the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 15 Aug 2024
Hillslope diffusion and channel steepness in landscape evolution models
Abstract. The streampower fluvial erosion (SP) model is the basis for many analyses and simulations of landscape evolution. It assumes that the rate of river incision into bedrock depends only on flow intensity and rock erodibility, and is insensitive to sediment flux. In two dimensions, the SP model is often coupled with diffusion processes, which together describe the coupled evolution of channels and hillslopes (SPD models). While it is implicitly assumed that channels in the SPD models retain their detachment-limited character, this has not been extensively tested. Here we show that the deposition component of hillslope diffusion has a substantial effect on channel slope and relief in SPD models, and present a new method to predict the channel steepness index from model parameters. We contrast the results with those of a mixed bedrock-alluvial river model coupled with a hillslope diffusion model that both track sediment mass balance, and suggest that the combination of mass-conservative hillslope processes and non-mass-conservative fluvial erosion in SPD models leads to unrealistic scaling behavior. We demonstrate this by examining several field sites where an SPD model adequately describes the spacing of first-order valleys, and show that it is inadequate to predict channel steepness.
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RC1: 'Comment on egusphere-2024-2418', Anonymous Referee #1, 13 Sep 2024
This review is of “Hillslope diffusion and channel steepness in landscape evolution models” by Litwin et al. In general I think it is a clearly written and elegant analysis of idealized models that the authors connect to real landscape applications and point out the challenges and limitations in doing so.
My main suggestion for improvement is to cite more landscape evolution modeling papers early on that have included sediment in different ways. If a reader was not very familiar with landscape evolution modeling done in the last 20+ years, they would be led to believe (until getting to the discussion section at least) that nobody had ever thought to incorporate effects of sediment coming off of hillslopes and into channels on channel incision. Cite the SPACE paper early on, and other works that have used it, and maybe Whipple and Tucker (2002), and many various other papers that have explored mixed bedrock alluvial models. I also suggest changing the intro to better explain why this approach of using what is an idealized, reduced complexity model (SPM, or SPD) is useful—to me anyway, the benefit is the simplicity, that it can be calibrated to real landscapes to infer parameters (and the paper nicely shows that parameter values can be misleading), and to not use a more complex model than is needed for a given task, while acknowledging that more complex equations do already exist.
Line 54: add comma before S(x)
103-104: Explain why these particular values were used. Why this particular ratio of diffusion to advection? Think of it from the point of view of someone not super familiar with the equations, and try to explain a little more.
145: “effective uplift rate” is an interesting idea. Is it really correct, directly comparable? I don’t think so but am not sure. Do you think of it as a conceptual idea or an actual way to evaluate channel steepness? Maybe at steady state. So if that’s the case, shouldn’t it be possible to “correct” for any best-fit bias in steepness (and K) parameters just by somehow adjusting the uplift rate? If the authors want to push effective uplift rate as a thing, I suggest explaining it more in the discussion, and clarifying it a little more here as well.
197: Either in main text or in another appendix, I think you should give the Ganti et al. (2012) nonlinear diffusion equation. Its helpful to see equations.
204-207: This description of cover effects being important in channels seems very incomplete, because basically no other landscape evolution models with cover have been cited or described (yet). Covering them in a paragraph in the intro would address this, or maybe here.
231-234, 241-244: Again, this is the content I think should be mentioned earlier.
253-255: While I think the use of the SPACE model in this way is technically correct, I think it should be explained more here, because its an odd usage of it. I’m fairly familiar with the SPACE equations and it took me a while to think through why it worked the way the authors show. The sediment is run with a high settling velocity (parameter given in the appendix), and low sediment erodibility, so its conceptually representing coarse sediment that acts like bedrock, and that deposits at some step in how the SPACE model is solved. So it includes a cover effect from this sediment. But then when it is entrained it effectively disappears. I think I’m right to say that the model could have just as easily been written to be numerically solved (with a different ordering of steps within a timestep) so that this wouldn’t work: If the model first checked Ff, and if Ff=1, then all the sediment would immediately be treated as washload that could not deposit, and would be equivalent to having a settling velocity of zero. I think; I could be wrong. My suggestion would be to explain a little more that this is an unusual combination of parameter the authors are using to enable the model to do something it wasn’t exactly designed for, though probably works.
