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**Earth Surface Dynamics**
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**Research article**
20 Apr 2020

**Research article** | 20 Apr 2020

Drainage divide networks – Part 2: Response to perturbations

^{1}GFZ German Research Centre for Geosciences, Section 3.3, Telegrafenberg, 14473 Potsdam, Germany^{2}Institute of Geological Sciences, Freie Universität Berlin, 14195 Berlin, Germany^{3}Institute of Environmental Sciences and Geography, University of Potsdam, 14473 Potsdam, Germany

^{1}GFZ German Research Centre for Geosciences, Section 3.3, Telegrafenberg, 14473 Potsdam, Germany^{2}Institute of Geological Sciences, Freie Universität Berlin, 14195 Berlin, Germany^{3}Institute of Environmental Sciences and Geography, University of Potsdam, 14473 Potsdam, Germany

**Correspondence**: Dirk Scherler (scherler@gfz-potsdam.de)

**Correspondence**: Dirk Scherler (scherler@gfz-potsdam.de)

Abstract

Back to toptop
Drainage divides are organized into tree-like networks that may record
information about drainage divide mobility. However, views diverge about how
to best assess divide mobility. Here, we apply a new approach of
automatically extracting and ordering drainage divide networks from digital
elevation models to results from landscape evolution model experiments. We
compared landscapes perturbed by strike-slip faulting and spatiotemporal
variations in erodibility to a reference model to assess which topographic
metrics (hillslope relief, flow distance, and *χ*) are diagnostic of
divide mobility. Results show that divide segments that are a minimum
distance of ∼5 km from river confluences strive to attain
constant values of hillslope relief and flow distance to the nearest
stream. Disruptions of such patterns can be related to mobile divides that
are lower than stable divides, closer to streams, and often asymmetric in
shape. In general, we observe that drainage divides high up in the network,
i.e., at great distances from river confluences, are more susceptible to
disruptions than divides closer to these confluences and are thus more
likely to record disturbance for a longer time period. We found that
across-divide differences in hillslope relief proved more useful for
assessing divide migration than other tested metrics. However, even stable
drainage divide networks exhibit across-divide differences in any of the
studied topographic metrics. Finally, we propose a new metric to quantify
the connectivity of divide junctions.

How to cite

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How to cite.

Scherler, D. and Schwanghart, W.: Drainage divide networks – Part 2: Response to perturbations, Earth Surf. Dynam., 8, 261–274, https://doi.org/10.5194/esurf-8-261-2020, 2020.

1 Introduction

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Divide migration is a time-dependent process that is difficult to quantify.
While the effects of regional-scale drainage captures may be preserved
within sedimentary archives (e.g., Clift et al., 2006), this is unlikely for
smaller-scale drainage captures or gradual divide migration. In such cases,
most studies rely on topographic indicators. Mobile divides are typically
inferred from post-drainage-capture evidence: distorted drainage structures,
low divides (wind gaps), or high tributary junction angles (e.g., Clark et
al., 2004) (Fig. 1). However, divide mobility may
also be expressed in the topography without major drainage captures or flow
reversals but as a result of the more gradual migration of divides. Willett
et al. (2014) inferred drainage divide mobility from across-divide
differences in *χ* values, a proxy for steady-state river channel
elevation (Perron and Royden, 2013). They argued that changes in drainage
area within mountain ranges, e.g., due to tectonic strain of the crust (Yang
et al., 2015), may commonly lead to relative differences in incision rate
and the formation of low-relief landscapes that are bordered by migrating
divides. Whipple et al. (2017a, b), however, argued that the timescale of such
changes is too short to profoundly affect mountainous landscapes. Instead,
they argued that transient low-relief landscapes, such as those in
southeastern Tibet, are more likely to be formed by regional changes in rock
uplift rate and upstream propagation of knickpoints between the adjusted and
unadjusted parts of landscapes. They also cast doubt on the ease of
comparing across-divide differences in drainage network geometry (i.e.,
*χ* values) where the common base level is far and opposing rivers
may incise into areas of different rock types, different rock uplift rates,
or different climates. Whipple et al. (2017a, b) instead proposed that the
shape of drainage divides themselves holds clues about their mobility
(Fig. 1). Amongst the topographic parameters that
they tested are across-divide differences in channel elevation at a
reference drainage area, mean headwater hillslope gradient, and mean
headwater local relief.

