Geometric constraints on tributary fluvial network junction angles

Pelletier, Jon D.; Hayes, Robert G.; Hoch, Olivia; Fenerty, Brendan; McGuire, Luke A.

doi:https://doi.org/10.5194/egusphere-2024-1153

Preprints

https://doi.org/10.5194/egusphere-2024-1153

Preprints

25 Apr 2024

| 25 Apr 2024

Geometric constraints on tributary fluvial network junction angles

Jon D. Pelletier, Robert G. Hayes, Olivia Hoch, Brendan Fenerty, and Luke A. McGuire

Abstract. The intersection of two non-parallel planes is a line. Howard (1990), following Horton (1932), proposed that the orientation and slope of a fluvial valley within a tributary network are geometrically constrained by the orientation and slope of the line formed by the intersection of planar approximations to the topography upslope from the tributary junction along the two tributary directions. Previously published analyses of junction-angle data support this geometric model, yet junction angles have also been proposed to be controlled by climate and/or optimality principles (e.g., minimum-power expenditure). In this paper, we document a test of the Howard (1990) model using ~10⁷ fluvial network junctions in the conterminous U.S. and a portion of the Loess Plateau, China. Junction angles are consistent with the predictions of the Howard (1990) model when the orientations and slopes are computed using drainage basins rather than in the traditional way using valley-bottom segments near tributary junctions. When computed in the traditional way, junction angles are a function of slope ratios (as the Howard (1990) model) predicts, but data deviate from the Howard (1990) model in a manner that we propose is the result of valley-bottom meandering/tortuosity. We map the mean junction angles computed along valley bottoms within each 2.5 km x 2.5 km pixel of the conterminous U.S.A. and document lower mean junction angles in incised late-Cenozoic alluvial piedmont deposits compared to those of incised bedrock/older deposits. To understand how this finding relates to the geometric model of Howard (1990), we demonstrate that, for an idealized model of an initially unincised landform, i.e., a tilted plane with random microtopography, lower ratios of the mean microtopographic slope to the large-scale slope/tilt are associated with lower mean junction angles compared to landforms with higher such ratios. Using modern analogs, we demonstrate that unincised late-Cenozoic alluvial piedmonts likely had ratios of mean microtopographic slope to large-scale slope/tilt that were lower (i.e., ~1) prior to tributary drainage network development than the same ratios of bedrock/older deposits (≫1). This finding provides a means of understanding how the geometric model of Howard (1990) results in incised late Cenozoic alluvial piedmont deposits with lower mean tributary fluvial network junction angles, on average, compared to those of incised bedrock/older deposits. This work demonstrates that the topography of a landscape prior to fluvial incision exerts a key constraint on tributary fluvial network junction angles via a fundamental geometric principle that is independent of any climate- or optimality-based principle.

Received: 17 Apr 2024 – Discussion started: 25 Apr 2024

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Journal article(s) based on this preprint

12 Mar 2025

Geometric constraints on tributary fluvial network junction angles

Jon D. Pelletier, Robert G. Hayes, Olivia Hoch, Brendan Fenerty, and Luke A. McGuire

Earth Surf. Dynam., 13, 219–238, https://doi.org/10.5194/esurf-13-219-2025,https://doi.org/10.5194/esurf-13-219-2025, 2025

Short summary

Jon D. Pelletier, Robert G. Hayes, Olivia Hoch, Brendan Fenerty, and Luke A. McGuire

Interactive discussion

Status: closed

RC1: 'Comment on egusphere-2024-1153', Anonymous Referee #1, 07 Jun 2024

Referee report on

"Geometric constraints on tributary fluvial network junction angles"

by

Pelletier et al.
The authors present an analysis of ~10 million junction angles

for which they extracted the corresponding stream network across the

Contiguous United States using a 10m Digital Elevation Model. They

find that the branching angles closely follow the slope-angle relations

proposed by Horton in 1932 and later modified by

Howard in 1990, however with the twist, that they use the mean basin

slope of the two upstream basins rather than the valley slope of the

tributary channels at the confluence.
While the paper makes some interesting points on how topographic

features and geology may be expressed in a channel network's branching

geometry, several points need to be clarified and some

miss-perceptions of previous studies need to be resolved.
First, the fact that vertical features of topography like slope

