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the Creative Commons Attribution 4.0 License.
| 08 Oct 2024
A Fractal Framework for Channel-Hillslope Coupling
Abstract. Questions of landscape scale in coupled channel-hillslope landscape evolution have been a significant focus of geomorphological research for decades. Studies to date have suggested a characteristic landscape length that marks the shift from fluvial channels to hillslopes, limiting fluvial incision and setting the length of hillslopes. The representation of real-world landscapes in slope-area plots, however, makes it challenging to identify the exact transition from hillslopes to channels, owing to the existence of an intermediary colluvial valley region. Without a rigorous explanation for the scaling of the channel hillslope transition, the use of computational models, which are forced to implement a finite grid resolution, is limited by the scaling of the physical parameters of the model relative to the grid resolution. Grid resolution is also tied to the width of channels, which is undetermined without a rigorous explanation of where channels begin.
Building on existing work, we demonstrate the existence and implications of the characteristic landscape length and its relationship to grid resolution. We derive the characteristic landscape length as the horizontal length in a one-dimensional landscape evolution framework required to form an inflection point. On a two-dimensional domain, channel heads form in steady state at the characteristic area, the square of the characteristic length, independent of grid resolution. We present a box-counting fractal definition using the grid resolution, revealing that the dimension of the contributing drainage region on steady-state hillslopes is expressed as a multifractal system. In sum, channels have contributing drainage areas, therefore a dimension of two, whereas, by definition, unchannelized locations or nodes have a dimension between zero and two, so not a well-defined area. This conceptualization aligns with the observed scaling of channel width. It also importantly suggests that real-world landscapes have something analogous to the concept of a grid resolution, as this paper demonstrates. In doing so, our works clarifies several unresolved properties of channel-hillslope coupling, with potential for substantially improving the accuracy of coupled landscape evolution models in replicating landscape forms.
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RC1: 'Comment on egusphere-2024-2847', Tyler Doane, 05 Nov 2024
Review of “A Fractal Framework for Channel-Hillslope Coupling” by Kargere et al.Review by Tyler DoaneThis manuscript presents an interesting approach to understanding the critical hillslope to channel transition point. The authors develop a non-dimensional expression for a one-dimensional landscape evolution model to derive the hillslope-channel transition point. Then, the authors discuss the fractal dimension of contributing areas for all locations in a landscape. They demonstrate that the hillslope-channel transition point occurs when the contributing area has a fractal dimension of 2. Last, they perform morphometric analyses on the Gabilan Mesa to identify the hillslope-channel transition point, which they claim reveals the critical length scale regardless of grid resolution.This manuscript is well-written and is a nice contribution. I have a few comments that I hope the authors consider for clarity. But in general, I believe that this manuscript is close to being ready.Major Comments:It is not exactly clear to me how the non-dimensionalization presented earlier in the manuscript figures in to the later parts. If that connection can be made more explicitly, I think that the clever dimensional analysis that they did might have more impact. I note that r is consistent throughout, but to me, that seems to just be the critical contributing area.Landscapes have many fractal qualities and things like fractal dimensions are useful in many contexts. However, fractal definitions often break down at smaller scales because scaling relationships simply do not continue at smaller and smaller scales. One example, is that the authors refer to locations with parallel flow lines as having fractal dimensions of 1. However, at scales of decimeters to meters, landscapes are rough and flow lines can diverge and converge as flow lines respond to shrub mounds, tree throw pit-mound couplets, and many other topographic roughness elements. So, is there a lower limit for this analysis? What if we had a 1-cm grid resolution DEM? I suspect that there area contributing areas with fractal dimension of 2 that would be classified as “hillslope” in the field.Line Comments:Line 79: The flux does not involve the absolute value of slope for linear diffusion. Flux is a vector and therefore the sign of the slope matters.Line 87: U should have units L T-1, not H T-1Line 100: It seems like \hat{t} has units L T-1 so that d\hat{z}/d\hat{t} has units