Theoretical and numerical considerations of rivers in a tectonically inactive foreland
 Institut für Geo und Umweltnaturwissenschaften, AlbertLudwigsUniversität Freiburg, Albertstr. 23B, 79104 Freiburg, Germany
 Institut für Geo und Umweltnaturwissenschaften, AlbertLudwigsUniversität Freiburg, Albertstr. 23B, 79104 Freiburg, Germany
Abstract. Recent advances in the numerics of fluvial landform evolution models allow for largescale simulations of erosion and sediment transportover time spans of several million years. This study aims at finding out fundamental properties of rivers in a tectonically inactive foreland of a mountain range by investigating a simple reference scenario. This scenario consists of a mountain range and a foreland in a quasisteady state where the material eroded in the mountain range is routed through the foreland. In order to understand the properties of foreland rivers, a subdivision into two classes – carriers and redistributors – is introduced. Carriers originate in the mountain range and are thus responsible for the largescale sediment transport to the ocean. In turn,redistributors are rivers whose entire catchment is located in the foreland. Using the concept of carriers and redistributors, it is shown that the drainage network in the foreland permanently reorganizes, so that a steady state in the strict sense is impossible. However, the longitudinal profiles of carriers are described well by a steadystate approximation. Their concavity index is considerably greater than that of rivers in the mountain range. Carriers are predominantly depositing sediment at high rates, while redistributors are eroding at much lower rates. Despite the low erosion rates, the sediment flux from redistributors into carriers is a major component of the overall sediment budget and finally the main driver of the highly dynamic behavior of the carriers.
Stefan Hergarten
Status: final response (author comments only)

RC1: 'Comment on esurf202214', Anonymous Referee #1, 08 Apr 2022
General comments
The manuscript entitled ‘Theoretical and numerical considerations of rivers in a tectonically inactive foreland’ is a timely contribution to the discussion of how fluvial sediment transport dynamics influence the transfer of sediment through landscapes and the implications for interpreting sedimentary records. Hergarten usefully identifies two types of rivers contributing to erosion and sediment transport within an alluvial foreland and presents a clear methodology for analysing the net contribution of these respective river types to the overall sediment budget and network morphology. It is the detailed explanation of the modelling approach that is a great asset to this paper and a reason for why I believe this paper is well suited for the Esurf community. While the paper does not integrate any field or lab data to support its modelling, I appreciate the adaption of the model to integrate typical field observations, such as the changing erodibility of sediment surfaces as they increase in age. I think that the insights gained from this modelling approach are a useful baseline for future studies.
While the paper is generally well written and easy to follow, there are a few sentences that would benefit from rewording, of which some are listed below in the technical comments. The grammar and some of the sentence structure could be improved by editing from a native English speaker. The figures are excellent and easy to interpret and the mathematical formulae are correctly defined. The first half of the abstract would be improved by stating more explicitly the context of the paper, as at present it is a little vague and difficult to follow. There are a few recent references that could also be included to give credit to parallel approaches being developed in this field (Malatesta et al., 2017 Basin Research, plus those detailed below).
Specific comments
 In the introduction, it would be useful to explicitly define what you mean by steady state in the context of this work and why this definition is relevant.
 The expectation of high concavity values (>1) for these alluvial rivers was surprising for me, as I am not aware of these values being frequently observed in natural alluvial rivers (c.f. Wickert and Schildgen 2019 ESurf). Why is this model formulation acceptable for this case?
 I think the absence or distribution of accommodation space generation in the model needs to be specified and considered. The low deposition rates along the carrier rivers are perhaps to be expected if there is no accommodation space available for the sediment to be deposited in. I am wondering how the net fluxes and the interactions between the carrier and redistributing channels would shift as an accommodation space is defined. This maybe something to mention in the future work section of the conclusions as it has important implications for understanding the generation of physical and measurable parameters such as downstream grain size fining trends along alluvial rivers (for example, see Harries et al., 2019 ESPL). For the reference case in this paper, I would expect that any sizeselective grain size fining that would occur along the length of the carrier rivers would be solely introduced by the integration or recycling of older sediments by redistributing channels or the migration of carrier channels.
 Recent work has demonstrated that sediment transport along alluvial rivers is a nonlinear process, even at millennial timescales (Carretier et al., 2019 Sci Rep, Sinclair et al., 2019 Geology). In which case, net sediment transfer is not well characterised by mean values. It might be worth mentioning this work in the introduction and highlighting the usefulness of the reduced complexity approach presented. The observation that the storage time of sediment in the fan surfaces is highly variable and dependant on the distance between channels (section 9) aligns well with the findings of Carretier et al. (2020 EPSL).
Technical comments
Frequently, the phrase ‘on the mean’ or ‘in the mean’ is used (e.g. L289, L319 and L426). I think the desired phrase is ‘on average.’
L21: ‘modelling studies mentioned above’ – include more references to modelling studies
L2324: Reword
L55: by ‘tectonically inactive’ do you mean without subsidence?
L56 and L419: ‘exposed’ is perhaps not the right word for this context. You impose boundary conditions.
L214: Remove ‘in’
L289: Reword
L339: ‘one half to one widths’  reword
L342: Replace ‘rate’ with ‘range’
L429: Reword
L440: Reword

