Dear Eric Deal,

thanks a lot for your thorough and even inspiring review! I must apologize that I was indeed not aware of your latest paper in GRL. The first part of the theory was developed in a slightly different way, but is in fact basically the same. And you were definitely earlier, although I spent a lot of time on the numerics before submitting my manuscript.

There is only one point where I clearly disagree with your opinion. This is the way erosion models of the stream-power type are formulated for non-constant precipitation. Probably owing to the fundamental studies of Hack, erodibilities have been expressed in terms of catchment size instead of discharge until now. If we want to keep this convention, there is no way of avoiding the definition of a reference precipitation p0 and imagining that a given erodibility refers to p0 . Then we can replace the catchment size A by either q/p0 where q is the discharge or even by pA/p0 , but where p is the mean discharge over the upstream catchment. Both versions become

increasingly cumbersome when proceeding to the shared stream-power model and also for the fluvio-glacial version. Beyond this, I do not really like the concept where a mean upstream precipitation occurs in the erosion model and a local precipitation in the climatic component. Therefore, I do not like the version used by the Tübingen/Potsdam groups for some years, which also applies to the version with the product IA in your 2012 paper. My version of defining the ratio q/p0 as the catchment-size equivalent of the actual discharge provides a clear definition that keeps the equations similar to the original stream-power model. I am convinced that reminding

the reader of this definition at some occurrences of A is sufficient to avoid confusion. So there is no realistic chance to convince me here.

There is another point where I am not sure at all. In your final comment, you suggest to consider the version where the thickness is parameterized in terms of flux and slope instead of flux alone. However, my results already show that such a parameterization leads to a weird scaling of thickness vs. width in a steady state. While width is proportional to q^0.3 then, thickness is proportional to q^0.7. As far as I can see, your recent approach including deformation softens this problem, but thickness still increases more rapidly with flux than width. I guess that the problem already comes in when parameterizing the width by the flux alone without taking into account the slope. If so, it already affects the 2020 EPSL paper by Günther Prasicek where both of us were coauthors, but unfortunately I did not think about this at that time. So it would also be interesting to look at thickness vs. width in your recent results. But for my concept, this mainly tells me that we must consider all approximations as a package and compare it, e.g., to simulations with iSOSIA.

Best regards,

Stefan

Dear Flavien Beaud,

thanks for your comments! I hoped that some more researchers from the community of modeling glacial erosion would share their opinion, and that they would perhaps even go beyond keywords. I am convinced that your expertise as a reviewer when considering the proposed concept seriously would have been helpful to improve the paper.

Let me briefly comment on some of your points.

For example, in Section 2, l. 63-64, the derivation of the glacial stream power starts with the statement "If we consider a rectangular cross section [...]". Yet, we know glacial valleys are not rectangular and there is no explanation of that simplification.

This is just some kind of entry gate for the readers. Those who think this way should stop reading at this point and not waste their time.

I hope there will be some readers not arguing on this level, but are able to think about the consequences of this simplification, although

I am aware that not all will be able to solve the exercise of finding out for which shapes of cross sections it remains valid.

Another point of concern is the recurring citation of Prasicek et al. (2020) to justify numerous simplifications. In the Prasicek et al. (2020)

study, the goal is to assess timescales of respective processes: glacial erosion, climate and tectonic; not to reproduce landforms. The assumptions Prasicek et al. (2020) are making to assess the relative effectiveness of processes over time, become inadequate when applied to a model that aims at reproducing landforms themselves.

Sorry, I am not such an expert on this since I only developed parts of the theoretical framework of the Prasicek et al. (2020) paper. So far I thought it was about longitudinal equilibrium profiles for a dendritic topology and not about thime scales. But maybe I just missed the key point of that paper.

The paper cited to support such scaling (Bahr, 1997) refers to glaciers themselves, not glacial landscapes, and is therefore not applicable to

glacier erosion.

I thought it was about length-width scaling of glaciers and used nothing else from this paper. And in contrast to previous studies, I was honest enough to explain that transferring scaling relations from entire glaciers into individual glaciers is nontrivial.

In term of the specifics of glacial erosion, even assuming a simple relationship between erosion and sliding velocity to a power, this power

is likely > 1.

I think I mentioned this, and anyone should feel free to run the model with an exponent > 1. Some researchers may even prefer this for the fluvial regime since there were some studies that suggested exponents > 1 (although I think there are many pitfalls and artifacts there).

Using a shallow ice approximation has be shown to not work very well on steep landscapes (Egholm et al., 2011). These simplifications and their implications for landscape evolution results should be explained.

This is not just a problem of steep topographies, but applies to all combinations of shallow-water and shallow-ice equations with

incision-type erosion laws if lateral stresses are not taken into account. Iti s explained briefly in the paper, and in principle it

is an argument in favor of approaches where erosion is driven by a linear element.

Model of glacio-fluvial incision (l. 229-232): "While Beaud et al. (2016) developed a more elaborate model for the incision by meltwater within narrow channels, the meltwater component should preferably not introduce a level of complexity much beyond the simple models of fluvial and glacial erosion used here. So let us assume that erosion by meltwater can be described by the same formalism as fluvial erosion." That statement is in opposition to the results presented in the Beaud et al. (2016) study. Since that model is the only to date to describe such mechanism, dismissing the results or making different assumptions should be substantiated.

I cited this paper because it is apparently the first modeling approach towards bedrock incision by meltwater channels. However, I was neither able to recognize any fundamentally new theoretical concepts nor any results that could be generalized to larger scales. So it is not clear to me where the contradiction is. Did you find that meltwater incision in total independent of the meltwater discharge or find that the pressure gradient is totally decoupled from the gradient of the ice surface? Maybe I missed the key point again.