correlate with horizontal geometric properties of landscapes such as

branching angles should surprising. Ultimately, Earth's surface

evolves as a result of erosion and uplift and valley networks are

simply characteristic features embedded in this three-dimensional

topography.
More important than correlations between different topographic

features is the understanding of how the underlying processes, namely

erosion, deposition, and tectonic uplift are expressed in the

network's branching structure. Quoting from Howard's 1990 paper on the

grometric model: , "[this geometric model] was only sketchily related

to the assumed channel processes ...".
This point actually prompted Howard in 1990, to explore additional

hydrological arguments to constrain the angle-slope relation foundon

more process-based descriptions of how the flow in a channel may

modify its bed, even if the particular expression may result in a

slightly lower r-square, than the rpedictions of the geometric model.
Also note, that much of the theoretical basis of Howard 1990 was

mostly done 20 years earlier in [Howard 1971b, 1971c] as referenced in

the 1990 paper. Similarly, Hooshyar 2017 and Seybold 2017 try to

relate branching angles to channel-forming processes, namely overland

flow v.s. debris flow in the first and seepage flow v.s. overland flow

in the latter, while the optimality models of Strong & Mudd e.g. use

hydrological conservation arguments at the junction, similar to the

arguments in Howard 1990. I would like to note here, that while the

authors promote the impression that the observed slope-angle relation

contradicts the above-mentioned interpretations, however, these

conceptual frameworks are more complementary as they simply provide

different aspects for explaining how landscapes evolve.
Consequently, the authors may consider a more "inclusive tone" in

their argumentation, particularly because the author's own

explanations remain rather vague on process understanding giving too

much focus on the trivial angle-slope relations of the geometric

model.
An interesting point in the manuscript which I was initially excited

to read about was the alleged finding that incisions into alluvial

piedmonts exhibit different structural controls on branching angles

than channel networks incised into bedrock. Here, the role of geology

is an important point for channel erosion and has not been addressed

adequately in previous work, including the recent papers cited by the

authors.
However,the more I was disappointed when reading through the text to

find out that the narrowing of branching angles in alluvial piedmonts

is fully explained by the fact that piedmonts are (usually)

preferentially sloped while bedrock valley networks do not display a

preferential slope.
Thus, what is thought to be a structural effect is simply the higher

prevalence of regional gradients that narrow the branching angle

statistics.
This effect of regional slope and microtopography has been already

demonstrated by Casteltort in 2013 through numerical simulations and

also observed in multiple previous studies over the last decade.
Below I provide some minor comments/suggestions on specific parts of

the text, which may repeat some of the points raised above.
L.5: Ignoring some degenerated special cases, three planes, one for

each upstream tributary and one for the downstream channel can

intersect at any angle and slope and thus does not provide a geometric

constraint for branched valley networks. What actually constraints the

argument of Horton /Howard is the presence of fixed a general slope

(for all three basins) aligned with the downstream channel, while the

side valley is additionally inclined with respect to this dominant

direction.

L 9-10: Landscapes evolve through erosion/deposition and uplift

creating sloped valley networks. Thus I firmly object to the

implications that the geometric "model" actually contradicts the

process-based explanation by Hooshyar et al. or Seybold et al. as well

as the optimality principles proposed by Rinaldo, Strong & Mudd, and

many others. Earth's surface is shaped by processes and not geometric

features.

L. 11-12: "Junction angles are consistent with the geometric model..."

It would be good to have a 1:1 plot between prediction vs measurement

and a quantification of the spread and uncertainty of the model

prediction. Also, it would be favorable to see how the "classical

geometric model" compares to the "modified geometric model" in such a

comparison.
L.17-25: How does the type of geology (alluvial vs bedrock) relate to

the sloping of the topography?
The tilting effect on branching angles and basin shapes observed in

the modeling procedures of Casteltort 2013) which the authors draw to

explain their findings makes absolutely no assumption on the ground

surface material. Neither piedmonts nor bedrock is part of

Casteltort's model. Consequently, I see nothing new in the

interpretation besides the finding that alluvial piedmonts at mountain

fronts often display a general sloping tilt and thus have narrower

branching angles on average. However, this has been known for over 10

years (Casteltort 2013) and also observed in multiple network studies.
The authors boldly use the word "demonstrate" multiple times in the

abstract but hardly prove anything in the main text. It would be

interesting if the authors could at least demonstrate that the

geological impact of alluvial v.s. bedrock is an actual effect and not

a spurious correlation because alluvial piedmonts often follow a

general slope.
L. 27: "demonstrates independent[ce] of climate and optimality

principles": How can the authors ensure that the generation of slopes

in landscape evolution is not the effect of climate-driven

erosion/deposition processes (rainfall leads to discharge and channel

incision as a consequence) which follow optimization rules of

landscape evolution?