In summary, several different metrics have been proposed that may allow for the
quantification of divide mobility in both natural and modeled landscapes.
Forte and Whipple (2018) compared the performance of these metrics with a
landscape evolution model in which they induced divide mobility and
concluded that across-divide differences in relief or gradient better depict
divide motion than *χ*. In their analysis, however, they focused on
divide motion that is perpendicular to the regional drainage direction and
averaged divide migration rates as well as topographic metrics across the
entire width of the model domain so that each time step is associated with
single values for divide migration rate, erosion rate difference, and the
tested topographic metrics. In part 1 of this study (Scherler and
Schwanghart, 2020a), we presented a new approach for the automatic
identification and ordering of drainage divide networks in a digital elevation model (DEM), which
removes the necessity of manually selecting drainage divides for comparison.
Here, we present experiments with a numerical landscape evolution model that
we conducted to investigate how drainage divide networks respond to
different perturbations, including fault activity and differences in
erodibility. In contrast to previous studies that examined the response of
drainage divides to perturbations, we studied the entire drainage divide
network in an objective manner and examined how different portions of the
divide network respond to perturbations. In addition, we tested the utility
of a new metric that quantifies the connectivity of drainage divide
junctions.

2 Materials and methods

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We studied the response of divide networks to stream captures and divide migration using the TopoToolbox Landscape Evolution Model (TTLEM; Campforts et al., 2017). In our experiments, we modeled the topographic evolution of a 20 km × 20 km square block (50 m node spacing) subject to uniform rock uplift, stream-power-based fluvial incision (e.g., Howard and Kerby, 1983; Whipple and Tucker, 1999), and hillslope diffusion (e.g., Culling, 1963):

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{\mathrm{d}z}{\mathrm{d}t}}=U-{K}_{\mathrm{r}}{A}^{m}{S}^{n}-\mathrm{\nabla}{\mathit{q}}_{\mathrm{s}},\end{array}$$

where *z* is elevation (L), *t* is time (T), *U* is rock uplift rate (L T^{−1}),
*A* is upstream area (L^{2}), *S* is local channel slope (L L^{−1}), *K*_{r} is
a parameter of the efficiency of river incision (T^{−1}) (${K}_{\mathrm{r}}=\mathrm{1}\times {\mathrm{10}}^{-\mathrm{5}}$ yr^{−1}), and *m* and *n* are dimensionless constants with
values of 0.5 and 1, respectively. The last term on the right-hand side
depicts elevation change due to the divergence in diffusive hillslope
transport *q*_{s} (L^{3} L^{−1} T^{−1}),
which we consider to be a linear function of hillslope gradient:
${\mathit{q}}_{\mathrm{s}}=-D\mathrm{\nabla}z$, where *D* is the diffusivity
(L^{2} T^{−1}) of soil creep ($D=\mathrm{2}\times {\mathrm{10}}^{-\mathrm{3}}$ m^{2} yr^{−1}). All four edges of the block were fixed in elevation (*z*=0 m),
which forced rivers to flow outwards. The uplift rate (*U*=1 mm yr^{−1})
was constant in all models. Our choice of parameter values was guided by the
study of Whipple et al. (2017b), who tested a wide range of rock uplift and
erosional efficiency parameters and found almost no difference of divide
mobility in models with and without hillslope diffusion and for *n* values of 1
and 2.

