the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 17 Feb 2026
Lift or impact: modeling bedrock incision coupled with sediment dynamics
Abstract. We analyze how the process of bedrock incision by the impact of sediment grains can be described and coupled with sediment dynamics. We first point out that the key parameter is a bedrock dimensionless coefficient that describes the ratio between the volumes of impacting sediments and bedrock erosion. We then write the coupled equations by introducing a partitioning coefficient between sediment and bedrock erosions. It describes the time spent doing one or the other erosion process – or the proportion of depositing grains that impact bedrock. In a 1D along-stream system, the resulting equations lead to a similar description of the cover effect proposed by Sklar and Dietrich (2004), giving a rationale to their expression. In a second step, we extend the concept to lateral erosion or deposition fluxes. We develop analytical solutions for a river fed by uniform lateral sediment fluxes from hillslope and show why the sediment load can exceed the transport capacity. We then implement the equations in the numerical code River.lab/eros, where water depth and velocity, as well as erosion and deposition fluxes, are solved with the method of precipitons. As an example, we simulate the evolution of the Rheinfall at Schaffhausen, Switzerland, a prominent knickpoint along the Hochrhein. In contrast with sediment processes, where the knickpoint slope decreases by diffusion without upstream displacements, bedrock abrasion allows knickpoint to move upstream while retaining almost the same shape. This is consistent with detachment-limited behavior as emphasized in the theoretical part of the paper. The knickpoint shape (foot elevation and height) and retreat rates are highly dependent on the sediment load in river. Bedrock erosion first happens in a narrow canyon that propagates upstream, and then the river widens after the knickpoint passed by.
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Status: final response (author comments only)
- CC1: 'Comment on egusphere-2026-420', Takuya Inoue, 22 Mar 2026
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RC1: 'Comment on egusphere-2026-420', Anonymous Referee #1, 09 Apr 2026
General Comments
The authors consider the role of sediment dynamics in the erosion of bedrock rivers by partitioning the time sediment spends impacting bedrock and the time it spends deposited. They then apply the framework to investigate how a river profile evolves as a function of sediment supply and rock properties. The manuscript offers and interesting approach to incorporating the complex role of sediment cover in landscape evolution. The paper would be strengthened by providing more context related to similar work attempting to couple sediment dynamics to bedrock incision, even though the particular approach may be new. The terms in many of the equations sometimes lack definition or the fully articulated assumptions behind them, which increases the difficulty of evaluating the model.
Specific Comments
As the authors note in line 49, there is an existing body of literature from which these ideas emerge in addition to previous studies with comparable applications (e.g., Beaumont et al., 1992; Nelson & Seminara, 2012; Turowski & Hodge, 2017). This body of literature is somewhat underrepresented, for example, alternative approaches to accounting for the partitioning between sediment participating in cover and sediment impacting bedrock. Making this broader context more explicit in the paper would aid the reader’s ability to both trace concepts used here and to better appreciate this new contribution.
Careful explication of the concept of transport capacity as it is used here, in particular, would greatly support key claims in the paper. The authors define it in relation to an erosion rate and a length scale, but this leads to some seemingly unintuitive results, such as negative bedrock incision rates (see comment on line 423). I think it is likely that some of these apparent concerns would be assuaged by a more detailed set of definitions. Generally, there are many places where more careful definitions and statement of assumptions should be made.
Relatedly, a more thorough description of the x_s partitioning coefficient would be helpful because it currently leads to some seemingly contradictory consequences, including only deposition with a fixed sediment cover. Since some of the novelty lies in this partitioning coefficient, resolving some of these issues by developing the rational behind it would be beneficial.
Line Comments
Line 20: It should be either “allows a knickpoint” or “allows knickpoints”.
Line 24: Change to “has passed” in order to match tenses.
Line 28: Here, and in the lines 33 and 60-61, the citations should be in-text instead of parenthetical.
Line 41: Change to “…other complexities, such as … roughness, …”
Line 51: More explicit engagement with the origin of these concepts would help contextualize the contribution.
Eq. 2: There is more description in line 81, but it might make sense to define the parameter here where it first appears.
Eq. 5: It would be helpful to have a space between the comma and zero to make it clearer that it they are part of the two terms from which a maximum is selected rather than 1,0 as a single number.
Line 64: Is there a better name for \sqrt{RdD_s^3}. I know it is the standard normalization in the Einstein number, but I am not sure calling it a typical sediment flux itself is the best way to label it. Maybe characteristic sediment flux?
Line 68: Capitalize Eq. for consistency.
Line 68: Make “bedform” plural.
