outputs are. The paper would be more powerfull if a clear comparison could be made between your method and the classical FFT in sections 8 and 9, in order the reader can undertand was is the gain of this analysis : what does this new analysis bring to the user?
This could be more beneficial to the paper than section 10. Section 10 does not bring anything and it is suggested to keep it for another paper.
Little aditional comments are made in the document attached. My recommandation is to make the paper shorter, and limit the self-citation as much as possible, if possible.
With my best regards
Comments for author File: Comments.pdf
Author Response
See attached Word file.
Author Response File: Author Response.pdf
