Challenges in Description of Nonlinear Waves Due to Sampling Variability

Round 1

Reviewer 1 Report

This paper discuss the importance of the sampling variability to the nonlinear water waves. The authors used the numerical simulation to demonstrate the issue.  The problem considered in this paper is definitely of interest. It is of great current interest in the research community of coastal engineering. However, 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

We thank Reviewer 1 for the comments. We are glad to hear that Reviewer 1 has found the paper interesting and is recommending it for the publication in JMSE. 

The English has been improved in the revised version of the manuscript.

Reviewer 2 Report

 

General comment:

The paper deals with the analysis of sampling variability in the estimators of wave parameters, by means of numerical simulations.  As a general comment, the paper is well written, the methodology is well explained and it contains interesting data to be applied in maritime engineering. However, sometimes the explanation of the different parameters that appear in the manuscript is scarce. There are also some misprint to solve.

 

Specific comments:

1. Page 2, line 46: extra dot after the word “data”, just at the end of the line.
2. Page 3, from line 112 to line 128: there are several undefined parameters in the text, such as "gamma_p" (line 115), "phi" and "phi_p" (line 122), or N (equation 2), that not every reader needs to know previously. Please define each of the parameters that appear in the manuscript. Maybe a “list of symbols” at the end of the manuscript could help. 
3. Page 3, line 132: it seems that the sentence “Intrinsic wave variability is accounted for by changing random seeds for each run” is has a different format.
4. Page 4, line 145: please correct the word “paprameters”
5. Page 12, line 333: please correct the word “too” in the sentence “…estimator are due too limited…”
6. Page 12, Figure 10: please consider increasing the font size in the equations, especially in panel (b).
7. Page 12, line 344: it seems strange not having a conclusions section in the article. Please consider these two options: (1) rename section 4 as "discussion and conclusions," or (2) create a new section that includes the main conclusions of the article.

Author Response

Below the Reviewer 2 comments are listed in Italics, while the reply to them is included after the bullet points

General comment:

The paper deals with the analysis of sampling variability in the estimators of wave parameters, by means of numerical simulations.  As a general comment, the paper is well written, the methodology is well explained and it contains interesting data to be applied in maritime engineering. However, sometimes the explanation of the different parameters that appear in the manuscript is scarce. There are also some misprint to solve.

• We thank Reviewer 2 for the detail review of the paper; accounting for the given comments improved the quality of the manuscript. The description of the parameters has been improved as well as misprints have been corrected in the revised version of the manuscript. Some improvements have also been included in the conclusions.

 

Specific comments:

1. Page 2, line 46: extra dot after the word “data”, just at the end of the line.

•  Has been corrected.

2. Page 3, from line 112 to line 128: there are several undefined parameters in the text, such as "gamma_p" (line 115), "phi" and "phi_p" (line 122), or N (equation 2), that not every reader needs to know previously. Please define each of the parameters that appear in the manuscript. Maybe a “list of symbols” at the end of the manuscript could help. 

•  The parameters have been defined in the revised version of the manuscript.

 3. Page 3, line 132: it seems that the sentence “Intrinsic wave variability is accounted for by changing random seeds for each run” is has a different format.

•  The format has been corrected.

 4. Page 4, line 145: please correct the word “paprameters”

•  The word has been corrected.

 5. Page 12, line 333: please correct the word “too” in the sentence “…estimator are due too limited…”

• It has been corrected.

6. Page 12, Figure 10: please consider increasing the font size in the equations, especially in panel (b).

• The font has been increased.

 7. Page 12, line 344: it seems strange not having a conclusions section in the article. Please consider these two options: (1) rename section 4 as "discussion and conclusions," or (2) create a new section that includes the main conclusions of the article.

•  Thank you for this suggestion the title has been changed to “Discussion and conclusions”.

