the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 02 Apr 2024
Channel concavity controls plan-form complexity of branching drainage networks
Abstract. The plan-form geometry of branching drainage networks controls the topography of landscapes as well as their geomorphic, hydrologic, and ecologic functionality. The complexity of networks' geometry shows significant variability, from simple, straight channels that flow along the regional topographic gradient to intricate, tortuous flow patterns. This variability in complexity presents an enigma, as models show that it emerges independently of any heterogeneity in the environmental conditions. We propose to quantify networks' complexity based on the distribution of lengthwise asymmetry between paired flow pathways that diverge from a divide and rejoin at a junction. Using the lengthwise asymmetry definition, we show that the channel concavity index, describing downstream changes in channel slope, has a primary control on the plan-form complexity of natural drainage networks. An analytic model based on geomorphic scaling relations and optimal channel network simulations employing an energy minimization principle reveal that landscapes with low concavity channels attain stable plan-form configuration only through simple geometry. In contrast, landscapes with high-concavity channels achieve plan-form stability with various degrees of network complexity, including extremely complex geometries. Landscape evolution simulations demonstrate that the concavity index and its effect on the multiplicity of available geometries control the tendency of networks to preserve the legacy of former environmental conditions. Consistent with previous findings showing that channel concavity correlates with climate aridity, we find a significant empirical correlation between aridity and network complexity, suggesting a climatic signature embedded in the large-scale plan-form geometry of landscapes.
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RC1: 'Comment on egusphere-2024-808', Fergus McNab, 10 May 2024
This manuscript explores the origins of complexity in river networks. The authors make a persuasive case for the importance of river network complexity, which influences patterns surface runoff, erosion, sediment transport, relief generation, and the distribution of ecological niches. By analysis of natural landscapes, landscape evolution simulations, and the construction of energy-minimising networks, the authors show convincingly that network complexity is closely associated with the concavity index of associated river long profiles. They interpret these patterns in terms of the global energy associated with a water flow through a given network planform. When concavity is low, slope changes slowly with increasing drainage area, so that parts of the network with high drainage areas (hence high discharges) are still relatively steep. This combination of high steepness and high discharge results in high energy expenditure, so that the network will evolve to minimise drainage areas, and hence towards a simple planform. When concacivity is high, slope decreases more quickly with increasing drainage area, reducing the energy cost associated with higher drainage areas, and permitting more complex networks. With reference to further illustrative landscape evolution simulations, the authors discuss some interesting implications of this result, including the response of network planforms to changing climate and planform ‘memory’ of lithological structures that have long eroded away.
Overall I found the manuscript to be well argued, well written and well illustrated. The results and discussion are convincing, novel, of interest to the general geomorphological community, and important for those specifcially engaged in landscape evolution modelling. As such, I think the manuscript is certainly appropriate for publication in ESurf and could probably be accepted as it is. I do have some minor suggestions though which I would ask the authors to consider. These comments mostly concern: (a) descriptions of the methods used and (b) queries concerning some of the results and some suggestions for further discussion. I will give more details below, along with some technical corrections I found.
Lastly, I stress again that I consider these issues minor and that my comments are intended only to improve what I think is already a high quality manuscript. I congratulate the authors on what I believe will make an excellent contribution to ESurf!
Best wishes,
Fergus McNab
GFZ PotsdamSpecific comments
Methods
I felt several of the method descriptions would benefit from an additional sentence or two giving the reader an idea of the concepts behind the chosen methods. In most cases the authors point to the original literature, but I think providing a small amount of additional explanation will help the reader follow what is being done and why, and also make clearer what the underlying assumptions are.
For example, on lines 131-139, the authors describe how they measure the concavity index for the natural landscapes. They state that they use a ‘disorder scheme’, which ‘asses the extent to which the elevation-based order of channel pixels aligns with their order by their χ value’. I think the basic idea here is that the points in the landscape with the same χ value should be at the same elevation – but I think it could be stated more clearly for the benefit of readers less familiar with such analysis. I think this approach also assumes uniform uplift across the domain, so that ‘signals’ are introduced to the long profiles only at base level – this assumption is maybe reasonable given the authors’ choices of study areas, but could be stated explicitly (along with any other key assumptions behind the method).
Similarly, on lines 193-207, the authors describe the method for computing optimal channel networks. They introduce their ‘greedy’ algorithm, and contrast it to an ‘annealing’ algorithm that involves a ‘temperature dependent probability’. These terms do not mean much to me, and suspect not to the general reader either. So, if the distinction is important, I think it would be useful to explain the it in simpler terms here, if possible. I think that would be particularly important if the greedy algorithm the authors employ is new, which is ambiguous in the current description.
