Partitioning the Extreme Wave Spectrum of Hurricane Wilma to Improve the Design of Wave Energy Converters
Round 1
Reviewer 1 Report
This paper shows the relationship between wave parameters under hurricane conditions.
The results are interesting, but some questions I will address.
(1)The conclusions are summarized in Section 4.
It is known that the narrower the wave spectrum, the more nonlinearity there will be.
These conclusions are consistent with that.
These things are true for ocean surface waves in general, not just in the
case of tropical cyclones, are they not?
(2)Line 24: What is the dispersion effect here?
If the wavelength is longer in shallow water, is it considered to be a non-dispersive wave?
(3) When you estimate nonlinearity by partitioning the wave spectrum, does that mean that you don't consider, for example, the interaction between swells and wind and waves?
(4)Line 179: Is it the derivative with depth d? or group velocity?
(5) Line 189:
It is not zero even if the linear dispersion relation holds.
Does this mean that this term will be small when the linear dispersion relation holds?
(6) Line 198
I think that the spectra (or output from the model) at multiple stations are used for the partition.
Did you use data from a single point?
(7) Fig.9 and paragraph from Line 414.
How the wave group is determined (i.e., how red line are drawn, in particular, first and last individual waves) ?
(8) Fig.11: The definition of a wave grouping is unknown, but for the phenomenon in which a beat appears, such as in Fig. 9, it is likely to become more pronounced as the wave spectrum becomes narrower.
Both epsilon and Qp are parameters related to the distribution of the wave spectrum.
Therefore, taking one of them as a parameter of the wave grouping (Op in this case), isn't it obvious and meaningless to argue that ε is related to the parameters of the wave grouping ?
Author Response
Response to Reviewer 1 Comments
Point 1: The conclusions are summarized in Section 4.
It is known that the narrower the wave spectrum, the more nonlinearity there will be.
These conclusions are consistent with that.
These things are true for ocean surface waves in general, not just in the
case of tropical cyclones, are they not?
Response 1: Agreed. For surface gravity waves, the narrower the wave spectrum, the more nonlinearity. We have removed the statement “nonlinearity increased as the degree of wave grouping increased with decreasing bandwidth” in the Conclusions.
Point 2: Line 24: What is the dispersion effect here?
If the wavelength is longer in shallow water, is it considered to be a non-dispersive wave?
Response 2: The nonlinear interaction coefficient is given by
|
|
(8) |
For long waves or shallow water waves , , and the nonlinear dispersion term is a function of the wavenumber. Hence, in shallow water, the waves are dispersive waves. The same holds for the linear dispersion relation.
The dispersion effect is that the contribution of the nonlinear dispersion term in (8) exceeds the contribution of the wave-induced current term when , but the waves continue being dispersive waves.
We have improved the description of (8), removed the statement “dispersion effects can override the stabilizing effect of the wave-induced current” in line 24 and improved the abstract.
Point 3: When you estimate nonlinearity by partitioning the wave spectrum, does that mean that you don't consider, for example, the interaction between swells and wind and waves?
Response 3: True. The interaction between swells and wind and waves is not considered because there are no single-point wind measurements at the wave measurement site. Nonlinearity is estimated through the Benjamin-Feir Index, which essentially compares steepness (a measure of nonlinearity) and spectrum bandwidth (dispersive effects).
The partitioning of the wave spectrum is used to obtain the direction of the dominant waves. Then, a frequency spectrum, in that direction, is estimated by integrating for all frequencies within the low directional spread obtained from the partitioning. This accounts for the dispersion within wave groups.
It should also be noted that by integrating in the direction of the dominant waves (the peak direction), swells and wind-waves from different partitions are also considered.
Point 4: Line 179: Is it the derivative with depth d? or group velocity?
Response 4: Thank you for the question. It is the derivative with the wavenumber, that is, the group velocity. We have added this for clarity in line 179.
Point 5: Line 189:
It is not zero even if the linear dispersion relation holds.
Does this mean that this term will be small when the linear dispersion relation holds?
Response 5: Thanks. There is a transition from positive to negative values at . The nonlinear dispersion term does not hold for , and the nonlinear interaction coefficient is negative. For , the nonlinear dispersion term holds, and is positive. In this case, the dominant wave trains are unstable. We have added to the explanation of in the manuscript.
