Revetment Rock Armour Stability Under Depth-Limited Breaking Waves

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This study proposes a new wave energy-based formula to estimate the stable mass of rock armour for coastal revetments under depth-limited breaking waves. Comparisons with Hudson, van der Meer, and van Gent formulae suggest improved coherence with experimental data. However, the novelty is not adequately presented. The following comments should be addressed before reconsidering its publication.

- The manuscript heavily relies on unpublished project data without sufficient methodological details, please add more experimental setup details (e.g., wave generation methods, scaling criteria), and clarify data quality control and compatibility across labs (e.g., scaling effects, unit consistency).

- Explain why wave energy better represents destructive forces than traditional parameters.

- Line 230, the equation number should be 8. The empirical coefficients (300, 0.35) in the equation appear arbitrary, please explain its relationship with rock properties and hydraulic conditions.

- Line 293, table 3 should be replaced by table 4. Please add more discussion of experimental constraints, and the limitations of the proposed formula.

- The formula is not validated with field monitoring data, please add more evidence to enhance the formula feasibility, and the applicability of the formula in the prototype (e.g., scale effect, 3D flow field influence) needs to be discussed.

- The existing wave energy-based formulas should be well cited and compared to enhance the novelty.

Author Response

This study proposes a new wave energy-based formula to estimate the stable mass of rock armour for coastal revetments under depth-limited breaking waves. Comparisons with Hudson, van der Meer, and van Gent formulae suggest improved coherence with experimental data. However, the novelty is not adequately presented. The following comments should be addressed before reconsidering its publication.

• The manuscript heavily relies on unpublished project data without sufficient methodological details, please add more experimental setup details (e.g., wave generation methods, scaling criteria), and clarify data quality control and compatibility across labs (e.g., scaling effects, unit consistency).

Upon further study since submission, including transcription and analysis of van der Meer’s copious PhD data set, the writer has come to the realisation that the fronting seabed slope may have a significant effect on the incident wave energy impact on and, hence, the stability of rock armour. Therefore, the Mardie project data set, for which the seabed slope was flat, has been omitted from the final analysis now submitted. Hence, the final data set used now comprises two published data sets from very well-regarded university laboratories, being those at Delft Hydraulics, authored by Professor Marcel van Gent, and the University of NSW Water Research Laboratory, authored by Professor Doug Foster (deceased), both luminaries in this field, and each having used the same typical nearshore seabed slope of m = 0.033. The formula, which as a result has changed very slightly, and article no longer rely upon unpublished project data, which have been taken out of the article. The order of presentation of the data has been changed to that as presented in the introduction.

To equilibrate regular wave data to random wave data, “Reduced Test Data” for the UNSW WRL testing has introduced a modified wave height parameter. As the damage is occasioned by H_2%, as referenced to van der Meer, H_b ≈ H_2%, H_s = 0.7H_2%, N_{zero crossing} = N_2%/0.02.

• Explain why wave energy better represents destructive forces than traditional parameters.

Traditionally (Hudson 1958, van der Meer 1988, van Gent et al. 2003), the stability of rock armour related armour mass to wave height cubed. The only rationale proffered for this was the maintenance of dimensional integrity for the formulae proposed (Hudson 1958 p.612, van der Meer 1988 p.115). No other explanation has been documented. However, wave height is a scalar quantity whereas wave energy is the movement of water mass, which applies force to its boundaries, such as lift and drag, that can cause the boundaries (rock armour) to move. In researching toe scour of dune revetments and seawalls under breaking waves, the writer came across Henk Steetzel’s (1993) PhD thesis, which attributed toe scour to incident wave energy. Further research by the writer has confirmed this notion (Nielsen 2023). The article now embellishes the rationale regarding dimensional integrity at lines 104-106 as follows (what is added is italicised):

No explanation for this has been proffered other than it would appear to have been necessary to ensure dimensional integrity of the formulae [1p612,2p115].

and the rationale for adopting wave energy as the disturbing force at lines 110-111 as follows (what is added is italicised):

It is the energy of breaking waves, that is, the breaking velocity of fluid mass, that applies the disturbing forces of lift and drag to revetment armour that may cause it to shift, and it is the mass of that armour under the restoring force of gravity that secures or otherwise its stability.

• Line 230, the equation number should be 8. The empirical coefficients (300, 0.35) in the equation appear arbitrary, please explain its relationship with rock properties and hydraulic conditions.

