This is my second review of this manuscript. Overall, the manuscript presents an innovative approach towards understanding calving. I have a few minor points, but one remaining major point which I would like to see addressed, which either reflects a significant misunderstanding on my part or an error on the authors part. I would like to see this clarified in a revision.
Major Point of Confusion:
I’m still confused by the discussion related to the analytic calving relation in section 4.2 My understanding is that the authors are stuffing their parameterized estimate for the principle stress at the surface of the glacier into Equation (20) and integrating to find how long it takes damage to accumulate from an initial value to a critical value. So far so good. The `calving rate’ is then estimated as the distance to the stress maximum divided by the time for damage to accumulate to the critical value. This is where I’m confused and I have two problems with this calculation (and I apologize if these stem from my own misunderstanding). The first problem is that the time-to-failure calculated is the time-to-failure of a piece of ice (or element in the FEM formulation) at the surface of the glacier. This calculation, as the authors note, tells us how long it takes for a crevasse to form near the surface of the glacier. However, this calculation **doesn’t** tell us how long it takes for a crevasse to penetrate the entire ice thickness (or some fraction thereof). To calculate that one would presumably require an average vertical crevasse penetration velocity. The average vertical crevasse penetration velocity times the ice thickness (or fraction thereof) would then give the time-to-failure. Crudely speaking, we might roughly say that the vertical crevasse penetration velocity is analogous to Equation 19, but with a length scale given by the size of the nodes in the FEM model. This would result in a calving rate that includes an additional factor of ice thickness: it takes longer for a crevasse to penetrate the entire thickness if the glacier is thicker. It is possible that the authors are assuming that crevasse penetrate the entire ice thickness (or fraction thereof) as soon as the crevasse forms at the surface, but this was not clearly stated anywhere I could find. Moreover, because the tensile stress clearly decreases with depth, this assumption is tenuous and full-thickness calving will not occur just because a surface crevasse appears. (There are many observations of surface crevasses not associated with calving events). Furthermore, FEM models with damage computed similarly to that assumed here show that surface crevasses only penetrate a fraction of the depth. Fundamentally, the requirement that a surface crevasse forms seems to be an insufficient requirement for calving and I don’t understand how this gives you a calving rate. In general, I’m not opposed to heuristically arguing that one might attempt to fit an empirical law to the data available, but I think readers need to know what the assumptions are and what is “physics” and what is extrapolation/guess work.
Minor points:
Page 2, near line 55: I still think this is confusing two separate things: the yield strength of ice depends on the mode of failure. The authors are confusing tensile failure with shear failure. Observations indicate that the strength of ice is different for these two modes.
Equation 2 is wrong. The momentum equation is related to the **divergence** of the Cauchy stress tensor not the gradient. This also relates to a response to one of my previous comments where the authors erroneously state that the gradient of a vector is the divergence. The gradient of a vector is tensor. In fact the symmetric part of the gradient of the velocity is the rate of deformation tensor, which for small strains is equivalent to the strain rate tensor. The divergence of the velocity is a scalar and is related to incompressibility. Similarly, the divergence of the stress tensor is a vector. The gradient of the stress tensor, however, is a higher-order tensor.
Near Equation 4: You can always decompose the Cauchy stress into an isotropic and deviatoric component. This has nothing to do with incompressibility. For example, elastic materials are not incompressible and the stresses can be decomposed into deviatoric and isotropic.
Page 4, near line 115: I think “cartesian” should be capitalized??
Equation 11 is, I believe a condition on the traction at the calving front. I think you require a condition on the dot product of the stress tensor with the outward pointing normal vector. The normal component is, as the authors state, equivalent to the normal pressure of water, but you also need to enforce that the tangential traction vanishes. |

I have now received two new reviews based on the revised version of your paper as well as your answers to their first reviews. If reviewer #1 has no more comment, there is still an important one by reviewer #2 regarding the vertical propagation of crevasses in your approach that needs to be clarified before publication. Based on your answer to the 2nd review of reviewer #2 and the new version of your paper, I will then proceed to the final decision regarding your paper.

Best regards,

Olivier Gagliardini