Point 6: Line 198
I think that the spectra (or output from the model) at multiple stations are used for the partition.
Did you use data from a single point?
Response 6: We used single-point wave measurements, recorded every two hours. The directional wave spectrum was estimated from these wave measurements, every two hours. These are the spectra used for the partition. We have not used any spectra at multiple stations for the partition.
We have included an explanatory paragraph in line 198.
Point 7: Fig.9 and paragraph from Line 414.
How the wave group is determined (i.e., how red line are drawn, in particular, first and last individual waves) ?
Response 7: We consider that the dominant waves exist as wave groups. A characteristic of wave groups is that the elevation envelope decays away from the highest individual wave, close to the center of the group. We tried to obtain the envelope of the wave groups using the Hilbert transform, but it did not work well.
The wave groups were determined empirically, from the highest individual waves observed within the elevation record. The highest positive peak within a wave group was taken as a reference. Then, the decreasing peaks on both sides were followed until the smallest peaks in the wave group were found. The first and last individual waves were determined empirically, using the zero-upward-crossing wave elevation method.
We have reviewed and updated Fig. 9, and added the above explanation in line 414.
Point 8: Fig.11: The definition of a wave grouping is unknown, but for the phenomenon in which a beat appears, such as in Fig. 9, it is likely to become more pronounced as the wave spectrum becomes narrower.
Both epsilon and Qp are parameters related to the distribution of the wave spectrum.
Therefore, taking one of them as a parameter of the wave grouping (Op in this case), isn't it obvious and meaningless to argue that ε is related to the parameters of the wave grouping ?
Response 8: Wave groups are considered as the superposition of linear or nonlinear wave trains of similar frequency propagating in the direction of the dominant waves. We have estimated the frequency spectrum for the direction of the dominant waves. This is a reasonable approach to infer wave groups from spectral parameters. As the wave spectrum narrows, more wave groups are observed in the elevation records, but not necessarily more pronounced, nor with higher elevations. The elevation records of Hurricane Wilma have not been fully analyzed.
A narrow wave spectrum has a sharp peak, which corresponds to a pronounced group-structure. As Qp increases in the direction of the dominant waves, the spectrum becomes narrower. On the other hand, epsilon, the spectral bandwidth in the direction of the dominant waves, is a measure of the concentration of energy. As epsilon decreases, the spectrum becomes narrower. It is expected that, as Qp increases, epsilon should decrease. Indeed, we have estimated Qp and epsilon in the direction of the dominant waves. The relation between Qp and epsilon is therefore meaningful.
Fig. 11 shows the relation of Qp and epsilon in the direction of the dominant waves. As epsilon decreases, Qp increases. Hence wave grouping increases with decreasing bandwidth. This is the expected result.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper is organized in a good way. Its structure is clear, and the wordings are sufficient. The paper can be considered for publication after clarifying the mentioned questions
1. How can partitioned extreme wave spectrum data be used to inform the design of Wave Energy Converters?
2. What challenges exist when using Hurricane Wilma's wave spectrum to optimize Wave Energy Converters?
3. What are the potential benefits of partitioning Hurricane Wilma's extreme wave spectrum for Wave Energy Converters?
4. How was the energy spectrum of Hurricane Wilma partitioned in the shallow water study?
5. What was the maximum estimated significant height of the dominant waves?
6. What mechanisms were responsible for driving the wave groups in Hurricane Wilma?
7. How did dispersion effects affect the stabilizing effect of the wave-induced current in Hurricane Wilma?
Author Response
Response to Reviewer 2 Comments
Point 1: The manuscript should undergo extensive editing of English language and style.
Response 1: Thanks for the comment. We paid for the MDPI editing service, and the manuscript was then checked by a native English-speaker.
Point 2: How can partitioned extreme wave spectrum data be used to inform the design of Wave Energy Converters?
Response 2: The partition of the extreme wave spectrum provides the significant wave height, the wave steepness, the spectral bandwidth and the degree of wave grouping of different wind-generated wave systems. We consider the peak direction obtained for each partition as the direction of the dominant waves. Since the superposition of wave trains of similar frequency propagating in the direction of the dominant waves form wave groups, the peak direction corresponds to the direction of the wave groups. These spectral parameters make it possible to determine whether Wave Energy Converters (WEC) would be confronted with multidirectional wave groups of different heights and steepnesses simultaneously. These parameters can be used by designers of WECs to simulate the interaction of the WEC with the wave group, either numerically or in wave tanks, so that the WEC can survive extreme wave conditions.