We started from a flat surface with imposed random noise and ran the
experiment for 30 Myr until the topography reached a steady state. The
result of this model, which we termed “Initialize”, provided the starting
point for four other models that we ran for 10 Myr
(Fig. 2). The model “Reference” included no
further changes. In the model “Rotating”, we included a circular (10 km
diameter) left-lateral strike-slip fault that was active throughout the
experiment. Strike-slip faults are well known for enforcing drainage
captures and thus divide mobility (e.g., Castelltort et al., 2012; Duval and
Tucker, 2015). Although the rotating block has, to our knowledge, no
real-world equivalent, this model setup represents a convenient way of
simulating extended periods of strike-slip faulting, as the fault does not
intersect the model boundary (Braun and Sambridge, 1997). The fault slip
rate was fixed at 4 mm yr^{−1}, which corresponds to an angular velocity of
$\mathrm{8}\times {\mathrm{10}}^{-\mathrm{7}}$ rad yr^{−1}, resulting in ∼460^{∘} of
total rotation during the model run. We note that the rotating movement
requires interpolation and thus leads to numerical diffusion of elevations
within the rotating disk. However, the resulting change in total volume by
interpolation is <0.03 % of the volume uplifted during the same
time and therefore small. The model “Inclined” included 1 km thick and 5 km
spaced layers of 50 % reduced erosional efficiency of rivers (*K*_{r}),
dipping 30^{∘} towards northwest. The Inclined model is
representative of a landscape in which rivers incise into tilted sedimentary
rocks of nonuniform rock strength, similar to what has been studied by
Forte and Whipple (2018). During the experiment, the combination of surface
lowering and inclination resulted in the resistant layers regularly sweeping
from southeast to northwest across the simulated landscape. The model
“Spheres” included 30 randomly assembled spheres of 3 km diameter with
75 % reduced erosional efficiency of rivers (*K*_{r}). This experiment may
represent incision of a region that is characterized by country rocks with
more resistant magmatic intrusions. The expected behavior of this model is
similar to the landscape response to localized perturbations studied by
O'Hara et al. (2018).

We analyzed the modeled topography and the associated drainage divide
network. For each modeled topography and at each time step (d*t*=40 000 years), we first computed flow directions and flow accumulation, and
we subsequently identified the stream network using a drainage area threshold
of 0.2 km^{2}. We next derived the drainage divide network on the basis of
the stream network and using the algorithm proposed in Scherler and
Schwanghart (2020a). We calculated divide distances and divide orders
based on the Topo ordering scheme (Scherler and Schwanghart, 2020a). As
topographic metrics, we included elevation (*z*), hillslope relief
(HR), and horizontal flow distance to the stream network (FD). HR was measured as
the elevation difference between a point on the divide and the nearest river
location as measured by the distance along local flow directions. We also
computed *χ* values on either side of a divide using a reference area
*A*_{0} of 1 m^{2}, a reference concavity *θ*_{ref} of 0.45, and
setting the base level *x*_{b} to 0 at the edge of the model domain (e.g.,
Perron and Royden, 2013):

$$\begin{array}{}\text{(2)}& \mathit{\chi}=\underset{{x}_{\mathrm{b}}}{\overset{x}{\int}}{\left({\displaystyle \frac{{A}_{\mathrm{0}}}{A\left({x}^{\prime}\right)}}\right)}^{{\mathit{\theta}}_{\mathrm{ref}}}\mathrm{d}{x}^{\prime}.\end{array}$$