262, 265, 269: I know you say (if I understand correctly) that you have an analytical solution for the SPACE model steepness? Give the equations, either here or in the SPACE appendix. The reason is that you’ve already talked about analytical solutions for the SP model, and just calling them analytical solutions here is confusing. In other words its not clear to me that the authors are now using an analytical solution that includes the SPACE sediment. Or maybe they’re using a solution that includes hillslope diffusion instead? I note that Guryan et al. (2024) recently presented SPACE model steady state steepness analytical solutions; others may have as well.
274-275: This result has me confused. I understand with the SP vs SPD model why the diffusion of what is effectively bedrock deposition into the channel increases steepness even at steady state erosion. But in the SPACE model, sediment from over the whole pixel area (and all of the upstream pixels) should be accounted for regardless of whether the hillslopes are diffusing or assumed to lower at exactly the same rate as the channel. I realize the authors try to explain this, but I still don’t quite understand why it would be different. It makes me wonder if it’s a numerical method or assumption or rounding error in SPACE rather than a real effect. At steady state incision across a landscape, shouldn’t the amount of sediment coming into each model node in SPACE be the same regardless of hillslope diffusion? Any explanation would be helpful.
286-287: The authors could consider mentioning Guryan et al. (2024), who looked at how alluvial cover changes steepness and effective erodibilities by comparing the SPACE model to the SP model, in their case also considering layered rocks.
290-293: if bringing up tools effects, the authors could also mention other channel feedbacks such as width adjustments, which also mediate tools and cover effects.
333-334: This is just a stylistic suggestion, but I prefer present-tense, so “We show” rather than showed, even for the conclusions, since its pretty common for people to read the conclusions right after the abstract.
351: I think “limited applicability in the field” is too strong, since the level of complexity of a model depends on what you want to use it for. The same arguments could be applied to the SPD model. Just be a little more nuanced in the criticism.
Whipple, K. X., and G. E. Tucker, Implications of sediment-flux-dependent river incision models for landscape evolution, J. Geophys. Res., 107(B2), doi:10.1029/2000JB000044, 2002.
Guryan, G. J., Johnson, J. P. L., & Gasparini, N. M. (2024). Sediment cover modulates landscape erosion patterns and channel steepness in layered rocks: Insights from the SPACE model. Journal of Geophysical Research: Earth Surface, 129, e2023JF007509. https://doi.org/10.1029/2023JF007509
Citation: https://doi.org/10.5194/egusphere-2024-2418-RC1 -
RC2: 'Comment on egusphere-2024-2418', Charles Shobe, 21 Sep 2024
Review of Litwin et al.: Hillslope diffusion and channel steepness in landscape evolution models
The authors analyze a widely used landscape evolution model in an effort to better understand the previously identified steepening of channels due to delivery of hillslope-derived sediment. They show the relative importance of this effect across the model parameter space (Fig. 2), they come up with a way to predict the magnitude of the effect based on model parameters (appendix B), and then show how the effect diminishes when a model is used that incorporates some additional physical realism (the SPACE model in this case). Finally, they show three field examples illustrating the steepening effect and showing that it is not captured by the simple stream power/diffusion model.
This is a well-written and useful paper. Geomorphologists, and landscape evolution modelers in particular, have known anecdotally about the key effect the authors describe (diffusion steepening channels) for a long time, and I think it's fair to say that we pretty much already knew the cause (the hillslope-derived sediment “becomes” bedrock when it hits the channel; it is not hard to intuit why this is bad) in an informal sense. However, to my knowledge, the current paper is the first systematic investigation of the diffusion-induced-steepening phenomenon. For that reason, and for the elegant way in which the authors derive predictions for the effect as a function of model parameters, this paper will be useful to the community.