While I agree with the arguments that the geometry of valley networks

are sensitive to the initial conditions such as regional tilt and

micro topography, it is hard to separate fluvial incision from the

formation of the initial landscape.

L 35: parallel and sub-parallel networks in piedmonts v.s. dentritic

and rectangular basins incised in bedrock: Is this a geological effect

(alluvium vs bedrock) or simply an effect that the piedmont has a

dominant regional slope?
L. 40: Piedmont deposition: I don't buy the claim that the deposition

cycles created initially unincised low-relief landforms because the

timescales for deposition are rather similar to the timescales of

valley incision. Thus one can hardly separate the two processes as

they occur simultaneously. Here tha authors need to provide some

additional arguments which support their claim.
L.56-58: If this is true, then the narrow branching angles in

piedmonts are not the result of the underlying geology but a spurious

effect of the gently sloping terrain. This effect has been very

clearly demonstrated by Casteltort using landscape evolution models

which are completely independent of the underlying geology.
L 63: The authors fail to demonstrate that tortuosity is indeed the

driver for wider junction angles compared to the arguments of Howard

1990. "may promote narrower junction angles ..." is simply not enough

for a central result in a scientific publication.
L 104: the importance of piedmont deposits: As the authors have

already explained in a previous paragraph, it is not the piedmont

deposit (geology) that narrows the branching angle but the gentle

regional slope that often comes with the depositional landscape at

mountain fronts. Consequently, any sloped geology e.g. volcanic

bedrock in Hawaii would also lead to the same effect.
L. 133: How do the authors define the basin averaged slope in the case

of n-th order junctions with n>1? Do they follow the longest upstream

tributary up to its source, or do they average the slope of the whole

upstream network tree? This point needs to be clarified.
L 190: Same as above. Do the authors use the pixel-by-pixel slope of the

DEM and then average over all pixels of the upstream basin? What is

the upslope along directions? These points need to be clarified.
L. 229-231: "... suggests that junction angles may be systematically

lower, on average, in fluvial networks incised into late-Cenozoic

alluvial piedmont deposits than those incised into adjacent areas of

bedrock/older deposits." The authors need to show that the structural

control on junction angle is not a spurious slope bias in gently

sloped depositional landscapes such as piedmonts.
L. 316: comparison with NHD: Comparing USGS's medium-resolution blue

lines with a fixed flow accumulation threshold extracted channel

network is not a fair comparison, particularly as one can obtain any

drainage density by adjusting the drainage area threshold. NHDPlus (

and its higher resolution companion NHDPlusHR have been extensively

ground-checked.
For example, I was unable to identify many of the fine channel

networks from Fig. 6b/d around 32.3N/110.9W on aerial images on Google

Maps.
L. 560: variability in hillslope length: Minor point: While I agree

with the presented arguments it remains to be shown that hillslope

convergence is the dominant factor controlling drainage density.
L. 579: I agree with the AI~elevation argument which should be easy to

test. (correlation elevation~AI and correlation angle~elevation)
L. 595: higher or less infiltration in arid landscapes: I agree with

the authors that the question of groundwater recharge and its

dependency on environmental conditions is rather

controversial. See for example a recent Nature Climate Change which

argues that "Groundwater recharge is sensitive to changing

long-term aridity" which suggests higher recharge in humid landscapes

than in arid ones based on lysimeter measurements.
L. 611: "The result presented here could potentially be reconciled ..."

..." It should be easy to test and prove or disproof if the presented

results are consistent with Hooshyar 2017 or not.
L 615: I agree with the author's similarity arguments between minimum

power and MGM, but while MGM only relates different aspects of Earth's

tomography with each other, the optimal branching models seek to

explain WHY the slopes are as they are from a physical mechanistic

perspective of landscape evolution in general and channel erosion in

particular. Here, the topographic slope itself is only a feedback

variable for flow accumulation in the erosional term of the landscape

evolution equation.