For each divide edge, we computed these topographic metrics and the
erosion rate (Eq. 1) for the two neighboring pixels that belong to adjacent
drainage basins and denoted the across-divide minimum, maximum, sum,
difference, and average in any one metric *X* as *X*_{min}, *X*_{max},
∑*X*, Δ*X*, and $\stackrel{\mathrm{\u203e}}{X}$, respectively. Erosion rates were based
on the erosion rate of the first downslope stream pixel to reduce the impact
of local noise along hillslopes. Topographic metrics of entire divide
segments are based on those of the divide edges that it is composed of. For
quantifying across-divide differences in topographic metrics and
erosion rates, irrespective of the actual values, we used normalized indices
of the form $\mathrm{\Delta}X/\sum X$. One such index that we frequently used in
our study is the divide asymmetry index ($\mathrm{DAI}=\left|\mathrm{\Delta}\mathrm{HR}/\mathrm{\Sigma}\mathrm{HR}\right|)$, which is the absolute value of the
normalized hillslope relief difference and which ranges between 0
(symmetric) and 1 (most asymmetric).

The above-described across-divide differences in topographic metrics
essentially aim to quantify divide mobility. In contrast, Spotila (2012)
studied the stability of divides and argued that divide junctions and
pyramidal peaks are more stable than solitary linear divides and might
therefore act as anchor points for drainage divide networks. He proposed
that divide junctions are more difficult to erode than linear divides due
to their greater volume of topography per unit area, their greater
mechanical stability, and their reduction of confluent flows (Spotila,
2012). He also suggested that the stability of divide junctions is related
to the number of joining drainage divides. Because the divide junctions obtained
from our algorithm cannot connect more than four divide segments (Scherler
and Schwanghart, 2020a) – and most often connect three segments – we
introduce a new metric to quantify divide junction connectivity, *C*_{J}:

$$\begin{array}{}\text{(3)}& {C}_{\mathrm{J}}={\displaystyle \frac{\mathrm{d}x}{{d}_{\mathrm{d},\mathrm{max}}}}\sum _{i=\mathrm{1}}^{n}{\displaystyle \frac{{d}_{i}}{{d}_{\mathrm{d},i}}}.\end{array}$$

We define *C*_{J} to correspond to the sum of the ratios of the Euclidean
distance, *d*, and the divide distance, *d*_{d}, of all divide edges, *n*, within a
specified maximum divide distance, *d*_{d,max}, times the ratio of the cell
size, d*x*, and *d*_{d,max}. The dimensionless quantity *C*_{J} is sensitive to
the number of divides within a given divide distance from a junction
weighted by their orientation towards the junction
(Fig. 3). The value *d*_{d,max} reflects the
divide distance over which differences in junction connectivity are
measured. For junctions that connect a constant number of straight and
infinitely long divide segments, *C*_{J} is not sensitive to the value of
*d*_{d,max}. However, for actual junctions, *C*_{J} is typically sensitive to
the value of *d*_{d,max} because as *d*_{d,max} grows, increasingly more
junctions are at a distance ${d}_{\mathrm{d}}<{d}_{\mathrm{d},\mathrm{max}}$ of a specific
junction, and thus the number of divide segments grows with *d*_{d}
(Fig. 3a). In general, *C*_{J} will be sensitive
to the position of a junction within the drainage divide network if the
junction's maximum divide distance from an endpoint is smaller than
*d*_{d,max}. In other cases, *C*_{J} will provide a measure of how connected a
junction is within a network or, in other words, how prominent the junction
is compared to other junctions in the network. In this study, we used a
*d*_{d,max} value of 5 km.

3 Results of numerical experiments

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The simulated landscapes, along with their drainage divide networks at the end of the numerical experiments, are shown in Figs. 4 and 5, and in the “Video supplement” (Scherler and Schwanghart, 2020b) we provide movies of all simulations. To provide a measure of the mobility of drainage divides, we computed the percentages of drainage area that were exchanged during the simulations between individual catchments that drain to the margin of the model domain (Fig. 6). Except for the Reference model, all models are characterized by notable changes in drainage area and mobile drainage divides. Area changes in the Initialize model are large in the beginning but level off rapidly during the first 1 Myr. Although area changes are small after 1 Myr, they continue for another 20 Myr, during which they are mostly decreasing. In the Rotating model, large area changes appear as discrete pulses induced by drainage captures of major streams (Fig. 5b), whereas the background area changes during rotation and faulting are relatively small (<0.1 % per 40 kyr). Area changes in the Inclined model are moderate (∼0.25 % per 40 kyr) throughout the simulation and oscillate in conjunction with the passage of more resistant layers through the landscape (Fig. 5c). Area changes in the Spheres model are generally more pronounced if the resistant spheres appear in the course of rivers, which forces them to steepen and to induce surface uplift upstream as opposed to their appearance at drainage divides, which increases the height of the divide but does not induce drainage divide migration at a larger scale (Fig. 5d).

