Self-organization of channels and hillslopes in models of fluvial landform evolution

Hergarten, Stefan; Pietrek, Alexa

doi:https://doi.org/10.5194/egusphere-2022-605

Preprints

https://doi.org/10.5194/egusphere-2022-605

Preprints

25 Jul 2022

| 25 Jul 2022

Self-organization of channels and hillslopes in models of fluvial landform evolution

Stefan Hergarten and Alexa Pietrek

Abstract. Including hillslope processes in models of fluvial landform evolution is still challenging. Since applying the respective models for fluvial and hillslope processes to the entire domain causes scaling problems and makes the results dependent on the spatial resolution, the domain is explicitly subdivided into channels and hillslopes in some models. The transition from hillslopes to channels is typically attributed to a given threshold catchment size as a proxy for a minimum required discharge. Here we propose a complementary approach for delineating channels based on the discrete representation of the topography. We assume that sites with only one lower neighbor are channelized. In combination with a suitable model for hillslope processes, this concept initiates a self-organization of channels and hillslopes. A numerical analysis with a simple model for hillslope dynamics reveals no scaling issues, so that the results appear to be independent of the spatial resolution. The approach predicts a break in slope in the sense that all channels are distinctly less steep than hillslopes. On a regular lattice, the simple D8 flow routing scheme (steepest descent among the 8 nearest and diagonal neighbors) harmonizes well with the concept proposed here. The D8 scheme works well even when applied to the hillslopes. This property simplifies the numerical implementation and increases its efficiency.

Received: 06 Jul 2022 – Discussion started: 25 Jul 2022

Stefan Hergarten and Alexa Pietrek

Status: final response (author comments only)

RC1:
'Comment on egusphere-2022-605', Alan Howard, 18 Aug 2022

The detailed exposition of the modeling scheme and the results is generally clear. Most comments concern basic questions about the model implantation and definition of channels.

Line 32: If the comment about lack of sediment transport refers to the Howard (1994) model, this attribution is incorrect because sediment transport is considered as are alluvial channels. Even at locations where the channel is bedrock, the sediment is routed downstream and influences the gradient of any alluvial channel segment downstream.

Lines 37, 202: The description of simulations such as Fig. 2 as having “canyon”-like morphology seems inappropriate. It seems that “canyon-like” seems to be conflated with the appearance of a strongly elaborated drainage network. The common usage of “canyon” implies a valley sharply incised into a relatively smooth upland – often implying cliff-like slopes bordering the valley. I suggest using a different term and defining its meaning.

The issue of defining the channel network and, correspondingly, drainage density is not adequately discussed. In natural drainage networks one way of defining drainage density is the presence of actual channels with well-defined banks. This approach is, of course, not useful for landform evolution modeling at basin scale, at least at the level of process generalization in current LEMs. It also suffers in a more general sense that the drainage network so defined is time-dependent, because the channel network can expand and contract with flood events, land use change, and short-term climate changes that do not strongly affect drainage basin morphology as a whole. As the authors indicate, defining the channel system by a critical drainage area, A_c, is arbitrary. There are two ways to define A_c, and each has limitations. The first is straightforward imposing definition of the channel network to initiate at the critical area no matter what the modeled or actual landform processes are or what the landform morphology reveals. The second is to impose a process threshold in LEMs at a critical contributing area. That seems to be what the authors imply based upon discussion and simulations in section 7. There is some logic to this if the threshold is clearly defined as occurring at a critical process threshold, such as a critical fluvial shear stress for channel incision or a critical threshold for hillslope stability. Several studies have explored stochastic forcing of a critical fluvial shear stress (as a result of storm intensity) and its effect on drainage basin morphology. Both uses of A_c are relevant only to simulation modeling and not for determining drainage density in natural networks. The authors implement a more general scheme based upon the slope relationships between adjacent cells which is applicable to LEMs but probably error-prone for analyzing natural drainage basins. This seems to be a reasonable, but not unique approach for analysis of simulation models, although as discussed, it strongly dependent on the flow routing scheme and the use of square simulation cells. A possibly more general and less “noisy” method is to define a critical topographic concavity to define channel heads. Howard (1994), for example, used the gradient divergence divided by the basin-wide average gradient.