AC1: 'Reply on RC1', Stefan Hergarten, 25 Apr 2022
Dear Reviewer,
thanks for your constructive and encouraging comments! I am quite happy that all reviewers found the concept of carriers and redistributors useful. I also appreciate the additional references very much, although I am always afraid to pick those results of other studies that are in line with my own results. I have to read these papers thoroughly first, but I think that I can submit a revised manuscript quite soon.
Best regards,
Stefan

RC2: 'Comment on esurf202214', Anonymous Referee #2, 14 Apr 2022
This manuscript presents a numerical model and derives a conceptual framework to quantitatively explore sediment transport dynamics and drainage network evolution over large spatial and temporal scales in a foreland setting.
An emerging concept of the study is the division of foreland rivers into carriers and redistributors with welldefined geomorphic functions, long profiles, and sediment transport dynamics. The identification of the two groups of rivers is appealing and has the potential to serve other studies that explore sediment routing in foreland basins.
The manuscript presents only numerical results. This is a valid choice – numerical insights could be highly beneficial even when presented independently from a field, experimental, or previously presented numerical research questions or observation. However, I believe that the current manuscript could greatly benefit from some connection(s) to field observations, illustrating that the new concept of carriers and redistributors is meaningful.
Aside from a few odd phrasing choices (pointed out in RC1), the manuscript is overall wellwritten.
There are two major issues that I recommend considering, which could potentially increase the manuscript’s impact and usefulness.
First, the model is not described in sufficient detail. While the model was presented in a previous study, readers should be able to get the main message of how the model works without needing to consult the previous manuscript. This is particularly true since much of the inferred sediment and drainage dynamics appear to be specific to the model. Equation 1 has two unknowns: E and Q, and additional equations are needed to close the system. Both unknowns can be formulated as functions of local elevation, but this is not currently specified. Boundary conditions are also not fully specified. I.e., how does the model deal with Q at the highest node of each network? Choices pertaining to drainage dynamics are not presented and discussed. I.e., how the model deals with local slope reversals due to sediment deposition? Does the model assume steepest descent to induce drainage change? What is assumed about flow routing? Are lakes allowed? These choices likely directly influence model results, but readers are currently left in the dark.
The model is developed with nondimensional quantities, and a specific dimensional interpretation is proposed. This is a common practice that works in many cases. However, in the current manuscript, I found that the repeated dual interpretation (dimensionless and dimensional) is confusing. One way to overcome this issue is to formally present the scale factors once (it could also be interesting to see a nondimensional analysis of equation 1) and from that point on to present the results only with either dimensional or nondimensional form (the former is probably easier to read). On the same issue: why does it make sense to use two length scales? Why can’t the analysis work with a single length scale?
Some of the steadystate and long profile analyses presented in section 6 could be moved to the model description, providing muchneeded intuition of model expected behavior.
The second major issue is that I struggled to balance sediment mass and topography across sections 79. I assume that the model conserves mass (including sediments deposited in the ocean). However, I could not balance it myself (I didn’t try to balance it formally, but just in terms of sources, sinks, processes, and orders of magnitude). Figures 9 and 14 are confusing to me. How come sediments are incorporated from the foreland, but there is no arrow showing sediments being damped in the foreland? How come the percent sum of sediments deposited in the ocean is 100 while both the figure and the text refer to the total sediments as more than 100% (e.g., the 91% of the carriers is out of the 133% coming from both the mountain and the foreland).
Similarly, I’m unsure how to reconcile figures 10, 11, and the sediments transported to the ocean. Carriers’ deposition rate is larger by an order of magnitude with respect to the erosion rate of the mountain and by two orders of magnitude with respect to the erosion rate of redistributors. How can that be? Are the carriers two orders of magnitude smaller in area than redistributors? How is it possible that with such a high sedimentation rate, the foreland is not growing in topography but reaches a statistical steady state? How can this be reconciled with figure 9?
Perhaps a formal statement (with equations) of mass/volume conservation could help out here, clarifying which component out of the mass balance each figure analyzes?
Line comments
Line 1  First sentence of the abstract reads a bit detached.
Line 2 – ‘Fundamental properties’. At this stage, only fundamental numerical properties’.
Line 100 – Not sure there is a need to mention simulations that are not presented here.
Line 121 – Perhaps ‘representation’ instead of ‘coordinates’.
Perhaps a more informative title to section 2.
Figure 3 – Maybe variance would be a better description than relief.
Figure 5 – Maybe also show some profiles?