Line 70: \alpha is used again in line 99. Additionally, state that w_s is the settling velocity rather than just saying the second \alpha depends on settling velocity.
Eq. 7: There must be a notation error somewhere because the equation is currently dimensionally inconsistent.
Line 76: Should “Eqs” be “Eqs.”?
Eq. 8: I don’t think w_{si} is ever defined and \rho is not specified until line 140.
Line 84: Replace the less than sign with a period.
Line 85: Here, and in many places, there is no space between the period and the opening parenthesis in the equation abbreviation, but in other places there is. I suggest having the space in all occurrences.
Line 91: I suspect you mean completely eroded at a given location on the channel compared to completely eroded as in there is no sediment cover anywhere on the channel at a given instant. Can this be clarified?
Line 96: I think what is meant in the last clause is either “which is the product of the first two terms in Eq. 8” or “which is equivalent to the first two terms in Eq. 8,” but the construction with a that clause does not seem right.
Eq. 10: The numbering is on the wrong side.
Line 99: There might be a comma missing after is, depending on the intended meaning.
Eq. 11: It is evident from each derivative, but can labels be added to more readily indicate the respective sediment reservoir?
Eq. 12: I think this allows for the sediment cover to change with respect to the constituent grains but not with respect to its thickness. It is fair to model that, but the assumption should be stated very plainly.
Line 115: Add reference to Eq. 4.
Line 117: It may be worth noting the notational choice related to the prime to distinguish the integrating variable just to make it clearer, if that is what it is.
Line 185: The beginning bracket should be an absolute value sign.
Line 207: Throughout, the period should probably be a dot (\cdot command in LaTeX). It could be omitted in the units, but in Line 466, for example, the number has the same problem, which changes the numerical value.
Line 209: Which is meant: “directly supply the bedload” or “be supplied directly to the bedload”?
Line 235: This is an interesting claim, but I did not really follow where it came from. Relatedly, Figure 3 states that the top plot is for q_s^* but the axis is Q_s^*. I suppose they are equivalent for a fixed channel width, but is that an appropriate assumption over the distance which it takes to reach q_s^\inf?
Line 266: Remove the definite article before “Figure 2”.
Figure 4: The legends are illegible.
Figure 5: The legends are illegible.
Figure 9: I know it can be deduced from the other figures an is alluded to in the caption, but adding either a “Concentration =” next to the number or clarifying further in the caption would help the figure be independently readable, as done in the other figure captions.
Figure 10 Caption: The colon is missing.
Line 390: It should be “simulations with the same”
Line 394: The multiplication sign between the 4 and power of ten is missing.
Figure 12 Caption: “Evaluation” is not necessarily wrong, but was it meant to be “evolution”? The multiplication sign between the 5 and power of ten is also missing.
Line 423: The difference between the “classical transport capacity,” which you say can be exceeded and the “transport capacity,” which you say is physically justified is not very clear. Lines 435-437 imply that there may be no difference at all, but doesn’t that suggest negative bedrock incision via Eq. 8? How does a transport capacity that can be exceeded differ from a simply poorly defined transport capacity, or, I suppose more precisely according to equation 4, an incorrect choice of the hop length? If the negative value indicates deposition, then what does the criterion for deposition become if a river can transport more material than it is capable of transporting? If the river is only in a depositional state, that also seems to violate the fixed cover thickness that results from substituting Eq. 12 into Eq. 11. Based on Davy & Lague (2009), I think what is most likely meant is that transport capacity is not exclusively controlled by water discharge, but the use of the terms in this manuscript does not presently communicate that effectively and the questions about the term when “capacity” is exceeded remain.
Line 426: A comma is not needed after the hyphen.
Line 452: Use “has passed” instead.
Line 452: Insert “a” between in and canyon.
Line 456: Replace “of” with “to”.
Line 460: I think “from” rather “in” the Alps would sound more natural.
Line 461: Add “a” between as and reason.
Line 465: Should parameter be plural? Also, do you mean just mean high values or are they higher than something in particular?
Line 465: A more natural formulation may be “…which can lengthen the simulation time needed to obtain significant bedrock erosion.”
Line 473: While there is nothing wrong, the ending does seem quite abrupt by finishing talking about not making conclusions and things being beyond the scope of the paper.