Citation: https://doi.org/10.5194/egusphere-2024-1153-RC1
RC2:
'Comment on egusphere-2024-1153', Anonymous Referee #2, 07 Jun 2024
The authors present an analysis of approximately 10 million junction angles, for which they extracted the corresponding stream network across the Contiguous United States using a 10m Digital Elevation Model, suggesting that the topography of a landscape prior to fluvial incision exerts a key constraint on tributary fluvial network junction angles. This is due to the fact that different slope versus microtopography assemblages give rise to more or less sinuous patterns in the valley bed shape, as suggested by Lazarus and Constantine (2013). Indeed, when computed in the traditional way, junction angles are a function of slope ratios (as the Howard 1990 model predicts), but data deviate from the Howard model in a manner that the authors propose is the result of valley-bottom meandering/tortuosity.
I’ve read the paper with much interest and found it extremely well-written and fascinating. Only sometimes did I find myself a bit lost in the lengthy technical discussions, but this might be entirely my fault, as I am not an expert on the topic discussed here. Also, I appreciate that the literature and scientific debate on junction angles in tributary fluvial networks are extensive, so previous arguments need to be discussed in detail.
Being not an expert, I’ll refrain from delving into the debate regarding what drives the distribution of junction angles and how this study copes with or does not cope with earlier literature. The pros and cons of the approach are discussed in sufficient detail, and results are put into perspective with previous literature in a way that allows even a non-expert reader to navigate through the discussion, although it is very technical.
Still, I’d like to raise two possibly minor comments, one being purely terminological and the other perhaps more fundamental.
First, I’d suggest the authors avoid using the term “meandering” to describe sinuous valley patterns. Meandering refers to landforms that grow orderly rather than randomly. The fact that sinuous patterns can arise even from unordered growth, and that at low sinuosity random versus ordered shapes are not distinguishable (Limaye et al. 2021).
Second, and more importantly, the major limitation I see in the idea behind the paper is that valley sinuosity is assumed to remain fixed once their shape and sinuosity have been determined by the initial microtopography (in the way Lazarus & Constantine's idea illustrates).

Kwang et al. (2021) have demonstrated how lateral erosion/incision of rivers into bedrock is fundamental for the development of dendritic drainage networks, to the point that without lateral erosion, landscape evolution models cannot turn non-optimal, non-dendritic networks into optimal dendritic ones such as those observed in nature.

For the point being made in this paper (that valley tortuosity matters when it comes to junction angles) I think lateral erosion should be considered/discussed because it has the potential to:
Alter river valley tortuosity over time, thereby impacting junction angles.

Cause the system to lose memory of initial conditions (inducing persistent reorganization in dynamic steady-state network shape), thus critically diminishing (if not completely erasing) the effect of initial microtopography on which the authors anchor all their analyses and results.

In short, assuming that river networks evolve from initial conditions to a frozen state with no further modifications poses a strong limitations to the new perspective the authroes bring abotu regarding the drivers of junction angles in fluvial tributary networks. Later network modification by lateral river erosion into valleys, which can induce major drainage reorganization and long transience in landscape shape, should be discussed more thoroughly because it’s potentially tied to how tributary channels intersect each others.

CITED REFERENCES:
Kwang, J. S., Langston, A. L., & Parker, G. (2021). The role of lateral erosion in the evolution of nondendritic drainage networks to dendricity and the persistence of dynamic networks. Proceedings of the National Academy of Sciences of the United States of America, 118(16), 1–6. https://doi.org/10.1073/pnas.2015770118
Lazarus, E. D., & Constantine, J. A. (2013). Generic theory for channel sinuosity. Proceedings of the National Academy of Sciences of the United States of America, 110(21), 8447–8452. https://doi.org/10.1073/pnas.1214074110
Limaye, A. B., Lazarus, E. D., Li, Y., & Schwenk, J. (2021). River sinuosity describes a continuum between randomness and ordered growth. Geology, 49(12), 1506–1510. https://doi.org/10.1130/G49153.1
Citation: https://doi.org/10.5194/egusphere-2024-1153-RC2
AC1: 'Author response to referee comments on egusphere-2024-1153', Jon Pelletier , 05 Jul 2024

See attached pdf

Citation: https://doi.org/10.5194/egusphere-2024-1153-AC1

Interactive discussion

Status: closed

RC1: 'Comment on egusphere-2024-1153', Anonymous Referee #1, 07 Jun 2024

Referee report on

"Geometric constraints on tributary fluvial network junction angles"

by

Pelletier et al.
The authors present an analysis of ~10 million junction angles

for which they extracted the corresponding stream network across the

Contiguous United States using a 10m Digital Elevation Model. They

find that the branching angles closely follow the slope-angle relations

proposed by Horton in 1932 and later modified by

Howard in 1990, however with the twist, that they use the mean basin

slope of the two upstream basins rather than the valley slope of the

tributary channels at the confluence.
While the paper makes some interesting points on how topographic