Point 3: What challenges exist when using Hurricane Wilma's wave spectrum to optimize Wave Energy Converters?
Response 3: In shallow waters, waves generally come from a particular range of dominant wave directions. Submerged and semi-submerged WECs can probably be optimized to operate in stormy conditions. Regarding shorelines with multidirectional waves, identified using Hurricane Wilma´s wave spectrum, the WEC should self-tune or tuned remotely to adjust and operate efficiently to changes in wave direction and wave height. This could reduce energy lost in harnessing wave energy. Anchoring and mooring systems should also withstand local severe storms.
Hurricane Wilma was an extreme tropical cyclone. There are storm events that are not as severe as Wilma when they approach the shore. WECs are usually turned into a secure mode to sacrifice energy harvesting in favor of device integrity. A challenge for WECs is whether they could be able to capture wave energy from certain storms. For example, atmospheric frontal systems, trough lines, squall lines and some tropical cyclones. The extreme wave conditions generated by Wilma illustrate the wave conditions that submerged and semi-submerged devices would have to contend with. The amount of wave energy driven by individual storm events may be worth a try.
Point 4: What are the potential benefits of partitioning Hurricane Wilma’s extreme wave spectrum for Wave Energy Converters?
Response 4: The survivability of Wave Energy Converters (WEC) is an issue, particularly during storms. The partitioning of Hurricane Wilma’s extreme spectrum provided parameters of multidirectional wave groups. For shorelines with multidirectional waves, submerged and semi-submerged devices should be designed to accommodate changes in wave direction and wave height. However, under extreme hurricane conditions, energy harvesting should have to be sacrificed in favor of device integrity.
The estimation of the spectral wave steepness in the direction of the dominant waves associated with breaking wave groups is another benefit for WECs. Since the spectral parameters are estimated at specific times, it is possible to see the elevation wave record at those specific times. This is important for assessing the response of WEC to wave slamming due to breaking waves, because the magnitude of the mooring loads depends on the location of the breaking. In this way, the durability of device components when hit by extreme wave groups can be improved.
Overall, the extreme wave conditions generated by Wilma represent a step towards improving the ability of WECs to survive storms.
Point 5: How was the energy spectrum of Hurricane Wilma partitioned in the shallow water study?
Response 5: The directional wave spectrum of Hurricane Wilma in the shallow water study was partitioned according to reference [38] and using the free software developed by [39].
The partition algorithm was carried out in the following eight steps.
(1) Filter the measured energy spectrum using double convolution. A 3x3 smoothing filter was used to get rid of the small peaks in the measured spectrum, to avoid it being interpreted as a partition.
(2) Identify all possible partitions using a watershed algorithm.
(3) Identify and combine all wind sea partitions using a wave age criterion.
(4) Merge mutual swell partitions.
(5) Select only partitions above noise level and below an energy threshold.
(6) Merge remaining swell partitions that do not have a valley between them and are separated by less than 90o.
(7) Keep only swell partitions with a significant wave values above a default minimum.
(8) Re-order partitions in terms of energy level in descending order.
Point 6: What was the maximum estimated significant height of the dominant waves?
Response 6: The maximum estimated significant height of the dominant waves was 5.5 m. In contrast, the maximum estimated significant heights of the partitions and the omnidirectional energy spectrum were 11 m (swell) and 12 m, respectively.
Point 7: What mechanisms were responsible for driving the wave groups in Hurricane Wilma?
Response 7: We believe that, for most of the time, the superposition of linear waves was responsible for driven groups in Hurricane Wilma.
At times, from the superposition of weakly nonlinear waves, wave groups in shallow water were developed, through the mechanism of modulation instability. This is a novel result since modulation instability is usually considered an open-sea phenomenon.
Point 8: How did dispersion effects affect the stabilizing effect of the wave-induced current in Hurricane Wilma?