The equation number has been corrected. The factor and exponent of the equation (300, 0.35) were indeed fitted arbitrarily by eye to encompass most of the data such that, for a design formula, there was sufficient conservatism to obviate failures. However, since refitting to data of consistent bed slope, the “best fit” factor and exponent became (310, 0.33) with minimal outliers, which were adopted, so this slight change to the formula is now proffered in the revised manuscript. The following explanation is included in the manuscript:

The data were used to quantify the calibration coefficient K’. The proposed wave energy formula was re-written as follows to make the calibration coefficient K’ the dependent variable:

                                                  (9)

Plotting  against , as done in Figure 5, gave the values for the factor a and exponent n by fitting a power curve. Essentially, the relationship becomes:

                                                  (10)

The resulting calibrated wave energy formula is:

                                                                     (11)

• Line 293, table 3 should be replaced by table 4. Please add more discussion of experimental constraints, and the limitations of the proposed formula.

The Table numbers have all changed resulting from removal of the flat seabed data. However, correct references are now made to all Tables. Experimental constraints are discussed further by including the following:

Application of the proposed predictor should consider the following experimental constraints:

• Data are available for only one fronting seabed slope of m = 0.033. A likely range of nearshore seabed slopes could be 0.025 < m< 0.05. Test data for these slopes are not available but are likely to result in a modified formula.
• The predictor is based on a paucity of data, with only 16 suitable test results available. A much larger data set is warranted.
• The shortest wave period used in the models was 1.3 s, which limits a small 1:50 Froude scale model to schematise only prototype peak wave periods shorter than 8s. For larger scale models the maximum prototype peak period schematised would be even shorter. Prototype peak wave periods may exceed 15 s.
• The smallest significant wave height modelled was 0.04 m, which limits a small 1:50 Froude scale model to schematise wave heights only up to 2 m. For larger scale models the maximum prototype significant wave height schematised would be smaller. Depth-limited prototype waves incident upon coastal revetments can exceed this.
• The lightest model armour rock was M₅₀= 0.047 kg, which limits a small 1:50 Froude scale model to schematise armour rock only up to 6 t. For larger scale models the maximum mass of prototype armour rock schematised would be smaller.
• Much of the data is based on around 1,000 zero-crossing waves, which limits a small 1:50 Froude scale model to schematise a storm duration of up to 2 hours. For larger scale models the maximum prototype storm duration schematised would be shorter. Revetments in depth-limited situations are likely to be subjected to a much larger number of waves.
• The predictor is based on results of 2-dimensional flume studies, which do not replicate any 3-dimensional processes that occur in the nearshore surf zone fronting revetments.
• The predictor is based on flume studies that did not consider wave overtopping.
• The formula is not validated with field monitoring data, please add more evidence to enhance the formula feasibility, and the applicability of the formula in the prototype (e.g., scale effect, 3D flow field influence) needs to be discussed.

3D flow field issues are now discussed as included above. The authors are not aware of any field validation data for rock armour stability on shallow water revetments. It was surprising to realise the dearth of flume data supporting the existing formulae, being mostly non-breaking waves, given the extent of their application to revetment design. This article would double the amount of published flume data supporting designs using the new formula. Discussion of Reynolds number scale effects, including references to Sakakiyama et al., Shimada et al. and Yagar, has been included as follows:

Reynolds number scale effects are possible. For example, if the armour rock was assumed to be spherical, to ensure similitude for drag and lift coefficients on a spherical object, the Reynolds number 2 × 10³ < Re < 2 × 10⁵ for both model and prototype (Figure 10). For Re < 2 × 10³ the drag and lift coefficient for spheres increases whereas for Re > 2 × 10⁵ the coefficient decreases.

Reynolds Number is Re = ρVL/μ or VL/υ where ρ is fluid density, μ is dynamic viscosity, υ is kinematic viscosity, the wave breaking velocity jet being  and the characteristic length . Should the model data be applied at normal scales for prototype concept design, the likely prototype ranges of Reynolds number would be Re > 2 × 10⁵ as indicated in Table 4 and Figure 10. Values of Re > 2 × 10⁵ in a prototype with spherical armour units, such as armour rock, may cause the drag and lift forces of the flow being relatively lower than that in the model, leading to conservative estimates for requisite armour mass. However, Figure 10 applies to spheres that are submerged in flow, such as under non-breaking waves, but its applicability to armour rock in aerated flows of breaking waves is not known. Nevertheless, similar scale effects occur with certain types of concrete armour units [23,24].