features and geology may be expressed in a channel network's branching

geometry, several points need to be clarified and some

miss-perceptions of previous studies need to be resolved.
First, the fact that vertical features of topography like slope

correlate with horizontal geometric properties of landscapes such as

branching angles should surprising. Ultimately, Earth's surface

evolves as a result of erosion and uplift and valley networks are

simply characteristic features embedded in this three-dimensional

topography.
More important than correlations between different topographic

features is the understanding of how the underlying processes, namely

erosion, deposition, and tectonic uplift are expressed in the

network's branching structure. Quoting from Howard's 1990 paper on the

grometric model: , "[this geometric model] was only sketchily related

to the assumed channel processes ...".
This point actually prompted Howard in 1990, to explore additional

hydrological arguments to constrain the angle-slope relation foundon

more process-based descriptions of how the flow in a channel may

modify its bed, even if the particular expression may result in a

slightly lower r-square, than the rpedictions of the geometric model.
Also note, that much of the theoretical basis of Howard 1990 was

mostly done 20 years earlier in [Howard 1971b, 1971c] as referenced in

the 1990 paper. Similarly, Hooshyar 2017 and Seybold 2017 try to

relate branching angles to channel-forming processes, namely overland

flow v.s. debris flow in the first and seepage flow v.s. overland flow

in the latter, while the optimality models of Strong & Mudd e.g. use

hydrological conservation arguments at the junction, similar to the

arguments in Howard 1990. I would like to note here, that while the

authors promote the impression that the observed slope-angle relation

contradicts the above-mentioned interpretations, however, these

conceptual frameworks are more complementary as they simply provide

different aspects for explaining how landscapes evolve.
Consequently, the authors may consider a more "inclusive tone" in

their argumentation, particularly because the author's own

explanations remain rather vague on process understanding giving too

much focus on the trivial angle-slope relations of the geometric

model.
An interesting point in the manuscript which I was initially excited

to read about was the alleged finding that incisions into alluvial

piedmonts exhibit different structural controls on branching angles

than channel networks incised into bedrock. Here, the role of geology

is an important point for channel erosion and has not been addressed

adequately in previous work, including the recent papers cited by the

authors.
However,the more I was disappointed when reading through the text to

find out that the narrowing of branching angles in alluvial piedmonts

is fully explained by the fact that piedmonts are (usually)

preferentially sloped while bedrock valley networks do not display a

preferential slope.
Thus, what is thought to be a structural effect is simply the higher

prevalence of regional gradients that narrow the branching angle

statistics.
This effect of regional slope and microtopography has been already

demonstrated by Casteltort in 2013 through numerical simulations and

also observed in multiple previous studies over the last decade.
Below I provide some minor comments/suggestions on specific parts of

the text, which may repeat some of the points raised above.
L.5: Ignoring some degenerated special cases, three planes, one for

each upstream tributary and one for the downstream channel can

intersect at any angle and slope and thus does not provide a geometric

constraint for branched valley networks. What actually constraints the

argument of Horton /Howard is the presence of fixed a general slope

(for all three basins) aligned with the downstream channel, while the

side valley is additionally inclined with respect to this dominant

direction.

L 9-10: Landscapes evolve through erosion/deposition and uplift

creating sloped valley networks. Thus I firmly object to the

implications that the geometric "model" actually contradicts the

process-based explanation by Hooshyar et al. or Seybold et al. as well

as the optimality principles proposed by Rinaldo, Strong & Mudd, and

many others. Earth's surface is shaped by processes and not geometric

features.

L. 11-12: "Junction angles are consistent with the geometric model..."

It would be good to have a 1:1 plot between prediction vs measurement

and a quantification of the spread and uncertainty of the model

prediction. Also, it would be favorable to see how the "classical

geometric model" compares to the "modified geometric model" in such a

comparison.
L.17-25: How does the type of geology (alluvial vs bedrock) relate to

the sloping of the topography?
The tilting effect on branching angles and basin shapes observed in

the modeling procedures of Casteltort 2013) which the authors draw to

explain their findings makes absolutely no assumption on the ground

surface material. Neither piedmonts nor bedrock is part of

Casteltort's model. Consequently, I see nothing new in the

interpretation besides the finding that alluvial piedmonts at mountain

fronts often display a general sloping tilt and thus have narrower

branching angles on average. However, this has been known for over 10

years (Casteltort 2013) and also observed in multiple network studies.
The authors boldly use the word "demonstrate" multiple times in the

abstract but hardly prove anything in the main text. It would be

interesting if the authors could at least demonstrate that the

geological impact of alluvial v.s. bedrock is an actual effect and not

a spurious correlation because alluvial piedmonts often follow a

general slope.
L. 27: "demonstrates independent[ce] of climate and optimality

principles": How can the authors ensure that the generation of slopes

in landscape evolution is not the effect of climate-driven

erosion/deposition processes (rainfall leads to discharge and channel

incision as a consequence) which follow optimization rules of

landscape evolution?