Response 8: The nonlinear dispersion term is a function of κh. For long waves or shallow water waves (κh<1), less energy is available with decreasing depth. In this case, the nonlinear term in (8) is negative, the nonlinear dispersion term tends to vanish and, therefore, the wave-induced current has a stabilizing effect because the nonlinear transfer of energy is small [33]. For a specific depth (h=20.73 m), the nonlinear term is a function of the wavenumber. A significant reduction of nonlinear transfer of energy is expected around κ=1.363/20.73 m=0.066 . Then, as long as the wavenumbers driven by Hurricane Wilma remained below this theoretical limit value, the dispersion did not have a pronounced effect on the stabilizing effect of the wave-induced current.
For short waves or deep-water waves (κh>1), more energy is available with increasing depth. The nonlinear dispersion term tends to one. In this case, , the wave-induced current tends to vanish and, therefore, dispersion tends to destabilize the stabilizing effect of the wave-induced current. For the specific depth of the single-point measurements in Hurricane Wilma, the dispersive effects were dominant in wavenumbers ranging between 0.1164 and 0.4986 . The modulation instability was triggered in shallow waters for these wave numbers because the threshold for instability was exceeded (κ > 0.066 ).
Hence, during severe storms in shallow waters, dispersive effects can override the stabilizing effect of the wave-induced current.
Author Response File: Author Response.pdf
Reviewer 3 Report
Ulloa et al. investigated the spectral parameters by partitioning the wave spectrum during Hurricane Wilma. A better understanding of features of such extreme wave events are useful for developing and maintaining the Wave Energy Currents which are deployed in those areas where extreme events can occur. Overall the manuscript is pretty well-written. Improvements are needed in the organizations such as combining the sentences into same paragraph instead of being an isolated sentence. Please see my comments below for details.
Major:
1. The abstract might need some improvement for two reasons. First, there is no information regarding the improvement of WECs as what they premise in the title. Second. Information in Line 18-21 might be useful, however, it does not provide anything new regarding partitioning the wave spectrum, nor closely related to the sentence in Line 17. Again, it should be emphasized/highlighted why partitioning the spectrum of real extreme wave events are important for WECs, with the evidence supported by the analyses.
2. It is not a secret that the extreme wave events should be taken into account for the design of WECs (Line 504-505). Also, why not analyze the bulk property such as significant wave height and significant wave steepness from the omnidirectional energy spectrum during Hurricane Wilma, but rather from each partition? After all the WECs are experiencing the combined effects of all partitions, not the individual partition each time. Will the three partitions well represent the extreme events? – For instance, in Figure 4c, by adding the sum of peak values of the three partitions, roughly it’s only about 30% of the total shown in Figure 4b.
3. The limitation and future direction of this study shall be clearly indicated in the conclusion section.
4. In Line 379, the authors mentioned in Line 376 that “but there is no enough data for a robust linear trend”, however, in Line 379 said “A linear trend, with Qp increasing with S” - This is not well supported by only providing a scatter plot. How do you reach this conclusion?
Minor:
1. Line 122: the audience might want to know what SUV represent and the reference is needed as well.
2. Line 127: why 64 is adapted?
3. Line 88-89: It is not necessary to only put one sentence in one paragraph (same comment for Line 479-480 and Line 484-485). Please check throughout the paper.
4. Line 487: Is this a new finding? This seems pretty common to me though.
Author Response
Response to Reviewer 3 Comments
Major Point 1: The abstract might need some improvement for two reasons. First, there is no information regarding the improvement of WECs as what they premise in the title. Second. Information in Line 18-21 might be useful, however, it does not provide anything new regarding partitioning the wave spectrum, nor closely related to the sentence in Line 17. Again, it should be emphasized/highlighted why partitioning the spectrum of real extreme wave events are important for WECs, with the evidence supported by the analyses.
Response Major Point 1: Thank you for the comments. We paid for the MDPI editing service, and the manuscript was then checked by a native English-speaker.
We have rewritten the abstract to link with the title. We have also emphasized in the abstract the importance of partitioning the spectrum of real extreme wave events for WECs.
