Motion of Submerged Body in a Frozen Channel with Compressed Porous Ice

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 

The formulation of the problem as well as the methodology are presented very clearly and as concisely as possible. The results are presented in detail. The conclusion is complete and appropriately defined considering the procedures covered in the paper.

 

 

 

Only, the relatively significant self-citation of the authors is questionable. I would kindly ask for an explanation as to whether so much self-citation is really necessary?

Author Response

Thank you very much for the review and positive opinion about our study. Our response is in the enclosed file. 

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The paper studies the problem of submerged body motion in a frozen channel. The submerged body is modeled as a dipole, whose potential in the channel is determined using the method of mirror images. The objectives of the study are clearly stated and the research method is scientific. I recommend the acceptance of the paper after the following comments are well addressed:

(1)       Page 11, line 354, how to determine the qcr=2.12 and 2.477?

(2)       In Fig 2 (a), the lowest phase speed for qx=2.477 is just close to zero, why?

(3)       Fig 3 (b) and Fig 5 (b), the label of vertical axis is missing.

(4)       The authors explained “The speed U = 3 m/s is closest to critical in the case qx = 2.12. So the ice deflections and strains in this case are the largest among all showed.” What is the critical boundary value for averaged qx? What if the averaged qx exceeds 2.47? More explanations are necessary.

(5)       Page 15, the reference to determine the porosity parameter and dimensionless strains needs introduction.

(6)       It is better to discuss the advantages of the proposed method. Also, the limitation of current research and future plan need to be demonstrated at the end of the conclusions.

Comments for author File: Comments.pdf

Author Response

Dear reviewer, 

Thank you very much for the review and positive opinion about our study. Our response is in the enclosed file

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper investigates the stresses in a frozen ice sheet above an inviscid fluid, in which there is a moving 3D point source with constant speed. The case of constant ice thickness, and variable ice thickness in the transverse direction were considered.

The results in the paper appear to be sound although i have not checked them all.

I recommend publication of this paper. A couple of comments the authors may wish to consider are below.

Lines 232-233. Here you say both sigma and delta are the dimensionless porosity parameters, but i don't see any sigma in the above equations.

For both the constant thickness ice and the variable ice cases it would be good to see some convergence results for the numerical schemes. The variable ice case needs sufficiently more modes for the generated results and it would be good to confirm to the reader that the level of accuracy is the same as for the constant ice case where many fewer modes are used.

Comments on the Quality of English Language

The English language is adequate, but there are a few places where the paper isn't clear. For example lines 216, 355 and 513.

A reading by a proof reader with English as a 1st language would soon sort out all the issues.

 

Author Response

Thank you very much for the review and positive opinion about our study. Our response is in the enclosed file.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The paper describes a mathematical method for constructing solutions that describe the wave response of ice-covered water to the steady motion of a submerged sphere. The computational domain of a fixed depth is bounded by vertical walls. The problem is solved by the method of the horizontal eigen modes with using the dipole representation of the water movement around a sphere. The solution is constructed in the form of infinite series of the eigen functions. The coefficients of the series are found from an infinite system of linear algebraic equations following from the boundary conditions. For numerical analysis, the system is truncated, and the coefficients are found numerically. Similar methods have been used previously to solve problems of floating ice dynamics. The list of references is given in the end of the paper. I believe that the research results described in the paper are mathematically correct. 

The main comments to the paper are related to the physical formulation of the problem and the numerical values of the physical constants used in ice modelling.

1.       Term “porous ice” is used in the paper to mean that water moves through the ice under the influence of dynamic water pressure below the ice. In line 315 authors specify parameters of fresh ice. It is not clear how fresh ice can be porous. Physically it may happen when the ice temperature is 0C. But for the ice formation the ice surface temperature should be below 0 C. In this case liquid water is absent inside the ice. Sea ice is porous, but not always permeable. There are many papers on the permeability of sea ice, but they are not referred in the present paper. Please describe what type of fresh ice in nature or in the lab is porous ice considered in the paper.

2.       Boundary condition (4) sets that the vertical velocity of water is different from the normal velocity of the ice because water can move through the ice. The speed of the vertical movement of water is proportional to the difference between the dynamic pressure of water under the ice and atmosphere pressure above the ice, assumed to be zero. According to the Darcy’s law the coefficient is proportional to the permeability. There are bending stresses inside the ice caused by ice deformations. Why these stresses are excluded from the Darcy’s law ? 

3.       The dimension of the permeability in the Darcy’s law is m^2. In Line 317 the permeability 0.7 10^-8 is given without dimension. Where does this value come from ? Please give references on the porosity of fresh porous ice.

4.       The retardation time introduced in Line 129 is dimensional quantity. The retardation times given in lines 480 and 481 are nondimensional. Where do these value come from ? Please give references.

5.       Please give references on the value 4.2 GPa of the elastic modulus of porous fresh ice. 

6.       In Lines 604-608 the authors discuss zero phase velocity depending on the ice compression. The instability of compressed plate is known as Eulerian instability under critical load. Typically, in ice this is achieved as a load range greater than the compression strength of the ice. Please check it for your estimates since compression strength is low in warm ice.    

 

I think that the physical formulation of the problem considered in the paper should be improved according to the comments made above. It may be easier to remove the word “porous ice” from the paper and use instead the term “porous plate” as in the paper by Meylan et al (Wave motion, 2017).

Author Response

We are grateful for your time and for the opportunity to improve the quality of our article. Our response is in the enclosed file.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

Please include dimension of the permeability in all places of the text 

Line 523: K=10^-12m²

Line 530: K in the range 10^-9-10^-7m²

Figure 7: Point out the dimension of K

Figure 8: Point out the dimension of K

Author Response

Dear Reviewer,

We have taken all your comments into account and specified the dimensionality of K in the text.
Thank you so much for helping us improve our work.

Best regards, Tatyana Khabakhpasheva