Many LEMs utilize both fluvial and slope processes within each cell, with the emergent landscape depending upon the relevant process balance in each cell. The authors criticize this approach without providing specific justification. In actual simulations there is generally a very abrupt downgradient break between cells in which slope processes dominate and fluvial processes are unimportant and the inverse. In fact, in natural landscapes both fluvial processes and mass wasting processes occur on slopes, and individual locations can temporally transition between being dominated by either process, justifying this approach. The relative dominance of fluvial versus mass wasting process determines drainage density (Howard, 1997, EPSL, 22, 211-227; Tucker & Bras, 1998, WRR, 34, 2751-2764) and inferentially channel network definition.

The introduction should be more explicit about the implied issue in LEMs about cell-size dependency of LEM simulations. If processes are scaled correctly, there should not be cell size dependence unless the cell size is too small to adequately represent slope processes and morphology.

Lines 61-96: The exposition here is clear. The use of combined sediment transport and bedrock erosion is fine, although in many natural channel systems transition from bedrock reaches to alluvial reaches is abrupt.

Line 105: The choice of only one downstream cell to define channels is fine for steady state incision, but what happens when channels aggrade and there are multiple downstream potential flow paths (e.g., alluvial fans)? I see that this is addressed as a limitation in Lines 108-110.

Section 2 general comment: The erosion model in this part of the paper only considers fluvial processes. In the absence of slope processes is not the entire network channelized by definition?

The use of A_h to emulate slope processes in what is essentially a fluvial-only model seems arbitrary. The justification seems to be primarily to allow computationally-efficient modeling by eliminating the complexities of explicit modeling of mass wasting. This lessens any general implications of scale independence beyond their specific LEM.

In general the paper is clearly written and illustrated. The conclusions of the modeling seem limited to a specific implementation of their LEM in terms of process modeling, channel network definition, flow routing and grid characteristics.

Citation: https://doi.org/10.5194/egusphere-2022-605-RC1
- AC1: 'Reply on RC1', Stefan Hergarten, 02 Sep 2022
  
  Dear Alan Howard,
  
  thank you very much for your review! Please allow us some short responses before addressing all aspects in more detail in a revised manuscript.
  
  Detachment-limited erosion:
  
  Of course, the reference to your 1994 paper does not refer to the lack of sediment transport. However, it seems that this paper was the first to specify the transport-limited end-member of fluvial erosion explicitly, while the concept was used implicitly in earlier work, e.g., on optimal channel networks. Anyway, if you can provide an earlier reference to this term, we will be happy.
  
  Scaling problems:
  
  We are a bit wary concerning your statement "If processes are scaled correctly, there should not be cell size dependence". If it refers to your 1994 paper, we have to keep in mind that it was hardly possible to check such a hypothesis numerically at that time. Reconciling the approaches mentioned in our manuscript (including your suggested scaling of the detachment-limited term with channel width) with Hack's relation for the concavity of rivers is difficult. If sediment transport is taken into account, it is even worse since two parallel channels with half the discharge have not the same total transport capacity as a single channel. So considering both fluvial erosion and hillslope processes in each cell and assuming one channel per cell is clearly not free of scaling issues, so that we disagree to your statement.
  
  Abrupt transition from bedrock reaches to alluvial reaches:
  
  OpenLEM can also switch to a fully transport-limited model as soon as sediment aggradation takes place. In its spirit, the concept is similar to your 1994 paper, but simpler. In the shared stream-power formalism, it is just setting the parameter Kd to infinity. This topic was recently addressed in an own paper (Hergarten 2022, doi 10.5194/esurf-10-672-2022), but it does not contribute much to the topic of the recent manuscript.
  