We first analyzed the response of the entire drainage network topology to the perturbations by quantifying the aggregated length of divide segments as a function of their order (Fig. 7). The first few million years of the Initialize model are characterized by large changes in divide lengths and orders. Initially, the divide network extends to orders as high as 100 but rapidly contracts as the drainage network becomes dendritic. After about 5 Myr, the highest orders are down to 60. Subsequent changes result in some scatter of the divide lengths but not in the range of divide orders. Compared to the Reference model, in which the divide network structure no longer changes, the Rotating, Inclined, and Spheres models exhibit changes in the divide network, mostly at divide orders greater than ∼20. This observation is related to the fact that low-order divides are distributed across the entire model domain and their number is accordingly high. Any of the perturbations we imposed only affect some of these divides, and thus the impact on their average length is rather small. In contrast, high-order divides are constrained to the highest parts of the modeled land surface and their numbers are much lower. The imposed perturbations typically affect a greater portion of them and hence the scatter in divide lengths is wider. In the Rotating and Spheres models, we also observed that maximum divide orders occasionally extend to higher values, but these changes are rather small. We note that the above observations also prevail when considering divide distance instead of divide order because the two are linearly related (Scherler and Schwanghart, 2020a), with divide distance ∼430 m × divide order. The 430 m corresponds to the mean length of the divide segments.

We next studied how the above-described disturbances affect drainage divide
metrics during the simulations (Fig. 8). For all
models, we computed the averages of the topographic parameters measured at
drainage divides of specific divide distance intervals
(Fig. 8a–d). As in the analysis of divide-segment
lengths by order, it should be kept in mind that the numbers of divide
segments, or their aggregated lengths (Fig. 7),
are much higher for low orders and distances compared to higher ones. For
reference, a divide order of 20 corresponds to a divide distance of
approximately 9 km. In the Initialize model, all of the studied metrics
attain a constant value that remains unchanged in the Reference model
(Fig. 8) and that may or may not depend on the
divide distance. For example, the mean elevation and junction connectivity
(*C*_{J}) clearly increase with divide distance, whereas the flow distance
exhibits only minor dependence on divide distance, and hillslope relief
appears unrelated to divide distance. The dependency of some metrics on
divide distance is partly explained by the model setup. Although divides
with low distances also occur at higher elevation, the bulk of them are near
the model edge, close to zero elevation. In contrast, divides at high
distances are exclusively found near the center of the model, where
elevations are also high. Similarly, the junction connectivity (*C*_{J}) is
high in the model center, where the divides are far from most of the
endpoints, which are more abundant near the edges of the model.

It is also worth noting that none of the normalized across-divide
differences in the topographic metrics attain zero values in the Reference
model. This means that even at topographic steady state, there are
residual across-divide differences in hillslope relief, flow distance, and
*χ*. In the case of *χ*, these also depend on the divide distance
and are greater closer to the model edge, where divide distances are low. In
the perturbed models, we observed fluctuations in all topographic metrics,
although of different magnitudes. For example, a comparison of across-divide
differences in erosion rate with differences in hillslope relief, flow
distance, and *χ* (Fig. 8e–h) shows that the
normalized difference in hillslope relief (i.e., the divide asymmetry index)
is sensitive to drainage divide mobility in all perturbed models, whereas
across-divide differences in *χ* and flow distance are sensitive to
divide mobility in the Rotating model but less so in the Inclined and
Spheres models. The junction connectivity (*C*_{J}) metric attains
temporally averaged values in the perturbed models that are quite similar to
the constant values in the Reference model. In many cases, the deviations
from the Reference model are greater the higher up in the divide network,
i.e., for higher divide distances. This pattern is particularly visible
in the Inclined model, wherein the amplitudes of the oscillations in all of
the parameters increase with divide distance.