Finally, a more minor example, on lines 147-150, the authors describe an elevation correction, but it is not explained why this correction is needed. Indeed, maybe I missed something, but I did not notice anywhere in the later analysis that elevation is used, so it is still unclear to me why this correction is applied.
Queries / discussion points
Below I list some queries that came up for me while reading the manuscript. I do not think any of these points represent significant issues that necessarily need to be addressed in a revised manuscript. But they could warrant comment or further discussion, if of interest to the authors.
Throughout the manuscript I wondered about the potential importance of drainage density/the relative importance of hillslope vs. fluvial processes for the network complexity as defined here (ΔL). Intuitively I would have thought that a landscape with fewer channels would be more likely have a simple network structure than one with a dense channel network. In landscape evolution models, the density depends on model parameters like erodibility, concavity (i.e. m & n), and hillslope diffusivity (see e.g. Perron et al., 2008, JGR). (A related question: do the DAC models shown here have a hillslope component? If so, it is not mentioned in the method description.) Theodoratos et al. (2018, ESurf) define a lengthscale, lc (their equation A8), which depends on m and describes the relative importance of hillslope vs. fluvial processes: if lc is small, fluvial processes dominate and a high channel density will develop; if lc is large, hillslope process dominate and a more diffuse landscape develops. So, I wonder if an alernative explanation for the dependence of complexity on θ would be that we are looking at landscapes with different relative contributions from hillslope and fluvial processes? Since there are no hillslopes in the energy-minimising network simulations, I think the authors’ interpretation of the results in terms of energy minimisation are probably sound. But perhaps this would be a useful/interesting point to test/discuss. For example, an additional set of experiments could vary m but also an additional parameter (erodibility or hillslope diffusivity) to keep lc constant. Alternatively, hillslope diffusivity could be varied while m is fixed to test if that has any influence on ΔL.
During the discussion of Hack’s law (Section 5.1), the authors show that the observed relationship between θ and ΔL can be explained by some variation in the Hack’s law parameters k and h. It is interesting that the data seem to fall roughly into a specific variation band (maybe ki – kj = c. 0.2, hi – hj = c. 0.1, the specific band varying a bit with theta). Does this result point to some characteristic local variability in Hack’s law parameters in natural networks, which could potentially be measured? And could it then form a future test of the authors’ interpretations? If so, it might be worth commenting on. The authors quote some references on variability in Hack’s law parameters but it is not clear whether these numbers apply on a global or local scale (the latter being the relevant one here).
Lastly, there are two definitions of a ‘stable’ network used in the manuscript – one says that χ should be equal across drainage divides, the other that the network is ‘optimal’ from an energy point of view. These two definitions seem to point in the same direction in terms of the relationship between concavity and complexity. But are they generally consistent with one another? i.e. do networks that satisfies one condition always satisfy the other? It may be worth noting and comparing to two definitions explicitly somewhere.
Technical corrections
Throughout the manuscript the authors hyphenate “plan-form”, which I have not seen before. I know “planform” as a word in itself, which I think could be safely used here. But perhaps the authors have their reasons for this choice.
L49. By “inclination” do you just mean “slope”?
L67-68. “An alternative effect relates the concavity index to the lengthwise asymmetry between paired flow pathways that diverge from a single divide and rejoin at a junction or a common base level.” This phrase comes out of the blue a bit and on first reading I did not understand it. I see now how it relates to the definition of complexity used later (ΔL). But perhaps it is not necessary not bring that up at this stage in the introduction.
L140. “… with distance from divide endpoints exceeding 1000.” What are the units here? Pixels? Metres?
L152. In what sense is the DAC model “process based”? It solves the stream power model which I would consider empirical/phenomenological.
L156. The equivalence of the stream power model and Flint’s law is only true at steady state; the statement here should be qualified.
L205-207. “… ΔL and Δχ are calculated only for neighbouring channel head pairs that drain to different outlets (boundary nodes).” I am a bit confused by this part. Outlets/boundary nodes sounds to me like the edge of the model domain at base level. But then if the channels drain to different outlets, ΔL and Δχ can’t be calculated. So I assume you are talking about the immediate downstream nodes? Perhaps this sentence could be rephrased to clarify.
L278. “Exploiting” is quite strong for me and implies agency on the part of the network which I don’t think is quite right – maybe “facilitated by small variations in Hack’s exponent and coefficient”?