  Best regards,
  
  Stefan Hergarten and Alexa Pietrek
  
  Citation: https://doi.org/10.5194/egusphere-2022-605-AC1
RC2:
'Comment on egusphere-2022-605', Anonymous Referee #2, 23 Aug 2022

In this contribution, the authors take on the issue of grid scale dependence in coupled channel-hillslope landscape evolution models. Using a recently published mathematical formulation along with a new definition of what constitutes a channel, they construct an LEM that seems to circumvent some of the potential issues with past LEM implementations: chiefly the issues of 1) needing to define arbitrarily channels versus hillslopes (some previous approaches do this, though many do not) and 2) grid scale dependence in models that couple river and hillslope evolution.

I find the manuscript to be useful to the community given that we are always looking for modeling approaches that allow us to circumvent known issues with our current techniques. I think the authors have been honest about where the utility of their advance begins and ends (i.e., maybe not useful for running simulations on real DEMs), and that they do a nice job of exploring the behavior of the approach they propose. My chief concern about the manuscript in its current form is that it is not well enough integrated into the LEM literature. Readers are not shown clearly and specifically the shortcomings of other approaches, and it is therefore hard to be sure as a reader exactly in what situations this new approach is a major advance. I would like to see this paper published in ESurf after some modifications—to the writing rather than the science—that 1) clarify the position of the current work in relation to the large body of LEM literature, and 2) clarify the utility of the current approach given its intricate relationship to the D8 grid and its current lack of applicability to non-model-generated grids. To be clear, I think this is a useful contribution, but I think the authors could increase their impact by expanding on these points.

For example, the paragraph beginning on line 34 states what I believe is a well-known limitation of Eq. 2, for example Kwang and Parker 2017 talk about this. I would like to see a citation to either that work, other relevant work, or a combination of the two that demonstrates to readers that this is a known issue. Similarly, in the following paragraph (line 40) there is no citation to papers describing the problem of grid scale dependence. Readers are left to wonder whether this is a problem inherent to the SPIM/diffusion model in every case, or whether a model could in theory be scaled correctly in order to avoid it. Following this discussion, in line 44, itwould be good to be more specific than to say that a given approach is “not free of problems.” What are the problems? Are they problems that the new method being introduced will solve, presumably?

Another example is in the paragraph starting in line 45. We miss references to the work of Garry Willgoose (e.g., 1991 papers), who if I am not mistaken was one of the earlier workers to explicitly separate channel and hillslope process representations dynamically. I like the idea behind this new paper (that we can define channels as nodes with one downslope neighbor), but I think the impact will be greater if the authors are more thorough about placing their work in the context of past work. I congratulate the authors on an interesting paper.

Please find line-by-line comments below.

31: I am not sure that the Howard 1994 citation is well-captured by this statement. It might be better to rephrase or to choose a reference like maybe Braun and Willett 2013 where sediment truly is not considered.

33: It would be worth pointing the reader to some of the foundational papers on modern sediment-flux-dependent river incision models, e.g. Sklar and Dietrich, 1998; Whipple and Tucker, 2002; Gasparini et al., 2007; Turowski et al., 2007, as well as the long history of models (some of which are cited later in this paper like Davy and Lague 2009) that have computed the sediment mass balance in addition to calculating river incision. We don’t want readers to get the impression that we don’t have options beyond detachment-limited treatments.

37: The use of “canyon-like” is unclear here. Do you mean landscapes in which channels become very steep at low drainage area (e.g., Kwang and Parker, 2017)? I think a new phrase is needed for clarity, or the phrase could simply be deleted and the sentence would stand as-is.

41-42: I am not sure all readers will understand intuitively why this happens. Another sentence or two describing why adding hillslope processes causes a spatial resolution dependence would be useful for setting up the problem.

44: Given that the problems associated with coupled channel-hillslope models make a major motivation for this work, I ask the authors to please summarize the problems discussed in Hergarten et al. (2020a) that they reference here.

50: Again here, given that this paper is a separate contribution, it would be good to restate for readers what actually is the approach proposed by Hergarten et al (2020a).

55: Is it possible to be more descriptive/clear than “weird?” I know that in some cases the issue with this approach is that slope-area data no longer reveal a smooth hillslope-channel transition that is observed in many real landscapes, for example. Are there other specific issues that could be brought up here? Are there citations that could be added that illustrate these issues?