of T L-1. That would make equation (3) incorrect.Line 116: Why are the authors using this expression for CI? Where does it come from?Line 136: The authors refer to multiple-flow direction and D-infinity. It seems like they mean D-infinity. They are different.Line 186: This paragraph is a bit tough to get through. Sentences like “Lines with no width, have width delta” are a bit challenging. Is there another way to say this that does not have apparently contradictory statements?Equation 11: Can this be explained a bit more? It seems like much of the take-away of this manuscript relies on readers understanding how the fractal dimension is calculated. It’s not clear to me where this comes from.Figure 7: This is an interesting take on a classic plot! I think that it could be made a bit clearer. How does this differ from what one might infer from a classic approach to identifying channel-hillslope transition points?Line 280: The same must be done for linear models, yes?Citation: https://doi.org/
10.5194/egusphere-2024-2847-RC1 -
RC2: 'Comment on egusphere-2024-2847', David Litwin, 07 Nov 2024
General Comments:
Kargère and colleagues present analyses of the streampower+diffusion landscape evolution model that offer new insights into the transitional zone between hillslopes and channels, and physical interpretation of the grid cell width. They conduct a 1D analysis of the streampower+diffusion model to derive a characteristic length scale similar to those in the literature, and then use that characteristic length scale and grid cell width to define a fractal dimension that varies across the landscape with specific drainage area. They show that a fractal dimension Df>=2 corresponds to channels and 1<=Df<2 corresponds to the colluvial transition zone between channels and hillslopes. The fractal analysis also suggests that the location of the transition from hillslope to colluvial (the rollover point in the slope-area relationship) is predictably dependent on grid cell width, while the transition from colluvial to fully channelized is independent of grid cell width. They test predictions from the analyses at Gabilan Mesa, and suggest that the analysis meaningfully informs channel-hillslope transitions there.
Overall, I think this is an interesting and useful paper in the continued work to understand intrinsic scales and resolution effects in landscape evolution models and topography. I have two main concerns that should not be too hard to address. First is that the paper needs overall revision to help the reader understand how the problem is related to the analyses conducted, and how the data comparison is related to the analyses. I think all the pieces are there, but it was sometimes difficult to follow. I have specific notes on this below.
Second, I think there is one important element missing to the discussion of characteristic scales at Gabilan Mesa. The authors estimate a characteristic length scale of 62 m, based on the rollover point in the slope-area relationship. They acknowledge that this may not be equal to the characteristic length scale, r, and explain that the parameter K in r is poorly constrained so they could not calculate it. Perron et al. (2009) got around this limitation with a topographic analysis that uses the whole streampower+diffusion model, and estimated the characteristic scale Lc=17.2 m at Gabilan Mesa. With the algebraic factor that relates Lc and r, r=22.5 m (I think), which is significantly smaller than that estimated from slope-area analysis. Using this value in the fractal analysis would give significantly different values of both the hillslope-colluvial and the colluvial-channel transition points. To my mind, your analysis and this discrepancy suggest that the rollover point in slope-area (at least at Gabilan Mesa) and r in the streampower+diffusion model are functionally similar scales, even if the values aren’t close. This is despite Gabilan Mesa being a sort of type-case site for the streampower+diffusion model. This is not too far from what you have! But I think it is worth presenting the Perron et al. value and elaborating on the discrepancy but functional similarity of r (or Lc) and the inflection point as characteristic scales.
Line comments:
16-19. The end of the abstract could use some more thought. It’s not clear what “observed scaling of channel width” is being referred to, nor why it is logical that “real-world landscapes have something analogous to the concept of grid resolution.” Remove “As this paper demonstrates.”
26-28. I’m not quite sure what this sentence is trying to say.
38. When you say a threshold for ‘slope-area’, are you referring to thresholds for streampower, and other terms with the same dimensions as A^m*S^n (e.g., Theodoratos & Kirchner, 2020)?
56. I think the connection between what you have been discussing what you say you will do could be clearer. What is the specific value of the 1D analysis? How does the box counting method and fractal dimension build on or relate to your 1D analysis?