L308. “… an exceptionally simple geometry that differs significantly from any random network.” I think this statement is too strong: simple geometries can arise from random processes, though they may be unlikely. Perhaps something like: “… an exceptionally simple geometry that is extremely unlikely to arise from any random process …”.
L316. I think an “a” is missing between “reveals” and “monotonous”.
Figure C1: I think the “concavity index, K” should be the “erodibility, K”, or similar.
Citation: https://doi.org/10.5194/egusphere-2024-808-RC1 - RC2: 'Comment on egusphere-2024-808', Anonymous Referee #2, 13 May 2024
-
AC1: 'Comment on egusphere-2024-808', Liran Goren, 01 Jul 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-808/egusphere-2024-808-AC1-supplement.pdf
Status: closed
-
RC1: 'Comment on egusphere-2024-808', Fergus McNab, 10 May 2024
This manuscript explores the origins of complexity in river networks. The authors make a persuasive case for the importance of river network complexity, which influences patterns surface runoff, erosion, sediment transport, relief generation, and the distribution of ecological niches. By analysis of natural landscapes, landscape evolution simulations, and the construction of energy-minimising networks, the authors show convincingly that network complexity is closely associated with the concavity index of associated river long profiles. They interpret these patterns in terms of the global energy associated with a water flow through a given network planform. When concavity is low, slope changes slowly with increasing drainage area, so that parts of the network with high drainage areas (hence high discharges) are still relatively steep. This combination of high steepness and high discharge results in high energy expenditure, so that the network will evolve to minimise drainage areas, and hence towards a simple planform. When concacivity is high, slope decreases more quickly with increasing drainage area, reducing the energy cost associated with higher drainage areas, and permitting more complex networks. With reference to further illustrative landscape evolution simulations, the authors discuss some interesting implications of this result, including the response of network planforms to changing climate and planform ‘memory’ of lithological structures that have long eroded away.
Overall I found the manuscript to be well argued, well written and well illustrated. The results and discussion are convincing, novel, of interest to the general geomorphological community, and important for those specifcially engaged in landscape evolution modelling. As such, I think the manuscript is certainly appropriate for publication in ESurf and could probably be accepted as it is. I do have some minor suggestions though which I would ask the authors to consider. These comments mostly concern: (a) descriptions of the methods used and (b) queries concerning some of the results and some suggestions for further discussion. I will give more details below, along with some technical corrections I found.
Lastly, I stress again that I consider these issues minor and that my comments are intended only to improve what I think is already a high quality manuscript. I congratulate the authors on what I believe will make an excellent contribution to ESurf!
Best wishes,
Fergus McNab
GFZ PotsdamSpecific comments
Methods
I felt several of the method descriptions would benefit from an additional sentence or two giving the reader an idea of the concepts behind the chosen methods. In most cases the authors point to the original literature, but I think providing a small amount of additional explanation will help the reader follow what is being done and why, and also make clearer what the underlying assumptions are.
For example, on lines 131-139, the authors describe how they measure the concavity index for the natural landscapes. They state that they use a ‘disorder scheme’, which ‘asses the extent to which the elevation-based order of channel pixels aligns with their order by their χ value’. I think the basic idea here is that the points in the landscape with the same χ value should be at the same elevation – but I think it could be stated more clearly for the benefit of readers less familiar with such analysis. I think this approach also assumes uniform uplift across the domain, so that ‘signals’ are introduced to the long profiles only at base level – this assumption is maybe reasonable given the authors’ choices of study areas, but could be stated explicitly (along with any other key assumptions behind the method).
Similarly, on lines 193-207, the authors describe the method for computing optimal channel networks. They introduce their ‘greedy’ algorithm, and contrast it to an ‘annealing’ algorithm that involves a ‘temperature dependent probability’. These terms do not mean much to me, and suspect not to the general reader either. So, if the distinction is important, I think it would be useful to explain the it in simpler terms here, if possible. I think that would be particularly important if the greedy algorithm the authors employ is new, which is ambiguous in the current description.
Finally, a more minor example, on lines 147-150, the authors describe an elevation correction, but it is not explained why this correction is needed. Indeed, maybe I missed something, but I did not notice anywhere in the later analysis that elevation is used, so it is still unclear to me why this correction is applied.
Queries / discussion points
Below I list some queries that came up for me while reading the manuscript. I do not think any of these points represent significant issues that necessarily need to be addressed in a revised manuscript. But they could warrant comment or further discussion, if of interest to the authors.
