Transient Shallow Water Wave Interactions with a Partially Fragmented Ice Shelf
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsPlease see attached file.
Comments for author File: 
Comments.pdf
Author Response
Comment 1: Equation (1c) appears to use the same value of depth H in all R while, as correctly the authors do in the following, under the ice shelf and its fragments the depth is reduced by an amount corresponding to the ice shelf draft and becomes \(\widetilde{H}\). This should be indicated
and explained in equation (1c).
Response 1: Thank you for pointing out this, this has been made clear in equations 1c,1d and the subsequent text and marked in red.
Comment 2: Line 126. The different depth values imply that the mass flux is continuous, but the derivative of the water wave potential (hence the horizontal velocity) is discontinuous.
Response 2: The reiewer is correct and this is what we wrote. It is unclear why this issue has been raised, perhaps because the tilde{H} was not defined (which has been fixed).
Comment 3: Section 2.5. energy conservation: Please note that another method for deriving the energy conservation equation from the variational form and solving the reflection-transmission
problem for an ice-shelf was presented in the relevant study:
T. K. Papathanasiou, Karperaki, A. E., and Belibassakis, K. A. (2019). On the resonant
hydroelastic behaviour of ice shelves. Ocean Modelling, 133:11–26
Response 3: This paper has been mentioned in the enregy consrevation section-line 208.
Comment 4: What are the effects of elasticity (flexure) on the reflection coefficient? The final figure of
the paper indicates that as the ice-shelf thickness increases the reflection increases,
which is reasonable. This increase is mainly due to the draft increasing or due to the
change in the flexure properties?
Response 4: A new figure investigating the elasticity parameter has been added.
Comment 5: A discussion of the flexure effects as the fragment length becomes smaller could be also
interesting. It might be the case that as Li decreases, the flexure effects might be
negligible and the fragment heaves as a rigid body.
Response 5: We expect this results but the investigation of this particular effect is out of the scope of this study.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe hydrodynamics are modelled using shallow water wave theory and the ice shelf vibration is modelled using Euler-Bernoulli beam theory. The ensuing multiple scattering problem is solved in the frequency domain using the transfer matrix method. This work serves as a fundamental step towards developing a comprehensive model to simulate the breakup of ice shelves.
This is an interesting work and derived from mathematical way.
Maybe this manuscript fits the scope of journal Mathematics/AppliedMath more than this target journal.
The following are comments:
While the introduction provides a good background, it could be improved by more directly stating the research gap this study aims to fill in the abstract section. Clarifying how this work advances beyond existing knowledge early in the introduction would help.
How to solve the Euler-Bernoulli beam PDE equations? I guess this was done be solving the matrix in equation 10 and 11.
The difference of solution method of semi-infinite ice shelf case and Multiple Scattering Solution case may be illustrated.
The limitation of this model, or say, the assumptions should be addressed.
A more parametric study is encouraged.
Author Response
Comment 1: Maybe this manuscript fits the scope of journal Mathematics/AppliedMath more than this target journal.
Response 1: We respectfully disagree. Water wave interaction with structures and geographic features are important areas of topics in fluid mechanics.
Comment 2: While the introduction provides a good background, it could be improved by more directly stating the research gap this study aims to fill in the abstract section. Clarifying how this work advances beyond existing knowledge early in the introduction would help. .
Response 2: A description of the research gap has been added to the abstract lines 7-8.
Comment 3: Clarifying how this work advances beyond existing knowledge early in the introduction would help.
Response 3: We have chosen to keep this to later in the introduction after we have described the current state-of-the-art.
Comment 4: How to solve the Euler-Bernoulli beam PDE equations? I guess this was done be solving the matrix in equation 10 and 11.
Response 4: The Euler-Bernoulli beam equation is not solved independently, but it is solved in conjunction with the shallow water hydrodynamics. As described in the paper, the system is solved in the frequency domain, where the problem reduces to that of finding the unknown coefficients. These are found by solving matrix equations (e.g. equation 10-11).
Comment 5: The difference of solution method of semi-infinite ice shelf case and Multiple Scattering Solution case may be illustrated.
Response 5: We have added a new figure (11) which illustrates this.
Comment 6:The limitation of this model, or say, the assumptions should be addressed.
Response 6: We understand that the model has significant limitations, which originate from the assumptions underlying the Euler-Bernoulli beam equation and the shallow water equations. These limitations have been discussed in the the first paragraph of section 2.1. The fact that the model has limitations is also acknowledged in the Conclusion
Comment 7: A more parametric study is encouraged.
Response 7: A new figure illustrating the elasticity parameter has been added.