Motivated by the observation of constant values in hillslope relief and flow
distance in the Reference model, as well as in actual landscapes (Scherler
and Schwanghart, 2020a), and by our expectation that small values in either
one would be found where one catchment loses area to another
(Fig. 1), we next compared how minimum hillslope
relief (HR_{min}), minimum flow distance (FD_{min}), and the divide
asymmetry index (DAI) vary with divide distance in the Reference model and the
three models with landscape perturbances (Fig. 9). In contrast to the average values in Fig. 8,
we provide these metrics for all divide edges during the last 1 Myr of the
model runs. Note also that we plotted data points in an order that brings
high DAI values to the front to better assess where asymmetric divides are
located, but in all four models, relatively high DAI points may plot on
top of low DAI points. In the Reference model, both minimum hillslope relief
and minimum flow distance reach relatively steady values (HR_{min}
∼250–350 m; FD${}_{min}\sim \mathrm{400}$–600 m) at a divide
distance of ∼5 km. At lower divide distances, both
HR_{min} and FD_{min} approach zero – simply because divides are defined to
start at the stream network – and these divides can become increasingly
more asymmetric. It is notable, however, that some of the highest
HR_{min} and FD_{min} values are also observed at low divide distances of
approximately 1–2 km. In the three other models, the transition between
quite variable divides at short distances and more steady “background”
values at higher distances appears to be preserved, but we observe generally
more variability. For example, in the Rotating model, we observe divides with
significantly lower HR_{min} and FD_{min} values at higher distances. These
divides are particularly prominent at a distance of ∼10 and ∼15–22 km and correspond to the position of the
strike-slip fault. Where HR_{min} and FD_{min} are low, DAI values are
relatively high (divides are highly asymmetric), although there are also
divides that have high DAI but regular HR_{min} and FD_{min} values. In the
Inclined and Spheres models, the HR_{min} and FD_{min} values are never as
low as in the Rotating model at divide distances >5 km, which
reflects the lack of drainage captures. Instead, we observe frequent
excursions to both higher and lower HR_{min} and FD_{min} values, either
across all divide distances (Inclined) or at specific locations (Spheres).
Deviations from the average values in the Reference model are greatest in
the Spheres model compared to all other perturbed models and always
correspond to peaks that grow where strong spheres are exhumed. In the three
disturbance models, DAI values are generally higher than in the Reference model
– although divides with high HR_{min} and FD_{min} values and at great
divide distances mostly appear to have somewhat lower DAI values.

The spatial pattern of divide junction connectivity (*C*_{J}) values at the
end of the landscape evolution experiments (Fig. 10) partly follows the pattern of divide distances
(Fig. 4). Junctions with higher *C*_{J} values
tend to occur at higher elevation and at greater divide distance
(*d*_{d}). In the Reference model, the highest *C*_{J} values occupy the
center of the model domain and the centers of the four quadrants of
the model domain, resembling the five on six-sided dice. In the Rotating
model, these clusters of high *C*_{J} values are maintained, but their
connection is disrupted by the strike-slip fault, which induces low
*C*_{J} values. The centrally located divide junctions occupy a similar range
in *C*_{J} values but are all shifted to higher elevations
(Fig. 4). Similar, although lower, offsets to
higher elevations occur in the Inclined and Spheres models, where junctions
coincide with rocks of reduced erodibility. In the Spheres model, the basic
structure of *C*_{J} values is similar to that of the Reference model, but
the highest *C*_{J} values are steered towards the less erodible spheres,
whereby they also attain *C*_{J} values that are distinctly higher than in any
of the other models (Fig. 10b). In general,
divide junctions with combinations of elevation and *C*_{J} values that are
outside the range of values observed in the Reference model are found in the
most disturbed parts of the landscape (Fig. 10c).
In summary, the perturbed models appear to induce mostly changes in junction
elevation, whereas changes in junction connectivity (*C*_{J}) are seemingly
constrained to the Spheres model.

