Consistent Models of Flexural-Gravity Waves in Floating Ice

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The relevant derivation processes are complete, the obtained results are reliable, and the corresponding conclusions are definite. The subject of this paper is within the aims and scope of Journal of Marine Science and Engineering, but the manuscript is acceptable after minor revision.

(i)The authors must discuss the advantages and disadvantages of their method in the Introduction. 

(ii) Describe the roots of the dispersion relation (6).

(iii)Authors must discuss about the validation of numerical results.

(iv)What is the Effects of water depth and ice thickness on the waves?

 

Comments on the Quality of English Language

Good

Author Response

(i) The authors must discuss the advantages and disadvantages of their method in the Introduction.

Response: Advantages and disadvantages of our approach are explained in a new paragraph, lines 77-83.

 

(ii) Describe the roots of the dispersion relation (6).

Response: The roots of the dispersion relation (6) are described in a new paragraph, lines 203-209.

 

(iii) Authors must discuss about the validation of numerical results.

Response: The obtained theoretical results are compared with the experimental ones from [33], see figure 2. It is seen that the ice-breaking model employed in our study well predicts the condition of breaking.

 

(iv) What is the Effects of water depth and ice thickness on the waves?

Response: The effects of water depth and ice thickness on the waves are discussed in the new paragraph, lines 73-76.

Reviewer 2 Report

Comments and Suggestions for Authors

The research presents a significant and well-structured contribution to the theoretical understanding of flexural-gravity waves (FGWs) in floating ice. By establishing physically consistent criteria—particularly through the use of yield strain and dispersion relations—it offers a clear classification of wave regimes and model applicability. The framework is valuable for both researchers and engineers working in polar environments, enabling more accurate selection of modeling approaches under varying ice and wave conditions.

 

1. What is the main question addressed by the research?

The study investigates the validity domains of simplified models (linear, broken ice, neglecting inertia or bending) and provides classification boundaries based on yield strain, wave amplitude, and wavelength.

2. Is the topic original or relevant to the field? Does it address a specific gap?

Yes, the topic is both original and highly relevant to the field of hydroelasticity, polar engineering, and wave-ice interactions. 

It addresses a significant gap by: Proposing a physically consistent framework to distinguish when ice can be modeled as continuous, broken, or irrelevant. Integrating yield strain as a constraint in determining model validity a factor often neglected or simplified in prior work. Clarifying under what conditions simplified models (e.g., neglecting bending or inertia) are justified. This is a valuable contribution for both theoretical research and practical modeling in Arctic and Antarctic marine engineering.

3. What does it add to the subject area compared with other published material?

This study provides:

A clear classification of wave regimes using theoretical bounds derived from physical parameters (e.g., ice thickness, strain limits, δ thresholds).

A visual framework (Figures 1–4) for determining when linear FGWs, nonlinear behavior, or gravity-wave approximations are appropriate.
Enhanced consistency criteria, combining dispersion relation and yield strain into wave modeling—bridging mechanical and hydrodynamic perspectives.
Experimental validation using previous field and laboratory studies, reinforcing the practicality of the proposed criteria.
In contrast to prior work, which often focuses narrowly on one regime, this manuscript gives a comprehensive map of the modeling domains.

4. Are the conclusions consistent with the evidence and arguments presented and do they address the main question posed?
Yes, the conclusions are logically derived and clearly supported by the theoretical derivations and experimental references. They directly address the central question of model applicability and provide practical insights into when and how simplified FGW models can be used with physical consistency.

5. Are the references appropriate?
Yes, the references are extensive and well-chosen, spanning foundational theory 

Author Response

The authors thank the reviewer for their comments which clearly and more sharply formulate the ideas and results of our study.

Reviewer 3 Report

Comments and Suggestions for Authors

The yield strain criterion of sea failure is not very typical. The main reason is that it is more difficult to measure absolute displacements in experiments than forces. If the flexural strength, elastic modulus and Poisson’s ratio are known then the yield stain is calculated from the Hook’s law. It is known that flexural strength of sea ice measured in the tests with floating cantilever beams depends on the liquid brine content (see e.g., Timco&O’Brien, CRST, 1994; Karulina et al, AOR, 2019). Elastic modulus of fresh ice also depends on the temperature and the frequency (Sinha, J. Glaciol, 1978). Elastic modulus of sea ice depends not only on the liquid brine content but also on the frequency (Marchenko, Ocean Modeling, 2024). It means that the yield strain in sea ice also depends on the liquid brine content and the frequency. There is a big difference in the flexural strength, elastic moduli, and Poisson’s ratios of natural sea ice and artificial saline ice. For example, the elastic modulus of natural sea ice in frequency range of about 10 Hz is 3-4 GPa, while the elastic modulus of model saline ice is of about 0.1 GPa (Timco, CRST, 1980). Typical flexural strength of natural sea ice is of around 0.3 MPa, while the flexural strength of model saline ice is of around 50 kPa. The yield strain criterion  used in the paper does not take into account the dependence from the liquid brine content of ice and the frequency. I didn’t find this value in the paper of Schulson (1999) referred in the manuscript. I recommend including in the paper yield strain analysis of saline ice depending on its liquid brine content and frequency. If the paper is focused mainly on model sea ice then the analysis could be performed for model saline ice only.         

Comments for author File: Comments.pdf

Author Response

The authors thank the anonymous referee for their helpful and inspiring comments that improved the quality of the manuscript.

The beginning of Section 2, lines 90-120, has been revised adding more references and discussions of the failure criteria. See also the very end of the Conclusion.

In Schulson (1999) paper, page 24, the section ‘Brittle Behavior’ starts with

Brittle behavior sets in at higher strain rates. Under tension (regimes TII \& TIII, Figures 3 and 4) ice breaks after lengthening 0.01–0.1% through trans granular Cleavage [38]. The tensile strength is rate independent and is only slightly thermally dependent,[39] rising by less than 25% upon decreasing temperature from –5°C to –20°C.

This gives us the range of the yield strain from 10^{-5} to 10^{-4}.