First, I would like to thank the reviewers for many constructive comments, and the authors for preparing and submitting a revised version of the manuscript. I think that the manuscript has been improved and it is on the right track.
However, that said, I also think that there are places where improvements and clarifications are needed before publication.
I recommend that the authors at this stage continue revision of the manuscript based on my comments to the revised version of the manuscript. My comments are listed below. In this process, the authors should use any potential there is to shorten the text, in particularly section 3, which I found difficult to read.
General comments:
As a first general comment, I must say that I agree with reviewer 1 in that the detailed level of the math hampers the reading of the paper, and I do not think the authors have done enough to meet the concerns of the reviewer here. In my view, the problem is not the equations themselves, nor is it the number of equations. However, the lengthy deviations are in places poorly motivated, and I had several times to read ahead in the paper to understand where the math was going.
It is particularly a problem in section 3, which I think could be significantly shortened, for example by moving the non-dimensional analysis in section 3.1 or the steady-state analysis in section 3.2 to the appendix. In stead, I suggest that more use could be made of the numerical solution to eqn. 13 (Fig. 8). Does the numerical solution coupled with intuition not illustrate most points (e.g. line 615, 780-782) made about steady-states and the transient transitions between them? I do not question the validity of the analytical solutions, but I wonder if they represent the best way to communicate your points.
In any case, you need to motivate the analytical solutions more and carefully explain the assumptions made. For example, what is the implication of assuming qs=0 in line 503? Under which conditions can this be assumed, which limitations does the assumption introduce, and what new insights can expect from studying this situation in particular.
Also, like reviewer 1, I was confused about Eqn. 13. At first glance it looks like a standard continuity equation, but with double treatment of deposition/entrainment. I fully understand the reviewer’s comment, and it was only after reading your answer to the reviewer’s comment that I understood that this is the balance of only the mobile mass. When introducing the equation in line 314 you write: “…we use the total sediment mass on the bed as a variable”. I recommend rephrasing this and explain carefully how the two reservoirs interact. I also recommend including a schematic figure as you suggest, and to move Eqn. 16 up next to Eqn. 13 so that both balance equations for the two sediment reservoirs are presented together. It is important here to carefully present the model to a wide group of readers, instead of relying on the readers’ knowledge of the Exner equation (which I did not know, and would not have remembered if I did).
Section 2 presents the probabilistic framework, and this section already reads well. However, there are some issues with the implied relation between Ms and Qs. This relation is the main subject of section 3, but it seems here that you are assuming something that the readers are not yet shown. For example in line 155, you link eq. 6 directly to eq. 1 even though one is using Qs and the other is using Ms. I failed the see why these two linear equations are identical. At least some clarification is needed.
Like for section 3, I suggest that you consider introducing a simple schematic figure also in section 2, perhaps of an evolving bed-cover in a channel. Such a figure could be used to illustrate the meaning of A*, Ms*, P etc. It would also nicely supplement the many curves presented in the other figures.
As a final general comment, there is a tendency in the manuscript to use quite subjective, and at the same time imprecise, descriptions of previous work. Some examples:
Line 720: “Despite short-comings, Aubert et at. (2015) presented…” What shortcoming? You need to either skip the reference to shortcomings or carefully describe them.
Line 749: “While the model is interesting and provides…” What is interesting about it? Be more specific please, or skip the descriptive wording.
Line 771: “…quickly in response to changing boundary conditions, a somewhat counter-intuitive notion for slowly-eroding channels” – If this is counter-intuitive or not must depend on the precise meaning of “quickly” and “slowly-eroding”. Again be more specific or skip it.
More specific comments:
line 24: “they” -> channels
line 79: “tended” – tend
lines 99-101: consider rephrasing this sentence; I’m not sure that I understand it.
lines 124-128: You could move some of the information given in lines 678-698 up here to better motivate why influence of other parameters such as bed topography and roughness is ignored.
line 162: move “,k,” up to after “…value”; it will make k more visible.
eq. 9: in eq. 2 you use exp here e – needs to be consistent.
line 245: “though” -> “through”?
line 254: What is the model step length? Also is the probability of deposition always 1? If so, please write so explicitly.
line 270: “… given out by the model…” please rephrase.
line 271: Is the CA model not also a probabilistic model? Its control parameters seem to be likelihoods of entrainment, whereas the “probabilistic framework” presented has likelihoods of deposition. But still, in essence both are probabilistic, right?
Fig. 3: The comments above also makes me think that there should be more direct relationships between P and A within your framework, and then pi and pc from the CA model?
Line 298: The relation between Qs* and Ms is not clear from eqn. 3.
Line 299: what do you mean by “muddled”?
Line 299: what is incorrect?
line 300: “bases” -> “basis”
line 312: skip “of course”
line 325: while non-dimensional variables are certainly useful, they also introduce complexity. I did not understand why they were necessary here. Please motivate the non-dimensional analysis better. Give the reader some clear hints as to where the analysis is going (and consider moving it to appendix).
line 335: “The rate of change of the stationary sediment mass…” seems a bit counter-intuitive text-wise. To me “stationary” means “not moving”. Could you replace “stationary sediment” with “deposited sediment”? You would have to change this throughout the text though – but the reading would be better I think.
line 497: “the equations” – which ones?
line 503: Please remind the reader about the meaning of qs* and qt*. What is happening when qs*=0 and what is the relevance of this case?
line 663: “1000s” -> “1000 s”
line 717: “depended” -> “depends” (if Aubert et al.’s results are still valid then consider using present tense in such places.
line 730-735: I could not follow these arguments about grain size and grid size.
line 751-753: I’m still not sure how this would work. Can you be more specific in how bed roughness could be implemented in the framework?
line 762: Also here you could be more precise. It is a general point of this manuscript that the probabilistic framework can me integrated with many types of data, but I’m still not sure how. More guidance would be useful.
line 771: “Cln” – “In” |

Dear Jens and Rebecca,
Thank you for your ESurf submission - I am delighted to inform you that it has now been accepted for final publication. The AE noted there may still be some minor copy editing to do, but this can be picked up in the proofreading stages.
All the best,
Tom |