References
Beaumont, C., Fullsack, P., & Hamilton, J. (1992). Erosional control of active compressional orogens. In K. R. McClay (Ed.), Thrust Tectonics (pp. 1–18). Springer Netherlands. https://doi.org/10.1007/978-94-011-3066-0_1
Davy, P., & Lague, D. (2009). Fluvial erosion/transport equation of landscape evolution models revisited. Journal of Geophysical Research: Earth Surface, 114(F3). https://doi.org/10.1029/2008JF001146
Nelson, P. A., & Seminara, G. (2012). A theoretical framework for the morphodynamics of bedrock channels. Geophysical Research Letters, 39(6). https://doi.org/10.1029/2011GL050806
Turowski, J. M., & Hodge, R. (2017). A probabilistic framework for the cover effect in bedrock erosion. Earth Surface Dynamics, 5(2), 311–330. https://doi.org/10.5194/esurf-5-311-2017
Citation: https://doi.org/10.5194/egusphere-2026-420-RC1 -
CC2: 'Comment on egusphere-2026-420', Jens Turowski, 10 Apr 2026
I would like to contribute four main comments on this manuscript.
- I agree with Takuya Inoue’s comment that the novelty of the paper needs to be clarified. The work is currently not well motivated. It has already been pointed out in the other comments that there are similar approaches in the literature that need to be cited and put into context. A highly relevant paper developing a model based on the entrainment-deposition framework that has not been mentioned yet is by Shobe et al., GMD 2017. You may also want to look at recent work by Rebecca Hodge. She has done a lot of very relevant flume and field work on the topic.
New equations are warranted to either fix theoretical or empirical problems with existing approaches, or to close a gap in predictive capabilities. The authors could summarize the existing approaches accordingly and point out where their formulation is intended to improve. In addition, they could provide an overview of how and where their new model differs from existing ones. - There seems to be an issue with the mass balance to me. Consider a control volume in the stream. A hallmark of the entrainment-deposition model is that it considers the mobile volume of sediment per unit area Vm explicitly separate from the stationary mass Ms. The rate of change of mass change is the dVm/dt = e – d – dqs/dx, where t denotes time, e the entrainment rate, d the deposition rate, qs the along-stream sediment flux, and x the downstream distance. This can be compared to the mass balance equation used by the authors, eq. (1) in the manuscript, and the equations are the same when the time derivative is set to zero. For alluvial systems and landscape evolution time scales, for which the authors’ development is intended, this can be justified by arguing that the relevant adjustment time scale are short in comparison to the timescales considered in the model (i.e., the forcing terms would be slowly varying in comparison to the adjustment timescales). In this case, the temporal variation of Vm can be neglected and each part of the river can be considered to be in a transport steady state.
In contrast, in a bedrock system, we need to explicitly consider the mass balance of the stationary sediment Vs (which, btw, determines the cover). This gives a second mass balance equation dVs/dt = d – e. While the authors do not explicitly formulate this equation, it is implicit that its time derivative is not set to zero, i.e., d can differ from e. Mathematically, steady states can be possible where e-d = dqs/dx. However, physically, this does not make sense to me. The reason is that the adjustment timescale of Vs is necessarily the same as that of Vm and the assumption of this timescale being short enough to be able to neglect the temporal derivatives would need to apply to both equations simultaneously. This steady state assumption thus leads to e = d always. - When reading through the introduction, it seems to me that the authors are unaware of a large body of relevant literature. For many of the statements, I can think of multiple other papers (see some suggestions below). I suggest to widen the literature search, include some additional key references, and add ‘e.g.’ to many of the citation lists to acknowledge that they are incomplete.
- The discussion is currently superficial. It fails to critically evaluate the new formulation with respect to its physical assumptions, current debates, and the existing literature on fluvial bedrock erosion and bedrock channel morphodynamics. It is unclear in how far the model differs from existing models and whether there is evidence to support one or the other. The case study feels incomplete (actually admitted by the authors, line 467) and its point is unclear.
28 better cite Sklar and Dietrich 1998 or 2004 here, and move the citation of the 2001 paper to line 29.
29 Sklar and Dietrich 2001 is a key citation for the experiments. There are other examples or experimental papers, so best add ‘e.g.’. There are also quite a few papers providing field evidence.
32 earlier mechanistic formulations exist, for example due to Foley or Ishibashi.
33 further changes are for example by Auel et al. 2017, Demiral et al. 2026, and Turowski et al., 2023.
36 again multiple further articles exist, important would be Miller and Jerolmack 2021 and Turowski et al. 2023.
40 The linear cover model was actually proposed by Sklar & Dietrich 1998 (as far as I am aware, there are two earlier conference abstracts mentioning that model, one by Sklar and Dietrich in 2016 and one by Slingerland et al. in 2017).
42 there is more relevant literature, for example, Hodge and Hoey 2012 and Johnson 2014.
45 The statement here is especially scarce in citations. The only reference cited at the moment, Lamb et al. 2008, is not even concerned with the cover effect, and is not really relevant here.