 

Reviewer 3 Report

The paper by Elzbieta M. Bitner-Gregersen, Odin Gramstad, Anne Karin Magnusson, and Mika Malila investigates the impact of the wave field nonlinearities and the effect of sampling variability on the wave temporal and spatiotemporal statistics in selected sea states. The topic of the research fits the area of JMSE very well and constitutes great importance from both fundamental and applicative points of view. The manuscript is well written and can be recommended for publication. However, several minor comments are suggested for consideration by the authors.

1 – It is mentioned that the HOSM numerical simulations were restricted to the 3^rd order of nonlinearity by selecting M = 3. The HOSM method is widely applied at much higher orders approaching the fully-nonlinear solutions. Why do the authors limit the analysis to the 3^rd order? This point is not clear.

2 – It is said in the manuscript that the physical domain was discretized by the grid having (1024 x 1024) and (512 x 512) points in x and y directions. Did the authors consider the grid independence of the solution? What is expected to happen if the resolution is refined or coarsened? Do the conclusions of the paper remain the same independent of the numerical grid resolution?

Is the grid resolution in the Fourier space maintained similar to the one in the physical space?

3 – The authors mentioned that 1000 repetitions were executed in the Monte-Carlo simulations of the unidirectional waves, while only 20 repetitions were used for the directional seas. The sampling period was considered to be 0.2s. What do authors expect if the number of repetitions increases or the sampling period reduces? Does it influence the conclusions of the paper?

4 - It is mentioned that the HOSM simulations are presented in green color in Figs. 1-3. The green color seems to be missed in those plots.

5 – The authors write on p. 11 line 316: “… it should be noted, that the results presented herein refer to unidirectional wave fields”. Also, on p. 13 line 375 it is indicated: “… the results of this study have been derived for unidirectional wave fields, therefore they should be applied with care in engineering applications ...”. What results do correspond to 2-dimensional (short-crested) wave states? It looks like only unidirectional cases were considered that contradicts to the introductory part of the paper.

6 – Several misprints in the references were noticed. See for instance reference [25], p.14 line 450 of the manuscript.

 

Author Response

Below the Reviewer 2 comments are listed in Italics, while the reply to them is included after the bullet points.

The paper by Elzbieta M. Bitner-Gregersen, Odin Gramstad, Anne Karin Magnusson, and Mika Malila investigates the impact of the wave field nonlinearities and the effect of sampling variability on the wave temporal and spatiotemporal statistics in selected sea states. The topic of the research fits the area of JMSE very well and constitutes great importance from both fundamental and applicative points of view. The manuscript is well written and can be recommended for publication. However, several minor comments are suggested for consideration by the authors.

• We thank Reviewer 3 for the comments and for finding the paper suitable for the publication in JMSE. Accounting for the given comments has improved the quality of the paper.

1. – It is mentioned that the HOSM numerical simulations were restricted to the 3^rd order of nonlinearity by selecting M = 3. The HOSM method is widely applied at much higher orders approaching the fully-nonlinear solutions. Why do the authors limit the analysis to the 3^rd order? This point is not clear.

• We have adopted M=3 in the analysis, as this order includes the leading order nonlinear dynamical effects, including the effect of modulational instability, as mentioned in the manuscript. From earlier experience, we do not expect that our results will be significantly affected by increasing M, and the conclusions will remain the same. Some comments are added in Sec. 4 “Discussion and Conclusions”.

2.- It is said in the manuscript that the physical domain was discretized by the grid having (1024 x 1024) and (512 x 512) points in x and y directions. Did the authors consider the grid independence of the solution? What is expected to happen if the resolution is refined or coarsened? Do the conclusions of the paper remain the same independent of the numerical grid resolution?

Is the grid resolution in the Fourier space maintained similar to the one in the physical space?

