the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Geometric constraints on tributary fluvial network junction angles
Abstract. The intersection of two nonparallel 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 junctionangle data support this geometric model, yet junction angles have also been proposed to be controlled by climate and/or optimality principles (e.g., minimumpower expenditure). In this paper, we document a test of the Howard (1990) model using ~10^{7} 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 valleybottom 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 valleybottom 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 lateCenozoic 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 largescale slope/tilt are associated with lower mean junction angles compared to landforms with higher such ratios. Using modern analogs, we demonstrate that unincised lateCenozoic alluvial piedmonts likely had ratios of mean microtopographic slope to largescale 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 optimalitybased principle.
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Status: final response (author comments only)

RC1: 'Comment on egusphere20241153', 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 slopeangle 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
missperceptions 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 threedimensional
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 angleslope relation foundon
more processbased descriptions of how the flow in a channel may
modify its bed, even if the particular expression may result in a
slightly lower rsquare, 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 channelforming 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 slopeangle relation
contradicts the abovementioned 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 angleslope 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 910: 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
processbased 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. 1112: "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.1725: 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 climatedriven
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 subparallel 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 lowrelief 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.5658: 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 nth 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 pixelbypixel 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. 229231: "... suggests that junction angles may be systematically
lower, on average, in fluvial networks incised into lateCenozoic
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 mediumresolution 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
groundchecked.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
longterm 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/egusphere20241153RC1 
RC2: 'Comment on egusphere20241153', 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 valleybottom meandering/tortuosity.
I’ve read the paper with much interest and found it extremely wellwritten 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 nonexpert 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 nonoptimal, nondendritic 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 steadystate 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/egusphere20241153RC2  AC1: 'Author response to referee comments on egusphere20241153', Jon Pelletier , 05 Jul 2024
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