Major Point 2: It is not a secret that the extreme wave events should be taken into account for the design of WECs (Line 504-505). Also, why not analyze the bulk property such as significant wave height and significant wave steepness from the omnidirectional energy spectrum during Hurricane Wilma, but rather from each partition? After all the WECs are experiencing the combined effects of all partitions, not the individual partition each time. Will the three partitions well represent the extreme events? – For instance, in Figure 4c, by adding the sum of peak values of the three partitions, roughly it’s only about 30% of the total shown in Figure 4b.
Response Major Point 2: We have analyzed the significant wave height and the significant wave steepness from the omnidirectional energy spectrum during Hurricane Wilma. However, such analysis does not provide specific spectral parameters in the direction of the dominant waves. The present work focuses on the possible interaction of WECs with wave groups associated with severe storms. This cannot be specifically achieved from the omnidirectional energy spectrum. While all WECs undergo the combined effects of all partitions, specific spectral properties associated with wave groups cannot be obtained from the omnidirectional energy spectrum.
The aim of this work is not to represent extreme waves with three partitions. The aim is rather to use the peak direction from the three more energetic partitions to infer properties associated with wave groups generated during Hurricane Wilma.
The frequency spectra shown in Fig. 4c are not the energy spectra of the first three partitions. This is why the peak values in Figure 4c do not represent the peak values resulting from the energy spectrum associated with the three partitions. These are only the peak values obtained in a specific direction. Hence, these peak values should not be added together to obtain the peak values in Figure 4b. It should be noted that the frequency spectrum in the direction of the dominant waves takes into account the combined effects of several partitions.
WECs also experience the effects of multidirectional wave groups, which can be unstable and focus wave energy through the mechanism of modulational instability. These effects cannot be inferred from the omnidirectional energy spectrum, but by partitioning the omnidirectional energy spectrum, to obtain a frequency spectrum in the direction of the dominant waves. This also aids in the discernment wave groups associated with wind-waves and swell.
In Fig. 6, we have added the scatter plot of the significant wave height and significant wave steepness estimated from the omnidirectional energy spectrum, as well as from the energy spectrum of the three partitions.
Major Point 3: The limitation and future direction of this study shall be clearly indicated in the conclusion section.
Response Major Point 3: The limitations of this study are the lack of single-point wind measurements, the interval between measurements (2 h), and discrete sampling of wave directions and frequencies (4o and 0.01 Hz).
Future directions from this study include analyzing wave groups in the elevation records, and identifying the signatures of individual breaking waves within groups. It would also be worthwhile to search for the modulational instability in records of other shallow water storms. The assessment of wave energy dissipation by depth-induced breaking would enable the estimation of the reduction in available wave energy that could be harnessed by WECs.
The information above has been included in the conclusion section.
Major Point 4: In Line 379, the authors mentioned in Line 376 that “but there is no enough data for a robust linear trend”, however, in Line 379 said “A linear trend, with Qp increasing with S” - This is not well supported by only providing a scatter plot. How do you reach this conclusion?
Response Major Point 4: Thank you for the comment. Line 379 was amended by including this sentence: “A binned linear trend, with Qp increasing with S, …”. The scatter plot in Fig. 8 is the best we can do with the available data.
Minor Point 1: Line 122: the audience might want to know what SUV represent and the reference is needed as well.
Response Minor Point 1: The name of the SUV means a combination of letters. It takes the letter “S” of AST and “UV” of the horizontal velocity components. The SUV method uses the AST data as well as instantaneously interpolated horizontal velocity components, vertically aligned with the AST, to calculate the directional distribution of the directional wave spectrum.
The following reference was added in line 122.
Pedersen, T.; Siegel, E. Wave Measurements from Subsurface Buoys. In Proceedings of the IEEE Ninth Working Conference on Current Measurement Technology, Charleston, SC, USA,17-19 March 2008. https://doi.org/10.1109/CCM.2008.4480872
Minor Point 2: Line 127: why 64 is adapted?
Response Minor Point 2: The default value used to smooth the wave measurements by Nortek software is 64, from a maximum value of 128. The value selected specifies the number of Fast Fourier Transform bins used at each frequency. As the number of bins increases, the spectrum appears smoother. The number of bins selected does not change the total energy, but may alter slightly the distribution of energy.
Minor Point 3: Line 88-89: It is not necessary to only put one sentence in one paragraph (same comment for Line 479-480 and Line 484-485). Please check throughout the paper.