48 The statement here begs for references.
In general, please define all the symbols when they are first used.
Eq. 8: here, the authors equate the saltation hop length from the original SD2004 model with the transport length from their model. This does not make sense to me, as the former one is defined for the process scale and the latter one for the landscape evolution scale. There is a statement on this in line 85, but the assumption should be justified before introducing the equation. Alternatively, introduce the equation with the saltation hop length first, then argue why this can be assumed to correspond to the transport length. I think a more honest way would be to make the two length scales scale with each other, rather than setting them equal. I would, however, expect that the scaling factor is not a constant.
Eq. 12: This seems to be quite a strong statement to me that seems odd from a mechanistic perspective. This is directly related to the construction of the model and should have a strong physical basis. Do I understand it correctly that this implies that the volume of eroded bedrock is set equal to the potential volume of sediment entrainment (corrected for porosity)? This seems implausible to me. The authors need to do some more work here to justify this.
It also does not make sense to me to add erosion products to the bedload (which can subsequently cause further erosion downstream). The erosion products produced by impact erosion typically are fine and would be part of the washload or at least suspended load.
415 The ad-hoc model is the linear model due to Sklar and Dietrich. Subsequent models have given physics-based (e.g., Aubert et al., 2016), reduced complexity (e.g., Hodge and Hoey, 2012) or stochastic frameworks (e.g., Turowski and Hodge, 2017).
435 The expression was not empirical, but an ad hoc formulation (essentially using the simplest function to connect the two endmembers of zero cover at zero sediment supply and full cover when supply matches transport capacity).
452 Increasing shear stress can also have the effect of increasing hop length, decreasing erosion rate. This has already been discussed in the Sklar and Dietrich 2004 paper. In addition, if the aspect ratio of the channel approach 1, shear stress is redistributed onto the walls, which can also reduce the bed erosion rate.
465 What is the point of including this into the paper if you do not do it properly?
469 Please cite the relevant literature.
References
Aubert, G., Langlois, V. J., and Allemand, P.: Bedrock incision by bedload: insights from direct numerical simulations, Earth Surf. Dynam., 4, 327–342, https://doi.org/10.5194/esurf-4-327-2016, 2016.
Auel, C., Albayrak, I., Sumi, T., Boes, R.M., 2017b. Sediment transport in high-speed flows over a fixed bed: 2. Particle impacts and abrasion prediction. Earth Surf. Proc. Land. 42 (9), 1384–1396. https://doi.org/10.1002/esp.4132.
Demiral, D., I. Albayrak, J.M. Turowski, R.M. Boes, 2026, Hydro-abrasion processes and modeling at hydraulic structures and steep bedrock rivers: 2. Hydro-abrasion model development and application, Journal of Hydro-environment Research, 64, 100690, https://doi.org/10.1016/j.jher.2025.100690
Hodge, R. A. and Hoey, T. B.: Upscaling from grain-scale processes to alluviation in bedrock channels using a cellular automaton model, J. Geophys. Res., 117, F01017, https://doi.org/10.1029/2011JF002145, 2012.
Johnson, J. P. L.: A surface roughness model for predicting alluvial cover and bed load transport rate in
bedrock channels, J. Geophys. Res., 119, 2147–2173, https://doi.org/10.1002/2013JF003000, 2014.
Miller, K. and Jerolmack, D.: Controls on the rates and products of particle attrition by bed-load collisions, Earth Surf. Dynam., 9, 755–770, https://doi.org/10.5194/esurf-9-755-2021, 2021.
Shobe, C. M., Tucker, G. E., and Barnhart, K. R.: The SPACE 1.0 model: a Landlab component for 2-D calculation of sediment transport, bedrock erosion, and landscape evolution, Geosci. Model Dev., 10, 4577–4604, https://doi.org/10.5194/gmd-10-4577-2017, 2017.
Sklar, L., and W. E. Dietrich (1998), River longitudinal profiles and bedrock incision models: Stream power and the influence of sediment supply, Rivers Over Rock: Fluvial Processes in Bedrock Channels, edited by K. J. Tinkler and E. E. Wohl, Geophys. Monogr. Ser., vol. 107, pp. 237– 260, AGU, Washington, D. C.