200: It is not clear why these small-scale persistent changes in the topography occur—could a sentence be added to explain more clearly?

202: Here it sounds like “canyon-like” just means “steep hillslopes.” I recommend re-phrasing for clarity.

209: This last sentence could use a little more detail to be clearer.

214-215: This feels like a stupid question, but: Why does the erosion rate decrease? Is this only for the case of the chosen fluvial versus hillslope parameter values, or is this universal?

218: Similar question for erosion rate increase. I am having trouble understanding, and I fear readers will too, what dynamics are occurring here. A few more details would help.

Section 5: I find this section very interesting. Do the authors expect the same result when m/n != 0.5? There are some applications in which 0.5 is a bit of a special value (Kwang and Parker, 2017) so it might be worth checking another ratio.

272: Like many of the literature references throughout, this one is quite vague. Could the authors add an extra sentence clarifying what salient points of that paper are relevant? I for example am aware of Hergarten 2021 but have not read it in any detail, so am a bit lost here.

328: Again it would be good to see multiple references here to demonstrate the extent to which this practice is “established.” Certainly this is an assumption in much topographic analysis of real DEMs, but in my understanding of the literature it has not (at least recently) been a favored approach for LEMs. If I am wrong, then that’s ok and the addition of several citations will settle the question.

Section 7 in general: I find this discussion somewhat difficult to follow, largely because I do not fully understand why the landscape is given to reorganization even under a largely steady state condition. Some more detailed explanation of the processes causing that behavior would clarify this section.

Conclusions: These in general represent the content of the paper well, but I would also like to see a brief addition (this could also be before the conclusions if the authors prefer) making the case for why and how future workers should take advantage of the advances provided by this paper given the limitations (which are already well-stated by the authors). What can we do with this new knowledge?

Thank you for the chance to review this interesting work!

Citation: https://doi.org/10.5194/egusphere-2022-605-RC2
- AC2: 'Reply on RC2', Stefan Hergarten, 02 Sep 2022
  
  Dear Reviewer,
  
  thank you very much for your encouraging comments! We will go deeper into detail when preparing a revised version quite soon.
  
  Since this paper is somewhat technical, we shortened the description of the respective modeling approaches compared to our earlier papers. However, since you feel that we should go more in detail at some points, we will think about this again.
  
  Basically the same holds for m/n != 0.5. Originally, we did not want to emphasize the specific model (m = 0 at hillslopes) too much and therefore did not go into detail concerning the ratio m/n. From some preliminary tests, we saw that m/n != 0.5 did not yield fundamentally different results. However, the analysis shown in Fig. 8 becomes a bit more complicated since horizontal and vertical scales are no longer independent then. Accordingly, the y-axis (catchment relief) must be rescaled. We will revisit this topic and think whether it is interesting enough to include it explicitly or just add a short statement.
  
  Best regards,
  
  Stefan Hergarten and Alexa Pietrek
  
  Citation: https://doi.org/10.5194/egusphere-2022-605-AC2
RC3:
'Comment on egusphere-2022-605', Anonymous Referee #3, 24 Aug 2022

This is an interesting manuscript which introduces a new idea of implementing landscape evoution simulations. I have found that many of my technical concerns are already commented by other referees.. and so try to add comments which were not mentioned yet. My major concern is that the focus of the manuscript is somewhat distracting. I understand the value of new modeling framework, but I am uncertain how this can lead to any new findings or scientific advances in self-organization processes. In particular, the OCN contents in section 3 are not well harmonized with the rest of the manuscript. I suggest in the revision that authors decide the focus of this manuscript sharply, and restructure the writing. Minor comments follow.

L25: If authors search for more literature, there is a much wider range of concavity index found in nature.

L35-39: This part needs to be rewritten in a much comprehensive manner.

L40: the linear diffusion equation would need a citation

Eq(3): This is a key governing equation in this study, and it requrires much stronger justification. It also requires relevant literature.

Figure 1: I was very confused when I first looked at the figure. I guess what authors mean on the x-axis is the 'channel forming area', not 'catchment area'?