Response Minor Point 3: Thanks. We have revised the paper throughout, to avoid this.
Minor Point 4: Line 487: Is this a new finding? This seems pretty common to me though.
Response Minor Point 4: Agreed. It is not a new finding. Line 487 was removed.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Point 2, Around Line 220:
In general, one would think that dispersion is due to the fact that the phase velocity depends on the wavenumber. Therefore, if the the area becomes shallow, the wave will be non-dispersive, as can be seen from c=(gh)^{1/2}.
Here, however, it is written as 'Hence, in shallow water, the waves are dispersive waves.' which is confusing.
The definition of dispersion seems to be different from that given above; please write a clear definition of dispersion (or dispersive wave) so that readers will not be confused.
Point 3.
I am sorry, but in my previous comment, I should have written interaction between swell and wind-waves.
Are you saying that the interaction between swell and wind waves is not significant ?
Please note the interaction between swell and wind-waves.
Author Response
Response to Reviewer 1 Comments
Point 1: Around Line 220:
In general, one would think that dispersion is due to the fact that the phase velocity depends on the wavenumber. Therefore, if the the area becomes shallow, the wave will be non-dispersive, as can be seen from c=(gh)^{1/2}.
Here, however, it is written as 'Hence, in shallow water, the waves are dispersive waves.' which is confusing.
The definition of dispersion seems to be different from that given above; please write a clear definition of dispersion (or dispersive wave) so that readers will not be confused.
Response 1: Thank you. You are right, dispersive waves are defined as any waves whose phase speed is a function of the wavenumber. In shallow water, and linear waves are non-dispersive waves.
Dispersion or the dispersive effects cannot be fully derived from the nonlinear interaction coefficient because it is not the nonlinear dispersion relation of a modulated wave train on a wave-induced current. The term “dispersive effects” is incorrectly used and has now been amended in the manuscript. We can only say whether or not the term associated with the nonlinear dispersion relation exceeds the term associated with the wave-induced current. Therefore, the statement in lines 219-220 “Hence, in shallow water, the waves are dispersive waves. The same holds for the linear dispersion relation” has been removed.
The nonlinear interaction coefficient is,
We found a mistake in writing . It was amended. We state that the calculations of were done correctly with the above expression. The nonlinear dispersion relation can be expressed as [34],
Here, is the wave energy for a single wave train with amplitude . For shallow water waves, and,
Since the nonlinear phase speed has a term that depends on the wavenumber, the nonlinear waves are dispersive waves in shallow water.
In addition, the maximum spectral period estimated was 14 s. L=1.56*T*T = 306 m. Then, h/L=0.07 for h=20.73 m. This wave propagates in intermediate waters. It is a dispersive wave in shallow water.
Point 2: I am sorry, but in my previous comment, I should have written interaction between swell and wind-waves.
Are you saying that the interaction between swell and wind waves is not significant ?
Please note the interaction between swell and wind-waves
Response 2: The interaction between swell and wind-waves is significant in the area where wind-waves are generated. However, these complex nonlinear wave-wave interactions are not explicitly considered in the context of the ongoing work.
The interaction of swell and wind-waves is taken into account through the formation of wave groups. The frequency spectrum estimated in the direction of the dominant waves considers the low dispersion spread, and contains frequencies associated with the swell and wind-waves. The superposition of these frequencies to form wave groups is also a way of considering how swells and wind-waves interact. They interact to form groups of waves.
The spectral steepness and bandwidth, as well as the Qp, are estimated using the frequency spectrum in the direction of the dominant waves. Thus, by partitioning the omnidirectional energy spectrum to estimate the frequency spectrum in the direction of the dominant waves, the nonlinear interaction between swells and wind-waves was estimated using the Benjamin-Feir Index.
We have included the above paragraphs in the manuscript.
Author Response File: Author Response.pdf
Reviewer 3 Report
The authors have addressed all my comments. Thanks for your efforts!
Author Response
Response to Reviewer 3 Comments
Point 1: (x) English language and style are fine/minor spell check required
Response 1: We have checked the entire manuscript and corrected the English language to the best of our ability.
Point 2: The authors have addressed all my comments. Thanks for your efforts!
Response 2: We appreciate all the comments provided. Thank you very much.