• The domain-size in Fourier-space was fixed so that k_x^(max)=k_y^(max)=8k_p in the fully de-aliased grid. This means that changing n_x and n_y affects the size of the domain in x/y-space, as well as the grid-spacing in wavenumber space. As the Reviewer 3 correctly points out, in order to obtain “converged” results it is important to have a sufficiently large physical domain as well as sufficiently fine resolution of the wavenumber space. Hence, larger values of n_x and n_y is generally an advantage. Based on our own experience with HOSM, as well as from numerous works using HOSM in the scientific literature, we know that our choice of discretization is sufficient to obtain “converged” results with respect to discretization. We have also previously carried out convergence studies with respect to this, but we have not done this specifically for the current paper.
• Another reason that it is important to use a sufficiently large computational domain is that the sample skewness and kurtosis are biased estimators of the real populations in the case that the samples are not independent, which is the case for wave surface data. This effect is more pronounced for small simulation domains; see Gramstad et al. (2018) referred in the manuscript. For the relatively large domain size adopted in the present study, the results are little affected by this bias.

• Some additional comments are included in the revised manuscript, in Sec.2 and Sec. 4 “Discussion and conclusions”.

3.- The authors mentioned that 1000 repetitions were executed in the Monte-Carlo simulations of the unidirectional waves, while only 20 repetitions were used for the directional seas. The sampling period was considered to be 0.2s. What do authors expect if the number of repetitions increases or the sampling period reduces? Does it influence the conclusions of the paper?

• From our experience with HOSM simulations we expect that the carried out number of simulations is sufficient to get accurate results, and the number of simulations will have negligible effect on the presented results. Increasing a sampling step alone will increase the number of data-points, but not the accuracy of the estimators of wave characteristics derived from them. This is already discussed in Sec.4 “Discussion”.

4.- It is mentioned that the HOSM simulations are presented in green color in Figs. 1-3. The green color seems to be missed in those plots.

• Thank you for this comment, it is corrected.

5.- The authors write on p. 11 line 316: “… it should be noted that the results presented herein refer to unidirectional wave fields”. Also, on p. 13 line 375 it is indicated: “… the results of this study have been derived for unidirectional wave fields; therefore they should be applied with care in engineering applications ...”. What results do correspond to 2-dimensional (short-crested) wave states? It looks like only unidirectional cases were considered that contradicts to the introductory part of the paper.

• We thank for this comment. The results presented are based on unidirectional simulations, but the examples of directional simulations are only used to support the conclusions. In Figure 10a results for directional data are plotted. In the updated version of the manuscript the text is revised to make clearer to which data we refer.

6.- Several misprints in the references were noticed. See for instance reference [25], p.14 line 450 of the manuscript.

• The miss prints are corrected in the revised version of the manuscript.

Reviewer 4 Report

This paper investigated the impact of nonlinearity of wave field by comparing the wave temporal and wave spatial using linear, 2^nd-, and 3^rd-order unidirectional numerical simulations and further, 3^rd- order data are simulated by a numerical solver based on the Higher Order Spectral Method. Also, the results analysis supported with the field data. The present work has interest to the researchers working in the same field to set a benchmark. The manuscript is well structured and well written.    

  On the other hand, the numerical results are supported by the published references and the presented results are well explained.

This reviewer recommends to accept this paper in the present form.

Author Response

Below the Reviewer 4 comments are listed in Italics, while the reply to them is included after the bullet points

This paper investigated the impact of nonlinearity of wave field by comparing the wave temporal and wave spatial using linear, 2^nd-, and 3^rd-order unidirectional numerical simulations and further, 3^rd- order data are simulated by a numerical solver based on the Higher Order Spectral Method. Also, the results analysis supported with the field data. The present work has interest to the researchers working in the same field to set a benchmark. The manuscript is well structured and well written.    

 On the other hand, the numerical results are supported by the published references and the presented results are well explained.

This reviewer recommends to accept this paper in the present form.

• We thank Reviewer 4 for the comments. We are glad to hear that Reviewer 4 has found the paper interesting and is recommending it for the publication in JMSE.
• The English has been improved in the revised version of the manuscript.