4 Discussion

Back to toptop
The analysis of stream networks has become a standard tool for inferring
tectonic forcing and landscape history (e.g., Wobus et al., 2006; Kirby and
Whipple, 2012; Demoulin, 2012; Schwanghart and Scherler, 2014). The divide
network holds the potential to record similar tectonic forcing, but also
other aspects of landscape history (e.g., Willett et al., 2014). The
question is which divide metrics are useful to analyze, and what do they tell
us about landscape history? Our Rotating model induced relatively sudden
drainage captures (Fig. 6). Because such events
are associated with the dissection of drainage divides, reliable indicators
are values of hillslope relief (HR) and flow distance (FD) that are much lower
compared to the values that divides ($>\sim \mathrm{5}$ km
divide distance) strive for (Fig. 9). More
gradual divide migration, however, likely lacks such simple diagnostic
criteria, and in those cases, across-divide differences in topographic
metrics may be more suitable indicators of divide mobility. The most
commonly used metric to infer drainage divide mobility is the across-divide
difference in *χ* (Willett et al., 2014). Although the utility of this
metric has recently received some critique (Whipple et al., 2017b; Forte and
Whipple, 2018), it has become a popular tool for studying drainage divides.
Whipple et al. (2017b) and Forte and Whipple (2018) instead advocated the
use of other topographic metrics, including mean gradient, mean local
relief, and channel bed elevation, measured at or upstream of a reference
drainage area. We note that these latter metrics are typically highly
correlated and very similar to the hillslope relief and DAI metrics that we
included in this study.

Figure 11 shows how normalized across-divide
differences in *χ*, hillslope relief (HR), and flow distance (FD) compare to
normalized across-divide differences in erosion rate (ER), evaluated for each
divide edge from the last 1 Myr of the landscape evolution experiments
Reference, Rotating, Inclined, and Spheres. We find that across-divide
differences in hillslope relief (HR; Fig. 11a) are
most sensitive to the disturbances included in the models, whereas
across-divide differences in *χ* are similarly sensitive to disturbances
in the Rotating model but less so in the Inclined and Spheres models
(Figs. 11b, 8).
Across-divide differences in flow distance (FD) are the least sensitive to
disturbances in the models and show the largest scatter when compared with
erosion rates (Fig. 11c). However, there is also
substantial scatter in the relationship between across-divide
differences in hillslope relief and erosion rate, which partly depends on
the divide distance. In general, we observe that the scatter is higher for
divide distances $<\sim \mathrm{5}$ km (dark blue in
Fig. 11), which corresponds to the value below
which we observe large variability in divide morphology, even in the
Reference model (Fig. 9). To quantify the
correlation of the normalized across-divide differences in topographic
metrics with normalized across-divide differences in erosion rate, we fitted
a linear model to all drainage divide edges from the entire model runs,
categorized into 1 km divide distance bins, and show the resulting
coefficients of determination (*R*^{2}) in Fig. 12. As already suspected from Figs. 8 and 11, the *R*^{2} values differ between
models and metrics and also depend on divide distance. In general, we
observe that all metrics perform poorly at divide distances $<\sim \mathrm{5}$ km and that across-divide differences in flow distance
perform poorly even at higher distances. The highest *R*^{2} values are
linked to across-divide differences in hillslope relief, whereas
across-divide differences in *χ* attain similar *R*^{2} values in the
Rotating model but some of the lowest *R*^{2} values of all metrics in the
Inclined and Spheres model. This difference may be explained by the fact
that in the latter two models, we introduced spatial variability in the
erosional efficiency of rivers (*K*_{r}) that we did not account for in our
across-divide comparison of *χ*, as would be required (Willett et al.,
2014). In natural landscapes, however, these values and their variability
are rarely well known.