Throughout the manuscript I wondered about the potential importance of drainage density/the relative importance of hillslope vs. fluvial processes for the network complexity as defined here (ΔL). Intuitively I would have thought that a landscape with fewer channels would be more likely have a simple network structure than one with a dense channel network. In landscape evolution models, the density depends on model parameters like erodibility, concavity (i.e. m & n), and hillslope diffusivity (see e.g. Perron et al., 2008, JGR). (A related question: do the DAC models shown here have a hillslope component? If so, it is not mentioned in the method description.) Theodoratos et al. (2018, ESurf) define a lengthscale, lc (their equation A8), which depends on m and describes the relative importance of hillslope vs. fluvial processes: if lc is small, fluvial processes dominate and a high channel density will develop; if lc is large, hillslope process dominate and a more diffuse landscape develops. So, I wonder if an alernative explanation for the dependence of complexity on θ would be that we are looking at landscapes with different relative contributions from hillslope and fluvial processes? Since there are no hillslopes in the energy-minimising network simulations, I think the authors’ interpretation of the results in terms of energy minimisation are probably sound. But perhaps this would be a useful/interesting point to test/discuss. For example, an additional set of experiments could vary m but also an additional parameter (erodibility or hillslope diffusivity) to keep lc constant. Alternatively, hillslope diffusivity could be varied while m is fixed to test if that has any influence on ΔL.
During the discussion of Hack’s law (Section 5.1), the authors show that the observed relationship between θ and ΔL can be explained by some variation in the Hack’s law parameters k and h. It is interesting that the data seem to fall roughly into a specific variation band (maybe ki – kj = c. 0.2, hi – hj = c. 0.1, the specific band varying a bit with theta). Does this result point to some characteristic local variability in Hack’s law parameters in natural networks, which could potentially be measured? And could it then form a future test of the authors’ interpretations? If so, it might be worth commenting on. The authors quote some references on variability in Hack’s law parameters but it is not clear whether these numbers apply on a global or local scale (the latter being the relevant one here).
Lastly, there are two definitions of a ‘stable’ network used in the manuscript – one says that χ should be equal across drainage divides, the other that the network is ‘optimal’ from an energy point of view. These two definitions seem to point in the same direction in terms of the relationship between concavity and complexity. But are they generally consistent with one another? i.e. do networks that satisfies one condition always satisfy the other? It may be worth noting and comparing to two definitions explicitly somewhere.
Technical corrections
Throughout the manuscript the authors hyphenate “plan-form”, which I have not seen before. I know “planform” as a word in itself, which I think could be safely used here. But perhaps the authors have their reasons for this choice.
L49. By “inclination” do you just mean “slope”?
L67-68. “An alternative effect relates the concavity index to the lengthwise asymmetry between paired flow pathways that diverge from a single divide and rejoin at a junction or a common base level.” This phrase comes out of the blue a bit and on first reading I did not understand it. I see now how it relates to the definition of complexity used later (ΔL). But perhaps it is not necessary not bring that up at this stage in the introduction.
L140. “… with distance from divide endpoints exceeding 1000.” What are the units here? Pixels? Metres?
L152. In what sense is the DAC model “process based”? It solves the stream power model which I would consider empirical/phenomenological.
L156. The equivalence of the stream power model and Flint’s law is only true at steady state; the statement here should be qualified.
L205-207. “… ΔL and Δχ are calculated only for neighbouring channel head pairs that drain to different outlets (boundary nodes).” I am a bit confused by this part. Outlets/boundary nodes sounds to me like the edge of the model domain at base level. But then if the channels drain to different outlets, ΔL and Δχ can’t be calculated. So I assume you are talking about the immediate downstream nodes? Perhaps this sentence could be rephrased to clarify.
L278. “Exploiting” is quite strong for me and implies agency on the part of the network which I don’t think is quite right – maybe “facilitated by small variations in Hack’s exponent and coefficient”?
L308. “… an exceptionally simple geometry that differs significantly from any random network.” I think this statement is too strong: simple geometries can arise from random processes, though they may be unlikely. Perhaps something like: “… an exceptionally simple geometry that is extremely unlikely to arise from any random process …”.
L316. I think an “a” is missing between “reveals” and “monotonous”.
Figure C1: I think the “concavity index, K” should be the “erodibility, K”, or similar.
Citation: https://doi.org/10.5194/egusphere-2024-808-RC1 - RC2: 'Comment on egusphere-2024-808', Anonymous Referee #2, 13 May 2024
-
AC1: 'Comment on egusphere-2024-808', Liran Goren, 01 Jul 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-808/egusphere-2024-808-AC1-supplement.pdf
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