While I agree with the arguments that the geometry of valley networks

are sensitive to the initial conditions such as regional tilt and

micro topography, it is hard to separate fluvial incision from the

formation of the initial landscape.

L 35: parallel and sub-parallel networks in piedmonts v.s. dentritic

and rectangular basins incised in bedrock: Is this a geological effect

(alluvium vs bedrock) or simply an effect that the piedmont has a

dominant regional slope?
L. 40: Piedmont deposition: I don't buy the claim that the deposition

cycles created initially unincised low-relief landforms because the

timescales for deposition are rather similar to the timescales of

valley incision. Thus one can hardly separate the two processes as

they occur simultaneously. Here tha authors need to provide some

additional arguments which support their claim.
L.56-58: If this is true, then the narrow branching angles in

piedmonts are not the result of the underlying geology but a spurious

effect of the gently sloping terrain. This effect has been very

clearly demonstrated by Casteltort using landscape evolution models

which are completely independent of the underlying geology.
L 63: The authors fail to demonstrate that tortuosity is indeed the

driver for wider junction angles compared to the arguments of Howard

1990. "may promote narrower junction angles ..." is simply not enough

for a central result in a scientific publication.
L 104: the importance of piedmont deposits: As the authors have

already explained in a previous paragraph, it is not the piedmont

deposit (geology) that narrows the branching angle but the gentle

regional slope that often comes with the depositional landscape at

mountain fronts. Consequently, any sloped geology e.g. volcanic

bedrock in Hawaii would also lead to the same effect.
L. 133: How do the authors define the basin averaged slope in the case

of n-th order junctions with n>1? Do they follow the longest upstream

tributary up to its source, or do they average the slope of the whole

upstream network tree? This point needs to be clarified.
L 190: Same as above. Do the authors use the pixel-by-pixel slope of the

DEM and then average over all pixels of the upstream basin? What is

the upslope along directions? These points need to be clarified.
L. 229-231: "... suggests that junction angles may be systematically

lower, on average, in fluvial networks incised into late-Cenozoic

alluvial piedmont deposits than those incised into adjacent areas of

bedrock/older deposits." The authors need to show that the structural

control on junction angle is not a spurious slope bias in gently

sloped depositional landscapes such as piedmonts.
L. 316: comparison with NHD: Comparing USGS's medium-resolution blue

lines with a fixed flow accumulation threshold extracted channel

network is not a fair comparison, particularly as one can obtain any

drainage density by adjusting the drainage area threshold. NHDPlus (

and its higher resolution companion NHDPlusHR have been extensively

ground-checked.
For example, I was unable to identify many of the fine channel

networks from Fig. 6b/d around 32.3N/110.9W on aerial images on Google

Maps.
L. 560: variability in hillslope length: Minor point: While I agree

with the presented arguments it remains to be shown that hillslope

convergence is the dominant factor controlling drainage density.
L. 579: I agree with the AI~elevation argument which should be easy to

test. (correlation elevation~AI and correlation angle~elevation)
L. 595: higher or less infiltration in arid landscapes: I agree with

the authors that the question of groundwater recharge and its

dependency on environmental conditions is rather

controversial. See for example a recent Nature Climate Change which

argues that "Groundwater recharge is sensitive to changing

long-term aridity" which suggests higher recharge in humid landscapes

than in arid ones based on lysimeter measurements.
L. 611: "The result presented here could potentially be reconciled ..."

..." It should be easy to test and prove or disproof if the presented

results are consistent with Hooshyar 2017 or not.
L 615: I agree with the author's similarity arguments between minimum

power and MGM, but while MGM only relates different aspects of Earth's

tomography with each other, the optimal branching models seek to

explain WHY the slopes are as they are from a physical mechanistic

perspective of landscape evolution in general and channel erosion in

particular. Here, the topographic slope itself is only a feedback

variable for flow accumulation in the erosional term of the landscape

evolution equation.