79. Check signs. Diffusion processes have flux inversely proportional to gradient (from Fick’s Law). The positive sign on the diffusion term in (1) comes from something like: dz/dt = U – Ef – Eh, Eh = -D nabla^2 z
93. Hillslopes do not just have parallel flowpaths, right? Divergence of flow paths is also an important part (Bogaart & Troch, 2006).
106. I think ‘3’ might be out of place.
139-143. You’ve already discussed the choice of m=1/2, n=1, so I would lead with (142) – the topic here is about the scaling of channel width.
145. Citation style.
150. This is slightly pedantic, but this is assuming unform generation of runoff, rather than just uniform precipitation.
152. “the contributing drainage of parallel locally parallel”
165-166. You could note this quantity (Pi1) is held constant by Theodoratos et al. (2018) and other papers.
176-179. You might want to be clear that this is an inflection point in topography, but a maximum in slope-area space (as shown in Figure 2). More generally, I’m having a hard time seeing what you describe in Figure 2. Maybe put some scaling lines to help guide the reader. Smaller points, and a binned trend like in Figure 7 could help.
Figure 2. “This value is resolution independent as an area, but not as a length.” It’s not clear to me what you mean by this.
Figure 3. Last sentence of the caption is not clear. The yellow dots show A>= r2 rather than A=r2.
225. Before talking about Gabilan Mesa, tell us what aspect of your results you are attempting to test or confirm, and how you will know if you were successful.
229. Citation style.
Figure 7. Need to find a way to clean up this figure. The dots are too big and too dark, especially relative to the thin binned median line. The caption also needs to say that these are data from Gabilan Mesa since this is the first figure with data rather than model output. At the moment, it just refers to following figure.
238. Estimating K this way also neglects the effect of hillslopes on channel steepness, both in the model (which is limited by assumptions made when coupling detachment-limited fluvial erosion and hillslope diffusion) and in real life. If you are interested, we also have a preprint in ESurf on this (Litwin et al., 2024). There is a way around this though, as shown in Perron et al. (2009). They did estimate something like r (their Lc), and found it to be much shorter, 17.2 m. See comments at the top about this.
235-236. The pieces are here, but as I mention before, we need a clearer indication of what you hypothesized and what you see. As I see it, your first hypothesis is that the transition from hillslope to colluvial is resolution-dependent and occurs at A = r*delta. Your second hypothesis is that channel head locations are resolution independent, and located at a distance related to the length r that you identified testing hypothesis 1, specifically A=r2. Optional, but if you wanted to add something for quantitative comparison to Figure 8, you could show a box and whisker plot of drainage area at the Pelletier channel heads for each resolution, with a horizontal line showing the length scale squared.
245-246. Again, I think it would be helpful to have scaling lines on one of these plots so we can see what m/n=1 and m/n=1/2 look like.
248. The “form A/w for channels” is vague.
250-251. Interesting to see the argument for grid cell width as channel head width here! We argued for more or less the same in Litwin et al. (2022), section 7.4, but in that case it was on the basis of subsurface water transport capacity that affects the point at which channel heads emerge from saturation excess overland flow.