AC2: 'Reply on RC2', Stefan Hergarten, 25 Apr 2022
Dear Reviewer,
thanks for your constructive and encouraging comments! I am quite happy that all reviewers found the concept of carriers and redistributors useful. It should be possible to improve the explanations without making the paper too excessive and I will hopefully be able to submit a revised manuscript quite soon.
Best regards,
Stefan

AC2: 'Reply on RC2', Stefan Hergarten, 25 Apr 2022

RC3: 'Comment on esurf202214', Jean Braun, 20 Apr 2022
Review of “Theoretical and numerical considerations of rivers in a tectonically inactive foreland” by Stefan Hergarten
Dear Stefan,
I have read your manuscript with great interest. It contains some very interesting results. In particular, I appreciated your division of channels in the foreland into carriers and redistributers. These are indeed very useful concepts to understand most of the dynamics of the quasisteady sedimentary system you study and that you labeled a “tectonically inactive foreland”. The analysis of the concavity of channels is very interesting and novel, the dual role played by the redistributers as well as the sourcetosink description of the system are all facilitated by the new nomenclature.
Although I support the publication of this material in ESURF, I express below some concerns that I have about the presentation of your results, their robustness and their applicability.
 The objectives of the manuscript are relatively well explained and certainly interesting for those of us that like to play with equations. I find, however, that you could improve your introduction by relating better your objectives (last paragraph of the introduction) to questions that are asked by sedimentologists, geomorphologists. Similarly, I believe your paper would gain much in its impact if you were to come back to these questions (and how your work has contributed to their resolution) in your discussion.
 Although the basic evolution equation has been presented elsewhere, I believe it is important for the comprehension and the flow of the manuscript to at least present the basic PDE that you are solving. Even though you focus on the quasi steadystate solution, you must solve an evolution equation, most likely expressed in terms of the vertical elevation of the topography as the main unknown. It is also important to give the form of this equation in the case where Kd tends to infinity, because this is the form that you have used for most of the results presented here (in the basin). Am I right in assuming that it then takes the form of a nonlinear diffusion equation? For both the mathematicallyoriented readers of your manuscript and the sedimentologist who might be interested in interpreting your results, it seems important to me that these equations (the full form and its asymptotic form when Kd tends to infinity) be presented.
 Although I fully support the need to use dimensionless variables when presenting model results, I do not agree with your approach to quote absolute values for basic parameters such as K or grid size and derive other length scales and time (or rate) scales out of it. I believe it would be much more useful to explain with some simple relationships how the dimensionalisation should be done, i.e., how one could apply your dimensionless results to a problem of known size and rate.
 In the model description, you mention that the algorithm you use is implicit and thus unconditionally stable. You do not, however, assess its accuracy, which we know must depend on the time stepping (and grid spacing). Can you please provide us with an estimate of this accuracy. I am concerned (see point 6 below) that the solution might be dependent on the step size. If your results are applicable to natural systems, which are characterized by finite avulsion rates, the model should be characterized by a characteristic time for channel geometry to change. I believe that you need to check whether the time step you are using is smaller than such a characteristic time for many of the conclusions you draw to be correct.
 You note that the foreland is made of two parts (a fan and what I will call an alluvial plain connecting the fan to the ocean). You also note that the behavior of the system is rather contrasted in these two sections. So what controls the size of the fan becomes an important factor in describing the system’s behavior. I recently demonstrated with a 1D version of a model identical to yours that it is the size of the mountain catchment area that controls the size of the fan (regardless of the value of Kf). This implies that the setup you have used (with a very small mountain) leads to a relatively peculiar situation that might not be representative of many forelands. May I suggest that you test the robustness of your finding against the size of the fan (by changing the size of the mountain area). It might lead to very similar results with a simple shift of some of your curves (as shown in Figure 10). But it might not. Furthermore, some of the numbers you quote in your “sourcetosink” section may be quite different for a different relative size of the fan.
 Let me now come to my main concern: I found the part concerning the time scale for drainage reorganization very interesting. However, I do not know how to interpret these results to understand how real (natural) systems behave. I am particularly concerned about how the spatial and temporal resolutions of your model experiments might influence your results. I think this needs to be investigated for your results to have the impact they deserve. As channels have no width, there is a possibility that you might not be able to extract an avulsion time scale out of the basic equations, in which case many of the results you present (for the time evolution of the system in its quasi steadystate) might be difficult to use to interpret natural systems.
 Another point of concern is your use of a single direction flow routing algorithm, which you should try to be better justify in your method description in a low slope system/environment controlled by continuously evolving states of deposition and erosion. Such natural systems are often characterized by nondentritic channel networks with flow splitting occurring as often as flow merging.
I also have some minor comments on the presentation of your results:
 Line 182: I am not sure about the use of the term “black and white scenario” nor to what it corresponds to.
 Figure 9 is not clear; you have two sets of arrows leaving the foreland into the ocean; I believe one must be associated to carriers and the other to redistributors. This should be indicated somewhere (in the caption?). Also I am not sure to what corresponds the set of vertical arrows in the foreland? In a steadystate solution, on average the flux in and out of the foreland must be nil.
 Please make sure that figure 13 that uses the classification of “regions” in the foreland as described/shown in figure 1 has a reference to it in the caption.
I hope you will find these comments and suggestions useful and hope to see a revised version of your manuscript.
Regards
Jean
Jean Braun