In summary, in its current form, I do not believe that the current study makes a convincing point that a stream power glacial erosion rule can, or should be used.

But unfortunately, other authors recently proposed a glacial stream-power law that is very similar to my version (Deal & Prasicek, Geophys. Res. Lett, 2020). They were definitely faster than me. Feel free to write a comment to their paper!

I hope you find my comments constructive and am available to answer further questions if necessary.

To be honest, not really. But I definitely appreciate your contribution to the discussion.

Best,

Stefan Hergarten

Dear Marc Jaffrey,

thanks for your comments! You speak a lot about "an extensive body of literature", "known glacier dynamics", and "the glaciology community". Unfortunately, it is difficult for me to recognize what you refer to. I tried to find out from you own work in this field, but only found a conference poster about basin-wide glacial erosion rates. Although I like the concept proposed there, I could not find out where the fundamentals of my work basically differ from what you assume, and I also cannot see how the concept of a mass balance differs as you claim.

Let me just comment on your point highlighted by boldface line numbers :

"60-65: This is false. The author makes a critical assumptions here defining the model presented which is a critical misunderstanding of glacier dynamics and the Shallow Ice-sheet approximation. The ratio of the depth averaged velocity and sliding velocity is categoricallly not controlled by ice thickness, not even a second order control. Sliding velocity is a complex issue which from a dynamic’s perspective, the principal first order controls are effective water pressure, bed temperature, bed roughness and a host of other bed parameters and processes. From a mathematical and numerical simulation perspective sliding velocity is treated independently with a sliding law, a Robin type boundary condition, which is typical defined in terms of the aforementioned bed processes, not ice thickness. The author seems to be unaware of glacier dynamics. Since the assumptions in 60-65 are crucial to the theoretical underpinning of the model the rest of the work, regardless of the care taken with the numerical simulation, is unfounded."

This is just what is typically assumed for the sliding and deformation velocities (not "depth averaged velocity" as you state). I agree that the factors of proportionality in Eqs. (4) and (5) depend on what you mention, but nevertheless the (quite old) model ICE-CASCADE and to some extent also the state-of-the-art model iSOSIA use these relations. They are also used in more theoretical considerations, e.g., Prasicek et al. (2018), doi 10.1029/2017JF004559 and Deal & Prasicek (2021) doi 10.1029/2020GL089263. You are free not to like this approach and propose alterantive concepts. In general, however, I would find it more useful to bring your own ideas to the scientific audience than just tearing down other researchers' work on a keyword level.

Best regards,

Stefan Hergarten

Dear Marc Jaffrey,

thanks for the more specific version of the review and in particular for the personal email you sent to me. This makes your point about some of the numerical models used on this field more clear to me. I am fully with you that it is a serious problem of science that things become "true" just by publishing and citing repeatedly.

Nevertheless, it seems indispensable for me to state that your arguments about the equations are completely wrong in my opinion.

1. Concerning Eq. (4), you state: "Equation 4 is incorrect:  See chapter 7, eq's 7.6, 7.10, 7.17, 8.35, 8.36, 8.65 and sections 8.1, 8.4, 8.5 and 8.6 in Cuffey, Kurt M., and William Stanley Bryce Paterson. The physics of glaciers. Academic Press, 2010. Sliding velocity cannot be reduced to ice thickness and slope under any approximation. As Cuffey and Paterson in discussion the Shallow Ice Approximation say in chapter 8,  `The rate of basal slip must be specified directly or through a relation to bed stress such as Eq. 8.25'"

   Equation 4 just requires v_s ~ τ^ψ/σ (Budd et al., J. Glacial.,23, 157-170, 1979) where τ = shear stress and σ = effective normal stress. According to the shallow ice approximation, τ ~ hS (h = ice thickness, S = slope of the ice surface). In this form, it yields v_s ~ h^ψ/(h-p) S^ψ where p = water pressure. Among the references you provided, at least Harbor, Hallet & Raymond, Nature 333, 347-349 (1988) and Tomkin, Geomorphology 103, 180-188 (2009) assumed p ~ h, which exactly yields Eq. 4, v_s ~ h^ψ-1S^ψ. You may question the assumption that the fluid pressure at the bed is proportional to the ice thickness and claim that coupling with a distinct model for melting and for the flow of melt water may yield better results, and that such models are available. However, your statement "Sliding velocity cannot be reduced to ice thickness and slope under any approximation" is wrong.

2. About the type of the differential equation: "Ice thickness is not a diffusion process. See section 8.5.5 and equation 8.65 , 8.70, 8.77, 8.78 and 8.79 in Cuffey and Paterson. In the Shallow Ice Approximation ice flux q~h not the partial derivative of h wrt to x. There is a divergence relationship, first order partial derivatives, but diffusion is second order in the spatial partial derivatives. Without substantial justification, ice thickness cannot be treated as diffusion with strong diffusivity."

   Sorry, but the shallow water equations and its derivates (shallow ice equations and Savage-Hutter equations for granular flow) assume a free surface and hydrostatic pressure in vertical direction. Then the horizontal force is proportional to the gradient of the free surface and not to the gradient of the bed (as you presumably assume in your reasoning). As a consequence, the velocity in the shallow ice equation also depends on h and on the gradient of the ice surface. Then the divergence term from the mass balance yields second-order spatial derivatives of h in total. The resulting spatial differential operator is elliptic, so that the entire time-dependent equation is a parabolic equation (= diffusion type for the broader readership).

Best regards,

Stefan