Turowski, J.M., R.A. Hodge, 2017, A probabilistic framework for the cover effect in bedrock erosion, Earth Surface Dynamics, 5, 311–330, doi: 10.5194/esurf-5-311-2017
Turowski, J.M., G. Pruß, A. Voigtländer, A. Ludwig, A. Landgraf, F. Kober, A. Bonnelye, 2023, Geotechnical controls on erodibility in fluvial impact erosion, Earth Surface Dynamics, 11, 979–994, https://doi.org/10.5194/esurf-11-979-2023
Citation: https://doi.org/10.5194/egusphere-2026-420-CC2 - I agree with Takuya Inoue’s comment that the novelty of the paper needs to be clarified. The work is currently not well motivated. It has already been pointed out in the other comments that there are similar approaches in the literature that need to be cited and put into context. A highly relevant paper developing a model based on the entrainment-deposition framework that has not been mentioned yet is by Shobe et al., GMD 2017. You may also want to look at recent work by Rebecca Hodge. She has done a lot of very relevant flume and field work on the topic.
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EC1: 'Comment on egusphere-2026-420', Rebecca Hodge, 14 Apr 2026
Thanks to all reviewers for their constructive comments. This comment is to note that CC1 is from one of the invited reviewers, and so this preprint has recieved the requisite number of reviews for the discussion to be closed at the end of the discussion period.
Citation: https://doi.org/10.5194/egusphere-2026-420-EC1
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- 1
In this study, the authors propose a numerical analysis model for analyzing elevation changes in mixed bedrock-aluvial channels using a different formulation than previous studies. I feel that their formulation is challenging and novel. However, it is unclear what new capabilities have been achieved as a result. Furthermore, there are several unclear points in the modeling. Below I list a few major comments for the review along with some line-by-line comments.
Major comments:
1. A two-dimensional model considering bedrock erosion (including lateral erosion) and alluvial deposition (including bar formation) has been proposed by Inoue, Parker and Stark (2017, ESPL), Inoue et al. (2021, JGR), and Cho and Nelson (2025). The novelty needs to be further clarified. It is insufficient to simply state that the fundamental equations are new; the authors need to clearly define their novelty in terms of the phenomena they can address.
2. I find the explanations for transport length, saltation length, deposition rate and impact velocity problematic (Lines 85-87). Transport length is often defined as the distance a sediment travels from the moment it starts moving until it stops. This is different from the distance of a single jump (i.e., saltation length or hop length). Also, d (deposition rate) in Equation 2 is the volume of deposited particles divided by the transport length multiplied by the river width, which is clearly a different definition from the particle collision velocity. If you claim these are similar, you must clearly state the previous research that supports that relationship.
3. Please provide sufficient explanation regarding particle size, roughness, and the amount of sediment supplied from the upstream end in in the calculation conditions (Section 4.2). Regarding roughness in particular, you describes the importance of roughness on alluvial cover in Introduction, but there is no mention of how roughness is treated in your model or numerical calculations.
4. Providing longitudinal and transverse profiles of the alluvial layer thickness on the bedrock "hs" would help to more easily understand the cover effects.
Line-by-line comments
Eq. 2: Add an explanation for ξ to the text following the equation. L71 is too far away.
Eq. 5: Add an explanation of α to the text following the equation.
L67: D-->Ds
L68: Include references that support the idea that A* is between 0.04 and 0.1. For MPM, qs = 8(τ*-τ*c)^1.5sqrt(RgDs^3), and the value of A* depends on τ*c and particle size.
Eq. 7: x*-->x. Otherwise, the dimensions of the left and right sides do not match.
L84: Please add explanations for ρs, wsi, and kv.
L91: Does this mean that the alluvial cover ratio pc = 0? or pc <1?
Eq.10: Since α is used in equation 5, please use a different symbol.
Eq. 12: Substituting equation 12 into the second equation of equation 11, the right-hand side becomes 0. This is strange, as it means that the alluvial layer thickness does not change over time.
L125: Do not write a statement regarding conflicts of interest in Figure 1.
Eq 20: If ξb is the transport length and ξ is the saltation length, then equation (2) should use ξb.
L176: Recently, Inoue et al. (2025, GRL) experimentally demonstrated that bedrock lateral erosion due to particle impacts increases with increasing lateral slope. Please refer to their work. Their experiments also showed that secondary flow suppresses lateral erosion. Please describe how the effects of secondary flow are considered in this model.
L262: In the appendix, please provide the basic equations for two-dimensional flow, the method for calculating shear stress, and the methods for calculating sediment transport in the x (streamwise) and y (lateral) directions. Also, indicate whether there is any double counting of the bed elevation change due to sediment transport in the y direction and the lateral erosion shown in Figure 2.
L296: Provide information regarding the grain size of the bed material. Also, describe how the sediment inflow from the upstream end was determined.
Fig. 4: Add legends to the two top figures of the color map. One has one, but the text is too small to read. For the bottom figure, indicate the dimension on the vertical axis.
Fig. 5: The legend in the color map is too small to read.
Fig. 11: Add legend to the color maps.