Citation: https://doi.org/10.5194/egusphere-2024-1153-RC1
RC2:
'Comment on egusphere-2024-1153', Anonymous Referee #2, 07 Jun 2024
The authors present an analysis of approximately 10 million junction angles, for which they extracted the corresponding stream network across the Contiguous United States using a 10m Digital Elevation Model, suggesting that the topography of a landscape prior to fluvial incision exerts a key constraint on tributary fluvial network junction angles. This is due to the fact that different slope versus microtopography assemblages give rise to more or less sinuous patterns in the valley bed shape, as suggested by Lazarus and Constantine (2013). Indeed, when computed in the traditional way, junction angles are a function of slope ratios (as the Howard 1990 model predicts), but data deviate from the Howard model in a manner that the authors propose is the result of valley-bottom meandering/tortuosity.
I’ve read the paper with much interest and found it extremely well-written and fascinating. Only sometimes did I find myself a bit lost in the lengthy technical discussions, but this might be entirely my fault, as I am not an expert on the topic discussed here. Also, I appreciate that the literature and scientific debate on junction angles in tributary fluvial networks are extensive, so previous arguments need to be discussed in detail.
Being not an expert, I’ll refrain from delving into the debate regarding what drives the distribution of junction angles and how this study copes with or does not cope with earlier literature. The pros and cons of the approach are discussed in sufficient detail, and results are put into perspective with previous literature in a way that allows even a non-expert reader to navigate through the discussion, although it is very technical.
Still, I’d like to raise two possibly minor comments, one being purely terminological and the other perhaps more fundamental.
First, I’d suggest the authors avoid using the term “meandering” to describe sinuous valley patterns. Meandering refers to landforms that grow orderly rather than randomly. The fact that sinuous patterns can arise even from unordered growth, and that at low sinuosity random versus ordered shapes are not distinguishable (Limaye et al. 2021).
Second, and more importantly, the major limitation I see in the idea behind the paper is that valley sinuosity is assumed to remain fixed once their shape and sinuosity have been determined by the initial microtopography (in the way Lazarus & Constantine's idea illustrates).

Kwang et al. (2021) have demonstrated how lateral erosion/incision of rivers into bedrock is fundamental for the development of dendritic drainage networks, to the point that without lateral erosion, landscape evolution models cannot turn non-optimal, non-dendritic networks into optimal dendritic ones such as those observed in nature.

For the point being made in this paper (that valley tortuosity matters when it comes to junction angles) I think lateral erosion should be considered/discussed because it has the potential to:
Alter river valley tortuosity over time, thereby impacting junction angles.

Cause the system to lose memory of initial conditions (inducing persistent reorganization in dynamic steady-state network shape), thus critically diminishing (if not completely erasing) the effect of initial microtopography on which the authors anchor all their analyses and results.

In short, assuming that river networks evolve from initial conditions to a frozen state with no further modifications poses a strong limitations to the new perspective the authroes bring abotu regarding the drivers of junction angles in fluvial tributary networks. Later network modification by lateral river erosion into valleys, which can induce major drainage reorganization and long transience in landscape shape, should be discussed more thoroughly because it’s potentially tied to how tributary channels intersect each others.

CITED REFERENCES:
Kwang, J. S., Langston, A. L., & Parker, G. (2021). The role of lateral erosion in the evolution of nondendritic drainage networks to dendricity and the persistence of dynamic networks. Proceedings of the National Academy of Sciences of the United States of America, 118(16), 1–6. https://doi.org/10.1073/pnas.2015770118
Lazarus, E. D., & Constantine, J. A. (2013). Generic theory for channel sinuosity. Proceedings of the National Academy of Sciences of the United States of America, 110(21), 8447–8452. https://doi.org/10.1073/pnas.1214074110
Limaye, A. B., Lazarus, E. D., Li, Y., & Schwenk, J. (2021). River sinuosity describes a continuum between randomness and ordered growth. Geology, 49(12), 1506–1510. https://doi.org/10.1130/G49153.1
Citation: https://doi.org/10.5194/egusphere-2024-1153-RC2
AC1: 'Author response to referee comments on egusphere-2024-1153', Jon Pelletier , 05 Jul 2024

See attached pdf

Citation: https://doi.org/10.5194/egusphere-2024-1153-AC1

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Jon Pelletier on behalf of the Authors (03 Aug 2024) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (13 Aug 2024) by Kieran Dunne

RR by Anonymous Referee #1 (12 Sep 2024)

Suggestions for revision or reasons for rejection

Referee report on the revised manuscript

"Geometric constraints on tributary fluvial network junction angles"

by Peletier et al.