Table 4. The range of model Reynolds numbers for the flume data presented herein and the range of likely prototype Reynolds numbers for the model data applied at two different scales

	Dn50		htoe		Re
	Min.	Max.	Min.	Max.	Min.	Max.
	(m)	(m)	(m)	(m)	(-)	(-)
Model	0.026	0.029	0.060	0.200	2.0E+04	4.1E+04
Prototype for Scale 1:10	0.26	0.29	0.60	2.00	6.3E+05	1.3E+06
Prototype for Scale 1:50	1.3	1.5	3.0	10.0	7.1E+06	1.5E+07

Figure 10. The variation with Reynolds number of the drag coefficient for spheres submerged in uniform flow [25]. The range of model values and likely prototype values is indicated.

• The existing wave energy-based formulas should be well cited and compared to enhance the novelty.

Comparisons with other formulae are made in Figure 7. A second reference to using wave energy is cited at line 107 as follows, the first reference being a Delft PhD thesis, the second a recent ASCE Journal paper (additions italicised):

However, other researchers have proposed sediment transport and revetment toe scour as a function of incident wave energy [13 p27, 14].

References:

Foster, D.N., Gordon, A.D. Stability of armour units against breaking waves. In Proceedings First Australian Conference on Coastal Engineering, Sydney, Australia, May 1973, 98‑107.

Hudson, R.Y. Design of Quarry-Stone Cover Layers for Rubble-Mound Breakwaters, USACE Research Report 2-2, 1958, 68pp. Laboratory investigations of rubble-mound breakwaters. ASCE Trans. Paper 3213, pp610-659, Waterways and Harbors Division 1959, 85, 93‑121.

Nielsen A.F. (2023). Design scour levels for dune revetments and seawalls. ASCE J. Waterway, Port, Coastal, Ocean Eng., 2023, 149(3): 04023005, 8pp, DOI: 10.1061/JWPED5.WWENG-1963.

Sakakiyama, T., Kajima, R. Scale effect of wave force on armor units. Proceedings 22nd International Conference on Coastal Engineering, Delft, The Netherlands, July 2, 1990, 1716‑1729.

Shimada, A., Fujimoto, T., Saito, S., Sakakiyama, T., Hirakuchi, H. Scale effects on stability and wave reflection regarding armor units. Proceedings 20th International Conference on Coastal Engineering, Taipei, Taiwan, Nov 9, 1986, 2238‑2252.

Steetzel, HJ (1993). Cross-shore Transport during Storm Surges. Thesis Tech. Univ. Delft, Published also as Delft Hydraulics Communication No. 476, September, 294pp.

Van der Meer, J.W. Rock slopes and gravel beaches under wave attack. PhD thesis, Tech. University Delft, 1988, also published as Delft Hydraulics Communication No. 396, 214pp.

Van Gent, M.R.A., A.J. Smale, C. Kuiper (2003), Stability of rock slopes with shallow foreshores, ASCE, Proc. Coastal Structures 2003, Portland.

Yager, R.J. Calculating drag coefficients for spheres and other shapes using C++. Report ARL-TN-612, US Army Research Laboratory, Aberdeen Proving Ground, MD 21005-5066, June 2014.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This paper presents a new formula for the stability of coastal revetment rock armour under depth-limited breaking waves. The formula is well-calibrated using empirical data from physical model tests and offers a more coherent approach compared to existing methods. However, some limitations, such as the scope of the data and the consideration of seabed slope, need to be addressed. Overall, the paper makes a valuable contribution to coastal engineering by providing a practical tool for designing rock armour in shallow water conditions. The following suggestions will further improve the author's article.

(1) Does the method of calculating wave energy in the new formula adequately take into account the energy loss of waves during the breaking process?

(2) Has an optimization analysis been performed on the effect of berm slope on the stability of the berm, and is there an optimal slope?

(3) Does the method of quantifying revetment damage take into account the effect of localized damage on overall stability?

(4) Is the new formula effective in distinguishing the effects of different wave breaking types (e.g., plunging and surging) on berms?

(5) Is the number of wave actions mentioned in the paper sufficient to model the long-term wave erosion effects in real engineering?

(6) Are formulas calibrated solely on the basis of existing physical model data reliable enough to be applied to different geographic regions?

(7) Please clearly define the scope of “shallow water depth” in the text.

(8) Is the effect of wave period on slope stability overemphasized? Are there other factors whose effects have been underestimated?