267. “Have shown that showed”
References
Bogaart, P. W., & Troch, P. A. (2006). Curvature distribution within hillslopes and catchments and its effect on the hydrological response. Hydrology and Earth System Sciences, 10(6), 925–936. https://doi.org/10.5194/hess-10-925-2006
Litwin, D. G., Tucker, G. E., Barnhart, K. R., & Harman, C. J. (2022). Groundwater affects the geomorphic and hydrologic properties of coevolved landscapes. Journal of Geophysical Research: Earth Surface, 127(1), e2021JF006239. https://doi.org/10.1029/2021JF006239
Litwin, D. G., Malatesta, L. C., & Sklar, L. S. (2024, August 15). Hillslope diffusion and channel steepness in landscape evolution models. Copernicus GmbH. https://doi.org/10.5194/egusphere-2024-2418
Perron, J. T., Kirchner, J. W., & Dietrich, W. E. (2009). Formation of evenly spaced ridges and valleys. Nature, 460(7254), 502–505. https://doi.org/10.1038/nature08174
Theodoratos, N., & Kirchner, J. W. (2020). Dimensional analysis of a landscape evolution model with incision threshold. Earth Surface Dynamics, 8(2), 505–526. https://doi.org/10.5194/esurf-8-505-2020
Theodoratos, N., Seybold, H., & Kirchner, J. W. (2018). Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model. Earth Surface Dynamics, 6(3), 779–808. https://doi.org/10.5194/esurf-6-779-2018
Citation: https://doi.org/10.5194/egusphere-2024-2847-RC2 -
AC1: 'Reply on RC2', Benjamin Kargere, 13 Nov 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2847/egusphere-2024-2847-AC1-supplement.pdf
-
AC1: 'Reply on RC2', Benjamin Kargere, 13 Nov 2024
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2024-2847', Tyler Doane, 05 Nov 2024
Review of “A Fractal Framework for Channel-Hillslope Coupling” by Kargere et al.Review by Tyler DoaneThis manuscript presents an interesting approach to understanding the critical hillslope to channel transition point. The authors develop a non-dimensional expression for a one-dimensional landscape evolution model to derive the hillslope-channel transition point. Then, the authors discuss the fractal dimension of contributing areas for all locations in a landscape. They demonstrate that the hillslope-channel transition point occurs when the contributing area has a fractal dimension of 2. Last, they perform morphometric analyses on the Gabilan Mesa to identify the hillslope-channel transition point, which they claim reveals the critical length scale regardless of grid resolution.This manuscript is well-written and is a nice contribution. I have a few comments that I hope the authors consider for clarity. But in general, I believe that this manuscript is close to being ready.Major Comments:It is not exactly clear to me how the non-dimensionalization presented earlier in the manuscript figures in to the later parts. If that connection can be made more explicitly, I think that the clever dimensional analysis that they did might have more impact. I note that r is consistent throughout, but to me, that seems to just be the critical contributing area.Landscapes have many fractal qualities and things like fractal dimensions are useful in many contexts. However, fractal definitions often break down at smaller scales because scaling relationships simply do not continue at smaller and smaller scales. One example, is that the authors refer to locations with parallel flow lines as having fractal dimensions of 1. However, at scales of decimeters to meters, landscapes are rough and flow lines can diverge and converge as flow lines respond to shrub mounds, tree throw pit-mound couplets, and many other topographic roughness elements. So, is there a lower limit for this analysis? What if we had a 1-cm grid resolution DEM? I suspect that there area contributing areas with fractal dimension of 2 that would be classified as “hillslope” in the field.Line Comments:Line 79: The flux does not involve the absolute value of slope for linear diffusion. Flux is a vector and therefore the sign of the slope matters.Line 87: U should have units L T-1, not H T-1Line 100: It seems like \hat{t} has units L T-1 so that d\hat{z}/d\hat{t} has units of T L-1. That would make equation (3) incorrect.Line 116: Why are the authors using this expression for CI? Where does it come from?Line 136: The authors refer to multiple-flow direction and D-infinity. It seems like they mean D-infinity. They are different.Line 186: This paragraph is a bit tough to get through. Sentences like “Lines with no width, have width delta” are a bit challenging. Is there another way to say this that does not have apparently contradictory statements?Equation 11: Can this be explained a bit more? It seems like much of the take-away of this manuscript relies on readers understanding how the fractal dimension is calculated. It’s not clear to me where this comes from.Figure 7: This is an interesting take on a classic plot! I think that it could be made a bit clearer. How does this differ from what one might infer from a classic approach to identifying channel-hillslope transition points?Line 280: The same must be done for linear models, yes?Citation: https://doi.org/
10.5194/egusphere-2024-2847-RC1 -
RC2: 'Comment on egusphere-2024-2847', David Litwin, 07 Nov 2024
General Comments:
Kargère and colleagues present analyses of the streampower+diffusion landscape evolution model that offer new insights into the transitional zone between hillslopes and channels, and physical interpretation of the grid cell width. They conduct a 1D analysis of the streampower+diffusion model to derive a characteristic length scale similar to those in the literature, and then use that characteristic length scale and grid cell width to define a fractal dimension that varies across the landscape with specific drainage area. They show that a fractal dimension Df>=2 corresponds to channels and 1<=Df<2 corresponds to the colluvial transition zone between channels and hillslopes. The fractal analysis also suggests that the location of the transition from hillslope to colluvial (the rollover point in the slope-area relationship) is predictably dependent on grid cell width, while the transition from colluvial to fully channelized is independent of grid cell width. They test predictions from the analyses at Gabilan Mesa, and suggest that the analysis meaningfully informs channel-hillslope transitions there.
