Dear Johannes,

The revised manuscript has now been seen by third referee as well as by referee 1 (the latter in light of the quite diverging nature of evaluations of the original manuscript.

The review of Referee 3, which is quite positive, is given below:

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This paper presents a dimensional analysis for the Shallow Shelf Approximation (SSA) equation, which is commonly used by glaciologists to model the ice dynamics of sliding dominant flow like those of tidewater glaciers and ice shelves. As a result, the authors obtain some relationships between geometrical, ice flow and mass balance parameters under the assumption that some dimensionless factors characterizing the flow are constant under scaling. Some of the relations are validated by numerical experiments.

This paper presents new theoretical results, which will certainly be of interest to ice flow modelers, but also a more general audience of glaciologists since the relationships derived in this paper can lead to new interpretations about the interaction between the main processes involved in ice sheets dynamics. In particular, the relation of the time response with respect to geometrical scaling factors, namely Eq. (31), is a key achievement, which justifies alone that this paper can be published in TC. As often, theoretical results are first obtained after simplifying the model set-up, and this gives some intuition on how strong the result is conditioned by the original assumptions, and whether these assumptions can be overcome or not. For this reason, and despite the fact that the applicability of isothermal SSA is very limited in real modelling cases, the assumptions made on flow (as isothermal) should not be an obstacle to the publication of this original work. In addition, I don't think that "thicker ice is usually softer than thinner ice" implies that "A^(1/n) decreases with the ice thickness". Softening is directly induced by an increase of strain-rate in response to thicker ice (Glen's law) independently of any change in the rate factor A. Finally, I don't think that the model assumes a priori any dependence between mass balance and friction. The dependence of the two comes a posteriori from the scaling analysis as expected.

With that said, I had a few (rather minor) concerns when reading the paper, which could potentially lead to some improvements in the final version.

* To me the scaling relationships (14)-(17) are the main achievement of the paper, and those must be emphasized. By contrast, the consistency with the boundary-layer theory is somehow expected (otherwise the computations would have been algebraically wrong) since this boundary-layer theory is also SSA-based. Thus, this is to me a weaker result, which is maybe over-stated in the paper.

* If I understand correctly, Fig 5 and 6 are the (only) connection between theory and experimental (to validate (31)). By contrast, Fig. 7, 8 and 9 are simple plots of the relationships you derived in Section 2.2. This order of result's presentation is a bit disturbing since Fig. 7, 8 and 9 could have be drawn right after Section 2.2.

* Dimensional analysis are by nature demanding in term of numbers of variables to be introduced. However, I have the feeling that the paper can still be more efficient and more easily-readable by renaming variables in a more intuitive way so that the reader remembers more easily that, for instance, xi is related to sliding, delta is related to mass balance, ect.
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Referee 1 states the following:

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I have read the new version of the paper as well as the reply of authors to both reviews. I am happy with this new version and have only noticed few minors typos.

- page 2, line 92: where vx and vy are the components in x and y direction of the horizontal velocity vector v, respectively, H...

- page 3, line 7: delete then the definition of the velocity vector here.

- page 3, line 12: behaviorS

- page 3, line 25: the fact that z is the vertical axis should be mentioned in the previous section.

- Figure 2 : give for which time are the different plot?
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Referee 3 remarks: "To me the scaling relationships (14)-(17) are the main achievement of the paper, and those must be emphasized. By contrast, the consistency with the boundary-layer theory is somehow expected (otherwise the computations would have been algebraically wrong) since this boundary-layer theory is also SSA-based. Thus, this is to me a weaker result, which is maybe over-stated in the paper." I would therefore suggest to move the latter (related to proving that your derivation is consistent with the boundary layer theory based on SSA) to an Appendix (it is useful to show, but it is expected to agree). The three referees pointed to this.

Another remark stemming from the first review round concerns the comparison with the Vialov profile. Any SSA model with high friction will produce an ice cap profile in agreement with the SIA Vialov profile, but the physical model does not agree with the deformation model on which it is based. And this should be clearly stated. It might be good to check the paper by Bueler (2014) An exact solution for a steady, flowline marine ice sheet, J. Glaciol. Vol.60, No. 224, 2014 doi: 10.3189/2014JoG14J066.

For a revised manuscript, please take into account the remarks of both referee 1 and 3 and the recommendations I made.

Kind regards,

-Frank

Dear Johannes,
Thank you very much for your revised manuscript that is now fit for publication in TC.
Kind regards,
-Frank