AC3: 'Reply on RC3', Stefan Hergarten, 25 Apr 2022
Dear Jean,
thanks for your constructive and encouraging comments! I am quite happy that all reviewers found the concept of carriers and redistributors useful. For me, developing this idea and pointing out that the redistributors are important for the dynamics of the alluvial plain is the main point of the paper, and I am aware that the numerical model used here has still several caveats. I think that I can submit a revised manuscript quite soon.
The numerical accuarcy  your main concern  is indeed an issue at some points. The value dt = 1e3 was coming from an earlier series of simulations with focus on the scaling properties of depostion rates. There I used a much smaller dt = 1/16384 and measured rates over different time intervals, finding a strong effect of avulsions only for time intervals considerably greater than 1e3. However, dt = 1e3 might indeed be too large for the analysis of the time scale of network reorganization. Since rivers are only one pixel wide, the area covered by the carriers in a given time interval is also limited by the number of steps and thus by dt. This may indeed become a problem in the range close to the ocean. I started a simulation with a smaller dt in order to test whether the results shown in Fig. 13 are strongly affected by the value of dt.
In turn, I do not fully agree to your statements about the alluvial fans and would be happy to discuss this with you. You mention that you demonstrated that the size of the mountain catchment controls the size of the alluvial fan. In your recent ESurf paper, however, it rather seems to me as if you enforce this result by a very specific assumption on the catchment size. If I read it correctly, you apply Hack's law to the part of the rivers outside the mountain range alone and then simply add the catchment size A0 of the part located in the mountain range (your Eq. 10, A = A0+kx^p). If we did this for any point in a "regular" catchment, it would be completely wrong. Here it may work. Your approach predicts a very weak increase in A close to the mountain range (which makes sense to me), but how fast the catchment size "recovers" depends directly on A0, k, and p in your approach. So I am a bit afraid that your result on the size of the fans is more related to your very specific Hack's law than to the erosion model.
Anyway, it is not the point to question your result here. However, taking it for truth without further discussion for validating my simulations would not be sound. Without having tested it explicitly, I am quite sure that my simulation would predict that the size of the alluvial fans is propertional to the spacing of the biggest rivers leaving the mountain range and thus also proportional to the square root of their catchment size. So we would finally arrive at similar results, but for different reasons. In my opinion, this question should not be reduced to a few sentences in the recent manuscript, but requires a thorough consideration.
Best regards,
Stefan

EC1: 'Comment on esurf202214', Sagy Cohen, 29 Apr 2022
Dear Dr. Hergarten,
Thank you for submitting "Theoretical and numerical considerations of rivers in a tectonically inactive foreland" to Earth Surface Dynamics. As you have seen, we received three thorough reviews. All three reviews are favorable, recommending different levels of revisions. The reviewers raised important points that require attention, and I would like to see those addressed in the revised manuscript. As some revisions may require additional analysis, I return the manuscript to you for Major Revisions. Details regarding the submission process and timeline will be provided separately.
Best, Sagy Cohen
Stefan Hergarten
Model code and software
A simple and efficient model for orographic precipitation: codes and data Hergarten, Stefan and Robl, Jörg https://doi.org/10.6094/UNIFR/219131
Rivers in a tectonically inactive foreland: codes Stefan Hergarten https://doi.org/10.6094/UNIFR/219131
Stefan Hergarten
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