Overall, I think this is an interesting and useful paper in the continued work to understand intrinsic scales and resolution effects in landscape evolution models and topography. I have two main concerns that should not be too hard to address. First is that the paper needs overall revision to help the reader understand how the problem is related to the analyses conducted, and how the data comparison is related to the analyses. I think all the pieces are there, but it was sometimes difficult to follow. I have specific notes on this below.
Second, I think there is one important element missing to the discussion of characteristic scales at Gabilan Mesa. The authors estimate a characteristic length scale of 62 m, based on the rollover point in the slope-area relationship. They acknowledge that this may not be equal to the characteristic length scale, r, and explain that the parameter K in r is poorly constrained so they could not calculate it. Perron et al. (2009) got around this limitation with a topographic analysis that uses the whole streampower+diffusion model, and estimated the characteristic scale Lc=17.2 m at Gabilan Mesa. With the algebraic factor that relates Lc and r, r=22.5 m (I think), which is significantly smaller than that estimated from slope-area analysis. Using this value in the fractal analysis would give significantly different values of both the hillslope-colluvial and the colluvial-channel transition points. To my mind, your analysis and this discrepancy suggest that the rollover point in slope-area (at least at Gabilan Mesa) and r in the streampower+diffusion model are functionally similar scales, even if the values aren’t close. This is despite Gabilan Mesa being a sort of type-case site for the streampower+diffusion model. This is not too far from what you have! But I think it is worth presenting the Perron et al. value and elaborating on the discrepancy but functional similarity of r (or Lc) and the inflection point as characteristic scales.
Line comments:
16-19. The end of the abstract could use some more thought. It’s not clear what “observed scaling of channel width” is being referred to, nor why it is logical that “real-world landscapes have something analogous to the concept of grid resolution.” Remove “As this paper demonstrates.”
26-28. I’m not quite sure what this sentence is trying to say.
38. When you say a threshold for ‘slope-area’, are you referring to thresholds for streampower, and other terms with the same dimensions as A^m*S^n (e.g., Theodoratos & Kirchner, 2020)?
56. I think the connection between what you have been discussing what you say you will do could be clearer. What is the specific value of the 1D analysis? How does the box counting method and fractal dimension build on or relate to your 1D analysis?
79. Check signs. Diffusion processes have flux inversely proportional to gradient (from Fick’s Law). The positive sign on the diffusion term in (1) comes from something like: dz/dt = U – Ef – Eh, Eh = -D nabla^2 z
93. Hillslopes do not just have parallel flowpaths, right? Divergence of flow paths is also an important part (Bogaart & Troch, 2006).
106. I think ‘3’ might be out of place.
139-143. You’ve already discussed the choice of m=1/2, n=1, so I would lead with (142) – the topic here is about the scaling of channel width.
145. Citation style.
150. This is slightly pedantic, but this is assuming unform generation of runoff, rather than just uniform precipitation.
152. “the contributing drainage of parallel locally parallel”
165-166. You could note this quantity (Pi1) is held constant by Theodoratos et al. (2018) and other papers.
176-179. You might want to be clear that this is an inflection point in topography, but a maximum in slope-area space (as shown in Figure 2). More generally, I’m having a hard time seeing what you describe in Figure 2. Maybe put some scaling lines to help guide the reader. Smaller points, and a binned trend like in Figure 7 could help.