Citation: https://doi.org/10.5194/egusphere-2022-605-RC3
- AC3: 'Reply on RC3', Stefan Hergarten, 02 Sep 2022
  
  Dear Reviewer,
  
  thank you very much for your encouraging comments! We will go deeper into detail when preparing a revised version quite soon.
  
  Before submitting the manuscript, we also thought about the explanation of the optimal channel network concept, which you presumably did not find very useful. However, we still believe that the entire approach would come to much "out of the blue" without the background/motivation from the optimal channel network concept. So we would prefer not to change the overall structure.
  
  Best regards,
  
  Stefan Hergarten and Alexa Pietrek
  
  Citation: https://doi.org/10.5194/egusphere-2022-605-AC3
RC4:
'Comment on egusphere-2022-605', Anonymous Referee #4, 06 Sep 2022

This paper proposes an alternative solution to identify the channel – hillslope domain in dynamic landscape evolution models. The topic is of interest to the community. In general, the authors can give somewhat more depth to this story by pointing out issues they generally declare and by supporting their statements with literature and examples. Also, the results and findings would benefit from a clearer description at several points.

Title: I do not find the title to be adequate. This paper is not about self-organization of channels or hillslopes but rather presents a new LEM, that is essentially a full-scale fluvial model where hillslopes are represented as a drainage area independent process. There is no backup of any of the findings by field observations and the authors declare themselves that more research is needed to underpin this work and potential consequences. Hence, I would suggest a more technical title like: “A new approach to delineating channels in Landscape Evolution Models.”

Line 40. What do you mean with ‘a scaling problem’? Please specify. Model components like SPACE (Shobe et al., 2017) have been used in combination with diffusion (Shobe et al., 2017). In theory, all processes should act everywhere on a landscape. Why would diffusion as a process not act over channels and vice versa for fluvial incision? Naturally, at small discharges (drainage area) diffusion would be dominant over fluvial processes. I have been asking myself this question at several points throughout the manuscript and find it critical to address this point. Referring to other work does not suffice since this assumption is at the heart of this story.

Line 63 Add SPACE (Shobe et al., 2017)

Line 70 explain Kd and Kt

Line 125 here diffusion is applied to the entire domain. Just curious how the afore mentioned scaling issues are altering the results here. Aha, it is mentioned in the next sentence I see. Still wondering what those scaling issues are. Also, is the D value dimensionless? How does this compare to actual diffusion values ? (e.g. m2/yr see e.g. (Godard & Tucker, 2021))

Line 150: energetically favorable means less energy, right? Maybe specify to help the readers a bit here.

Line 155. “In turn, we need a model for hillslopes that does not favor dendritic networks energetically” Not sure I understand why not, please explain better.

Line 171: This might be true for the shared stream power model, but in the Carretier solution, a threshold slope is still used to calculate transport lengths. Please specify what you mean exactly.

Line 209. The river is shorter, where? Explain better.

Line 210: belongs

Line 211: 5000. How do we see that on the figure? Catchment A only goes up to 400 (dimensionless?)

Figure 3: Explain in the subscript what Ah is. Makes the figure readable on itself.

Figure 3-6: Are all these findings for non-dimensional values/axes? Please specify.

Line 236: “Owing to the dominance of parallel flow patterns at hillslopes: That is interesting. So, at A<Ah, flow patterns do not organize in ‘energetically favorable’ patterns? Would be good to elaborate a bit on this.

244: Again, it has never been explained clearly what the ‘scaling issues’ and ‘such problems’ are. This is critical to support the value of this work. It does not suffice to point to previous work.

Line 270: Would the authors expect differently when m/n is not 0.5?

Line 272: I find these kinds of sentences of very little added value. I have no clue what is meant here unless I go read this paper. Either explain what is meant or drop the sentence.

Line 278: This paragraph needs some more context to be of added value for the paper. Is the focus on slope breaks, or rather on the orientation of streams? I was expecting to read how this model adjusts the SA plot one expects to see based on observations where a transition from a hillslope domain into a debris-flow dominated into a alluvial channel domain occurs (Montgomery & FoufoulaâGeorgiou, 1993). Please elaborate on that. Do we not see any hillslope domain because the model is actually a fluvial incision model where hillslope erosion does not depend on A? Curious to know.