The authors clarified many issues raised in the reviews in their revised manuscript.
However, one central point remains unresolved: how the particular spatiotemporal settings of the Piedmont deposits are crucial for creating low branching angles.

Regarding the question of structural control:
Piedmonts are geologic structures. The caption of Fig. 11, for example, reads: "Junction angles tend to be lower in areas with late-Cenozoic deposits." Thus, if the authors argue that specific properties of Piedmont deposits cause narrow branching angles, this is a clear structural argument.

If not, it is simply a slope/roughness effect and should also be presented as such.

For reporting the pure observation that piedmonts are gently sloped surfaces with small surface roughness and thus more elongated drainage basins with smaller branching angles, a one-page technical note would be sufficient, rather than an extensive article.

Although Casteltort did not explicitly refer to branching angles, his convergence argument for drainage basins equivalently applies to the networks therein. Moreover, Seybold et al. explicitly discussed the slope dependence, although they neglected the influence of surface roughness.

Thus, the main novelty discussed in the paper is the effect of surface roughness in combination with slope on valley network branching angles.

We all agree, that piedmonts are gently sloped surfaces with small tolographic roughess. However, as the authors show by using Landscape Evolution Modeling and analyzing channel networks of aeolian deposits on the Loess Plateau, the specific ground into which the channels are carved is irrelevant for the narrower branching angle argument. In fact, the LEM used for the analysis does not seem to depend on any surface properties and simply applies a classical stream power incision law. Consequently, ANY landscape (Fig.12) that fulfills the slope and roughness conditions displays narrower branching angles.
This point must be made very clear in a revised manuscript and not simply brushed away by "Sl/Sr values of which we did not study and don’t feel qualified to comment on."

Several other technical points also need to be explained more clearly so that a reader can understand the author's procedures and arguments.

Below, the authors can find more specific comments referring to particular sections of the text:

L 13: "when orientations and slopes are computed using drainage basins rather than …"
This sentence is still confusing. What is a basin's orientation? How is the basin's slope defined? Is it a basin regional slope, a mean roughness slope …, etc.? The authors should be more precise in their description.

L 16/17 Contrast between Piedmont deposits and bedrock: The model argument following this sentence is purely based on the roughness/regional slope arguments and thus does not include any specific properties of bedrock vs deposits.

Consequently, the argumentation should be reversed. The authors identified that regional vs. roughness slope controls branching angles, and depositional piedmonts are more likely to have gentle slopes with low surface roughness than mountainous bedrock regions.

L.86: Climate does not form channels. As far as I remember, Seybold et al.'s argument is based on the relative dominance of diffusive channel-forming processes, which, in turn, vary systematically with climate.

L155: How do the authors define drainage basin averaged direction and slope? The authors should clearly describe their procedure.

L 189: The authors say that the AVB model uses a search distance upstream/downstream until an elevation change of 10m is reached. In some cases, this can be multiple Kilometers. How do the authors consider cases when there is another junction Ustream?

L 191: Please clarify the orientation and slope of a single pixel is? Orientation of the steepest descent in a D-8 flow Routing scheme?

Hide

RR by Anonymous Referee #2 (19 Sep 2024)

ED: Publish subject to minor revisions (review by editor) (30 Sep 2024) by Kieran Dunne

AR by Jon Pelletier on behalf of the Authors (28 Oct 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (04 Nov 2024) by Kieran Dunne

ED: Publish as is (06 Nov 2024) by Paola Passalacqua (Editor)

AR by Jon Pelletier on behalf of the Authors (13 Dec 2024) Author's response Manuscript

Journal article(s) based on this preprint

12 Mar 2025

Geometric constraints on tributary fluvial network junction angles

Jon D. Pelletier, Robert G. Hayes, Olivia Hoch, Brendan Fenerty, and Luke A. McGuire

Earth Surf. Dynam., 13, 219–238, https://doi.org/10.5194/esurf-13-219-2025,https://doi.org/10.5194/esurf-13-219-2025, 2025

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Jon D. Pelletier, Robert G. Hayes, Olivia Hoch, Brendan Fenerty, and Luke A. McGuire

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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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On the gently sloping landscapes next to mountain fronts, junction angles tend to be lower (more acute), while in bedrock landscapes where the initial landscape or tectonic forcing is likely more spatially variable, junction angles tend to be larger (more obtuse). We demonstrate this using an analysis of ~20 million junction angles for the U.S.A., augmented by analyses of the Loess Plateau, China, and synthetic landscapes.


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