Figure 2. “This value is resolution independent as an area, but not as a length.” It’s not clear to me what you mean by this.
Figure 3. Last sentence of the caption is not clear. The yellow dots show A>= r2 rather than A=r2.
225. Before talking about Gabilan Mesa, tell us what aspect of your results you are attempting to test or confirm, and how you will know if you were successful.
229. Citation style.
Figure 7. Need to find a way to clean up this figure. The dots are too big and too dark, especially relative to the thin binned median line. The caption also needs to say that these are data from Gabilan Mesa since this is the first figure with data rather than model output. At the moment, it just refers to following figure.
238. Estimating K this way also neglects the effect of hillslopes on channel steepness, both in the model (which is limited by assumptions made when coupling detachment-limited fluvial erosion and hillslope diffusion) and in real life. If you are interested, we also have a preprint in ESurf on this (Litwin et al., 2024). There is a way around this though, as shown in Perron et al. (2009). They did estimate something like r (their Lc), and found it to be much shorter, 17.2 m. See comments at the top about this.
235-236. The pieces are here, but as I mention before, we need a clearer indication of what you hypothesized and what you see. As I see it, your first hypothesis is that the transition from hillslope to colluvial is resolution-dependent and occurs at A = r*delta. Your second hypothesis is that channel head locations are resolution independent, and located at a distance related to the length r that you identified testing hypothesis 1, specifically A=r2. Optional, but if you wanted to add something for quantitative comparison to Figure 8, you could show a box and whisker plot of drainage area at the Pelletier channel heads for each resolution, with a horizontal line showing the length scale squared.
245-246. Again, I think it would be helpful to have scaling lines on one of these plots so we can see what m/n=1 and m/n=1/2 look like.
248. The “form A/w for channels” is vague.
250-251. Interesting to see the argument for grid cell width as channel head width here! We argued for more or less the same in Litwin et al. (2022), section 7.4, but in that case it was on the basis of subsurface water transport capacity that affects the point at which channel heads emerge from saturation excess overland flow.
267. “Have shown that showed”
References
Bogaart, P. W., & Troch, P. A. (2006). Curvature distribution within hillslopes and catchments and its effect on the hydrological response. Hydrology and Earth System Sciences, 10(6), 925–936. https://doi.org/10.5194/hess-10-925-2006
Litwin, D. G., Tucker, G. E., Barnhart, K. R., & Harman, C. J. (2022). Groundwater affects the geomorphic and hydrologic properties of coevolved landscapes. Journal of Geophysical Research: Earth Surface, 127(1), e2021JF006239. https://doi.org/10.1029/2021JF006239
Litwin, D. G., Malatesta, L. C., & Sklar, L. S. (2024, August 15). Hillslope diffusion and channel steepness in landscape evolution models. Copernicus GmbH. https://doi.org/10.5194/egusphere-2024-2418
Perron, J. T., Kirchner, J. W., & Dietrich, W. E. (2009). Formation of evenly spaced ridges and valleys. Nature, 460(7254), 502–505. https://doi.org/10.1038/nature08174
Theodoratos, N., & Kirchner, J. W. (2020). Dimensional analysis of a landscape evolution model with incision threshold. Earth Surface Dynamics, 8(2), 505–526. https://doi.org/10.5194/esurf-8-505-2020
Theodoratos, N., Seybold, H., & Kirchner, J. W. (2018). Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model. Earth Surface Dynamics, 6(3), 779–808. https://doi.org/10.5194/esurf-6-779-2018
Citation: https://doi.org/10.5194/egusphere-2024-2847-RC2 -
AC1: 'Reply on RC2', Benjamin Kargere, 13 Nov 2024
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2847/egusphere-2024-2847-AC1-supplement.pdf
-
AC1: 'Reply on RC2', Benjamin Kargere, 13 Nov 2024
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Benjamin A. Kargère
José A. Constantine
Tristram C. Hales
Stuart W. D. Grieve
Stewart D. Johnson
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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