Paragraph 8. As the authors seem to suggest this conceptual model seems to be disconnected from reality. Hence, it should be made clear what exactly the added value of this approach is. Why would one favor this method rather than just assuming continues processes of diffusion and incision (the latter maybe with an incision threshold)? If I would be to use a LEM; I am not convinced I would consider this approach in the way it is described now. Please summarize the benefits of this versus other approaches (other than the Ac method). It would also be good to connect this work to field observations. Yes, it does not work well in its current state, but are there ways to improve this? On a similar note: the authors show different simulations with various values of A_h. Are those values chosen arbitrarily? Can they be set using data or by using DEM-derived topographic metrics?

Line 384. ‘Serious problems’. That sounds a bit suspicious. Explain what the problems are and why they are assumed to be not seriously affecting model behavior.

Line 389: What does it mean, works quite well?

Refs:

Godard, V., & Tucker, G. E. (2021). Influence of Climate-Forcing Frequency on Hillslope Response. Geophysical Research Letters, 48(18), 1–11. https://doi.org/10.1029/2021GL094305

Montgomery, D. R., & FoufoulaâGeorgiou, E. (1993). Channel network source representation using digital elevation models. Water Resources Research, 29(12), 3925–3934. https://doi.org/10.1029/93WR02463

Shobe, C. M., Tucker, G. E., & Barnhart, K. R. (2017). The SPACE 1.0 model: a Landlab component for 2-D calculation of sediment transport, bedrock erosion, and landscape evolution. Geoscientific Model Development, 10(12), 4577–4604. https://doi.org/10.5194/gmd-10-4577-2017

Citation: https://doi.org/10.5194/egusphere-2022-605-RC4
- AC4: 'Reply on RC4', Stefan Hergarten, 22 Sep 2022
  
  Dear Reviewer,
  
  thank you very much for your review! Please allow us some short responses before addressing all aspects in more detail in a revised manuscript.
  
  Scaling problems:
  
  It seems that a considerable part of the landform evolution modeling community is not aware of the dependence of the results on the spatial resolution that occurs when applying fluvial erosion and diffusion to all sites. Owing to the numerical limitations of the widely used explicit time-step scheme, spatial resolutions are typically low, so that diffusion with realistic diffusivities (order of magnitude of 10 m^2/kyr) practically only affects sites at drainage divides. In this case, the dependence on grid spacing is not recognized (and not a big problem). From this point of view, it might even look as if we were solving a non-existing problem. We will try to explain a bit more in detail why this is not the case.
  
  Title:
  
  About the title, however, we disagree. The paper is, of course, not about self-organization of channels OR hillslopes, but about the self organization of channels AND hillslopes. The landform evolution model, which is the key point for you, is just a specific application of this concept. Perhaps we can add this application to the title in the way "... and its application to delineating channels", but the scientific result is our focus.
  
  Nondimensional properties:
  
  There are no dimensional properties anywhere in the manuscript. We guess that you were confused by the colorbar in Fig. 5. This colorbar is optimized for recognizing the transition from hillslopes to channels (0 <= A <= 400), while the size of the entire catchment is much larger (A = 5000). So the largest channels are in a range where the color scale is already saturated (dark blue).
  
  Best regards,
  
  Stefan Hergarten and Alexa Pietrek
  
  Citation: https://doi.org/10.5194/egusphere-2022-605-AC4

Stefan Hergarten and Alexa Pietrek

Model code and software

Self-organization of channels and hillslopes Stefan Hergarten https://doi.org/10.5281/zenodo.6794117

Stefan Hergarten and Alexa Pietrek

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Short summary

The transition from hillslopes to channelized flow is typically attributed to a threshold catchment size in landform evolution models. Here we propose an alternative concept directly based on topography. Using this concept, channels and hillslopes self-organize, whereby the catchment size of the channel heads varies over some range. Our numerical results suggest that this concept works better than the established idea of a strict threshold catchment size.


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