My suggested changes are fairly minor. Before getting into those, though, I want to follow up on the perceptive comments by the other reviewer about some of the SPACE model results. Their questions are good ones and, since I have spent a lot of time building SPACE and then trying to understand its behavior, maybe I can be helpful.
Reviewer 1’s points about SPACE:
Their comment on line 253-255: I understand what the reviewer is thinking here, but I think there may be confusion about what the F_f parameter means. F_f only applies to the entrainment of bedrock (in the SPACE Python code it is defined as the “fraction of permanently suspendable fines in bedrock”). So the idea that “when [sediment] is entrained it effectively disappears” when F_f = 1 isn’t true, regardless of when in the model solution entrainment is calculated. It would be accurate to say that “when bedrock is eroded by fluvial processes, the sediment produced from bedrock erosion effectively disappears” when F_f = 1, but that doesn’t affect the fluvial response to sediment diffusing in from the hillslopes. My understanding is that SPACE+LD in this configuration is acting basically the way the authors want—they are showing that SPACE can replicate the sediment-driven-steepening effect, and showing the effect with a model that explicitly simulates both bedrock and alluvium gives us some insight into the mechanisms behind the effect. I do think the authors could revise the text a bit to prevent confusion: for example in line 273, F_f = 0 is equated with conserving sediment mass. I see what the authors mean, but this is likely to confuse readers. Really F_f is about whether or not you conserve mass during the transition from fluvially eroded bedrock to sediment. Once rock has become sediment, SPACE conserves all sediment mass. So I can see where the confusion arose. My advice is just to lay out more clearly what F_f actually is/does before you start diving into the results of the SPACE work: sed mass is always conserved, but we can simulate detachment-limited fluvial erosion as in Fig. 7A-D by saying that the bedrock becomes wash load when eroded. That’s where F_f comes in.
Their comment on line 274-275: Reviewer 1 has a point here; I also find this behavior surprising. The key question I’d like to see this discussion address is: Why does adding diffusion increase the equilibrium sediment thickness in the channel? If Qs=UA is by definition satisfied at steady state, it feels like the SPACE analytical solution should already be accounting for the cover effect of all sediment passing through a given point on the channel network. My wild guess would be that this is not the result of an approximation within SPACE itself but a result of the coupling with LD. I’m sure the authors have put a lot of thought into this and I am only speculating; I would just ask, as reviewer 1 did, that they lay out the explanation in a few more steps so it’s clear to readers.
General points on the manuscript:
My main suggestions, which should not be terribly hard to address, are that 1) the paper could be better situated within the existing literature both old and newer, 2) that the text could be expanded (by pulling appendices A and B into the main text) and possibly reorganized to better lead readers through key derivations and results, and 3) that there should be a little more focus earlier in the paper on the physical aspects of the problem rather than just the mathematical aspects.
- This criticism could probably be leveled at any geomorphology paper published in the last 30 years: the current draft understates the extent to which a key concept was already recognized by Alan Howard. True, the authors do cite the 1994 paper in which Howard shows (his Fig. 5A), using basically the SPLD equation, the importance of the diffusion term in setting channel slope. But the current draft repeatedly presents identifying the existence of this effect as one of its central results, which I think also does the current paper a disservice because it does so much more! To me, the key value the authors are adding is in the following sentence about their impressive ability to predict the magnitude of this effect from model parameters, and in their careful analysis of why/when/how this occurs. Along these lines, some changes to the writing would help clarify what’s known and what’s new:
- change the wording in lines 38-41 to focus your justification for the paper: it’s not just that this effect hasn’t received much attention. It’s that we know there’s this issue, but we don’t have a way to quantitatively predict its magnitude. You’re solving that, which is great! Along with that, line 42 could become much more specific. You are doing some really cool stuff here, but your contributions are not really about recognizing that this effect exists so much as they are about quantifying what controls it.
- in the abstract, I think it would be fairer to say that we already knew about the diffusion effect on slope and that you are making the contribution of figuring out when and how much it affects modeled outcomes.
- In 3.1 (line ~102), make sure to cite Howard again and point out that your results are a (very useful and thorough) generalization of the point.