(9) Under what conditions is the new formula more advantageous than the Hudson and van Gent formulas?

Author Response

This paper presents a new formula for the stability of coastal revetment rock armour under depth-limited breaking waves. The formula is well-calibrated using empirical data from physical model tests and offers a more coherent approach compared to existing methods. However, some limitations, such as the scope of the data and the consideration of seabed slope, need to be addressed. Overall, the paper makes a valuable contribution to coastal engineering by providing a practical tool for designing rock armour in shallow water conditions. The following suggestions will further improve the author's article.

• Does the method of calculating wave energy in the new formula adequately take into account the energy loss of waves during the breaking process?

The formula for the total amount of wave energy is equation (5), being E = f(H²L). When the wave breaks the energy would be dissipated in a myriad of ways; in sound, moving rocks, wave runup, splashing, etc. How much is moving rocks we don’t know. Nevertheless, that is the purpose of the scale model calibration, to relate the damage to the variables in the wave energy equation as follows:

The proposed wave energy formula was re-written as follows to make the calibration coefficient K’ the dependent variable:

                                                  (9)

Plotting  against , as done in Figure 5, gave the values for the factor a and exponent n in equation (9) by fitting a power curve. Essentially, the relationship becomes:

                                                                           (10)

• Has an optimization analysis been performed on the effect of berm slope on the stability of the berm, and is there an optimal slope?

This work was done originally by Hudson (1958) and modified by van der Meer (1988), the latter adopted in this article (as stated in the article). No optimal slope has been defined; the flatter the slope the more stable the rock.

• Does the method of quantifying revetment damage take into account the effect of localized damage on overall stability?

No it doesn’t and, often, geotechnical global stability is overlooked by coastal engineers. Nevertheless, the scale of armour damage is small compared with the scales considered for geotechnical global stability. This issue would not be pertinent in this article.

• Is the new formula effective in distinguishing the effects of different wave breaking types (e.g., plunging and surging) on berms?

The new formula is applicable to waves shoaling on the relatively flat fronting seabed slope and breaking onto a steeper coastal revetment, a condition that the Shore Protection Manual describes as the most reasonable for revetment design. The new formula does not refer to waves shoaling and breaking on the steep revetment slope itself, as do the formulae of van der Meer that distinguish between plunging and surging. The formula has been based on both plunging and surging waves, although that did not become apparent in the calibration plots, so the new formula does not make that distinction, probably because only one seabed slope was modelled. The van der Meer formulae indicate that the latter are more critical for stability.

• Is the number of wave actions mentioned in the paper sufficient to model the long-term wave erosion effects in real engineering?

The number of waves used in the modelling of van Gent et al. (2003) was around 1,200, which could be representative only of a single storm. However, the number of waves as modelled by Gordon (1973) was far greater and could represent the impacts of several storms. Nevertheless, the damage has been normalised by the number of waves (D/N^0.5), which forms the basis of the calibration, so the formula can apply to however many waves the designer wishes to choose, although that would involve extrapolation as discussed in Experimental Constraints, which, with respect to the number of waves, includes:

• Much of the data is based on around 1,000 zero-crossing waves, which limits a small 1:50 Froude scale model to schematise a storm duration of up to 2 hours. For larger scale models the maximum prototype storm duration schematised would be shorter. Revetments in depth-limited situations are likely to be subjected to a much larger number of waves.
• Are formulas calibrated solely on the basis of existing physical model data reliable enough to be applied to different geographic regions?

Yes. The basic physics applies world-wide. However, as wave conditions vary over the globe there will be locations where the results from this modelling will have to be extrapolated rather than interpolated. Discussion around this has been greatly expanded with new sections on modelling constraints and scale effects:

Application of the proposed predictor should consider the following experimental constraints:

• Data are available for only one fronting seabed slope of m = 0.033. A likely range of nearshore seabed slopes could be 0.025 < m< 0.05. Test data for these slopes are not available but are likely to result in a modified formula.
• The predictor is based on a paucity of data, with only 16 suitable test results available. A much larger data set is warranted.
• The shortest wave period used in the models was 1.3 s, which limits a small 1:50 Froude scale model to schematise only prototype peak wave periods shorter than 8s. For larger scale models the maximum prototype peak period schematised would be even shorter. Prototype peak wave periods may exceed 15 s.
• The smallest significant wave height modelled was 0.04 m, which limits a small 1:50 Froude scale model to schematise wave heights only up to 2 m. For larger scale models the maximum prototype significant wave height schematised would be smaller. Depth-limited prototype waves incident upon coastal revetments can exceed this.
• The lightest model armour rock was M₅₀= 0.047 kg, which limits a small 1:50 Froude scale model to schematise armour rock only up to 6 t. For larger scale models the maximum mass of prototype armour rock schematised would be smaller.
• Much of the data is based on around 1,000 zero-crossing waves, which limits a small 1:50 Froude scale model to schematise a storm duration of up to 2 hours. For larger scale models the maximum prototype storm duration schematised would be shorter. Revetments in depth-limited situations are likely to be subjected to a much larger number of waves.
• The predictor is based on results of 2-dimensional flume studies, which do not replicate any 3-dimensional processes that occur in the nearshore surf zone fronting revetments.
• The predictor is based on flume studies that did not consider wave overtopping.

Discussion of Reynolds number scale effects has been included as follows:

Reynolds number scale effects are possible. For example, if the armour rock was assumed to be spherical, to ensure similitude for drag and lift coefficients on a spherical object, the Reynolds number 2 × 10³ < Re < 2 × 10⁵ for both model and prototype (Figure 10). For Re < 2 × 10³ the drag and lift coefficient for spheres increases whereas for Re > 2 × 10⁵ the coefficient decreases.

Reynolds Number is Re = ρVL/μ or VL/υ where ρ is fluid density, μ is dynamic viscosity, υ is kinematic viscosity, the wave breaking velocity jet being  and the characteristic length . Should the model data be applied at normal scales for prototype concept design, the likely prototype ranges of Reynolds number would be Re > 2 × 10⁵ as indicated in Table 4. Values of Re > 2 × 10⁵ in a prototype with spherical armour units, such as armour rock, may cause the drag and lift forces of the flow being relatively lower than that in the model, leading to conservative estimates for requisite armour mass. However, Figure 10 applies to spheres that are submerged in flow, such as under non-breaking waves, but its applicability to armour rock in aerated flows of breaking waves is not known. Nevertheless, similar scale effects occur with certain types of concrete armour units [23,24].

Table 4. The range of model Reynolds numbers for the flume data presented herein and the range of likely prototype Reynolds numbers for the model data applied at two different scales

	Dn50		htoe		Re
	Min.	Max.	Min.	Max.	Min.	Max.
	(m)	(m)	(m)	(m)	(-)	(-)
Model	0.026	0.029	0.060	0.200	2.0E+04	4.1E+04
Prototype for Scale 1:10	0.26	0.29	0.60	2.00	6.3E+05	1.3E+06
Prototype for Scale 1:50	1.3	1.5	3.0	10.0	7.1E+06	1.5E+07

Figure 10. The variation with Reynolds number of the drag coefficient for spheres submerged in uniform flow [25]. The range of model values and likely prototype values is indicated.

• Please clearly define the scope of “shallow water depth” in the text.

The following (italicised) has been included at line 120:

In shallow water, where the depth is less than 0.04L (0.016T²) [15 p2‑9],

• Is the effect of wave period on slope stability overemphasized? Are there other factors whose effects have been underestimated?

The requisite stable armour mass is directly proportional to wave period, as indicated in Figure 7, which shows that both the wave energy formula and the van der Meer formulae indicate that waves with period T≈13 s require twice the mass than do T≈7 s waves in the same depth. However, the very popular Hudson formula and that of van Gent have no dependence whatsoever on wave period. That the new wave energy formula supports the van der Meer formulae in this respect adds weight to the importance of considering wave period in design, which requires emphasis. Factors that have not been estimated at all include seabed slope, which is highlighted in the article and for which recommendations have been made. Further, the article emphasises the importance of the number of waves in design.

(9) Under what conditions is the new formula more advantageous than the Hudson and van Gent formulas?

The new formula has advantages over the Hudson and van Gent formulae in that it gives consideration not only to wave period but requires the designer to think also about water depths for design and, with respect to the Hudson formula, the number of waves, that is, storm duration and wave period.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The paper provides different insights on the issue of rock armour stability. The paper is well written, the figures and the references are appropriate. The formula proposed for coastal revetments seems effective and new. This contribute is worty to be published.

Author Response

No comments were given

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors answered all my questions carefully, it is recommended to be published in the present form.

Reviewer 2 Report

Comments and Suggestions for Authors

All comments have been positively addressed. This paper can be considered for publication.