- I agree with reviewer 1 that the paper could be better situated in the literature from the beginning. Late in the paper the authors point out that there was a long history of modeling that did treat sediment separately from bedrock, but to some extent that was put on the back burner in favor of simpler, more efficient approaches in recent years. I think it would be better to move that point up to section 2. I won’t prescribe what should get cited, but there’s the series of models out of the Beaumont group, there’s CHILD, and then this would also be a place to initially mention SPACE, which is to some extent a descendant of those approaches, so readers know where you’re headed later in the paper.
- Ideally this will be a paper with implications beyond just a single family of numerical models. To that end, another concept that I think should show up earlier in the paper is the mechanism(s) for the diffusion steepening effect. As the authors note late in the paper, this basically occurs because 1) sediment and bedrock aren’t distinguished and 2) sediment mass is conserved on hillslopes but not in channels in SPLD world. This is nicely shown in the paper, but bringing it up earlier and using it to frame the study will help the discussion stay grounded in the physics of Earth’s surface, rather than just a quirk of a set of equations.
- Appendices A and B are both short and so is the paper overall. I won’t insist on it, but I recommend bringing both appendices into the main text where the appendices are currently referenced so that equations and theoretical constructs don’t come out of nowhere for readers not intimately familiar with prior papers in this literature.
Line comments:
- 5-6: As noted in major comment above: try to be a little fairer about what was already known (this effect exists) and what is new (great quantitative constraints on how/why)
- 51/52/throughout: I prefer “SP model” over “SP law,” but just decide what you’re comfortable with and keep consistency.
- 70: a bit pedantic maybe, but there need not be a connection between adding diffusion and expanding to 2D. You can do one, the other, or both.
- Sec 2.3: Before diving into the tech stuff (Landlab, domain size, BCs, etc) it would be good to have a more general 1-2 sentences just saying what types of simulations you’re going to run, and for what purpose. This would be a nice bridge between the theory background and the numerical implementation.
- Sec 3.2: see overall comment about pulling text out of appendices A and B.
- 201-206: I agree with everything written here, but it might make more sense to first discuss the reasons for the steepening effect within the model itself (dubious assumptions about single materials and mass conservation across process domains), and then dive into how that does or doesn’t square with natural systems. This would involve just reorganizing the order of the discussion.
- 285: In line with the citation to Ott here, in Shobe et al., 2018 (JGR-ES; Fig. 10) we suggested from a modeling exercise that uplift-rate-dependent incision thresholds caused by delivery of hillslope-derived boulders to rivers made channel steepness less sensitive to erosion rate.
- 295-96: Recommend rewriting this sentence because elsewhere in the paper you are making the very valid argument that the SPLD behavior is basically due to bad/incomplete/simple/whatever assumptions. But here the tone sort of implies that you’re talking about a physically real set of relationships. To me, and I think to you as well, this sentence should be more about the fact that we’re getting the right answer (hillslope processes affect channel steepness) for the wrong reason (not doing sed/bedrock and mass conservation in our models).
- Figure 8: consider changing linestyle as well as color between ksn and ksnpred; I printed this draft in black and white to review and couldn’t tell them apart. Different linestyles would fix that.
- 343: indicate that this first result is basically a more systematic confirmation of past work, which allows you to then put the emphasis on the next couple of lines which report really important results from your study.
Citation: https://doi.org/10.5194/egusphere-2024-2418-RC2 - This criticism could probably be leveled at any geomorphology paper published in the last 30 years: the current draft understates the extent to which a key concept was already recognized by Alan Howard. True, the authors do cite the 1994 paper in which Howard shows (his Fig. 5A), using basically the SPLD equation, the importance of the diffusion term in setting channel slope. But the current draft repeatedly presents identifying the existence of this effect as one of its central results, which I think also does the current paper a disservice because it does so much more! To me, the key value the authors are adding is in the following sentence about their impressive ability to predict the magnitude of this effect from model parameters, and in their careful analysis of why/when/how this occurs. Along these lines, some changes to the writing would help clarify what’s known and what’s new:
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