We speculate that the influence of divide distance on topographic metric–erosion rate relationships may also account for the differences in scatter observed by Sassolas-Serrayet et al. (2019) in landscape evolution experiments similar to our Initialize model between larger and smaller basin areas. But even when excluding divides of low order or low divide distance, we still observe considerable scatter in the topographic metric–erosion rate relationships, which, at the very least, demands caution when interpreting divide morphology in terms of mobility. In this regard, studying Fig. 5 and the videos of the landscape evolution experiments (see the “Video supplement”; Scherler and Schwanghart, 2020b) is insightful: where drainage divides are migrating, one typically observes a range of across-divide topographic metric values that vary considerably during the migration. In other words, despite a continuous divide migration at a large scale, there is often small-scale variability in divide morphology that may in part be related to across-divide differences in topographic metrics lagging behind across-divide differences in erosion rate.

As a final note, we emphasize that the above observations are from our numerical experiments, which depict an idealized world. It is clear that the complexities present in nature, such as anisotropic and variable rock properties, hydroclimatic gradients, mass-wasting events, and biological influences on erosion processes and rates, can lead to landscape patterns that bias any of the above topographic metrics and need to be taken into account when inferring divide dynamics from divide metrics in natural landscapes.

Stream networks tend to attain configurations that are in equilibrium with the geological and climatic environment, given an initial condition (e.g., Rinaldo et al., 2014). Because drainage divides are defined by adjacent drainage basins, the geometry of divide networks should attain a similar equilibrium, which expresses itself in both the geometry of divides and the topology of divide networks. Our numerical experiments have shown that during the initial establishment of a stream network, on a relatively flat surface, both stream and divide networks are far from their steady-state configuration and characterized by networks that extend to high orders (Fig. 7) and long divide distances. During the subsequent extension and shrinkage of individual streams towards their steady-state configuration, the divide network contracts and primarily high-order divide segments shorten and become fewer, whereas divides of low orders maintain their frequencies (Fig. 7).

In general, divide segments of high order, i.e., at great distance from endpoints, appear to be the most responsive to landscape disturbances (Fig. 8). In the case of the Rotating model, this is in part expected because the inner rotating part of the landscape contains the highest-order divide segments (Fig. 4b). In the cases of the Inclined and Spheres models, it may be related to the increased probability of recording a disturbance because the adjoining basins cover a larger area compared to lower-order divides. In other words, if drainage captures happen somewhere within a drainage basin, this will most likely influence divides further upstream. Over a distance of less than ∼5 km from divide network endpoints, the divide segments transition from low interfluves at river junctions to high topographic ridges, as seen in the Reference model (Fig. 9). In the other models, most of the investigated morphometric parameters are quite variable over the same distance and can be seen to rapidly adjust to disturbances such as drainage captures or migrating divides. Such behavior is consistent with the observation that the timescale of a river's response to changes in drainage area increases with the distance from the divide to the outlet of a river (Whipple et al., 2017b). To reliably distinguish the morphologic effects of real disturbances from “noise” close to the river, a minimum divide distance of perhaps ∼5 km, as in our analysis of the Big Tujunga divide network (Scherler and Schwanghart, 2020a), appears appropriate. This minimum divide distance could be lower or higher, depending on factors like drainage density and average hillslope relief, for example.

Our new junction connectivity index (*C*_{J}) complements existing
topographic metrics in assessing divide network dynamics. For example, the
junction connectivity in our Rotating model is low along the fault
(Fig. 10), consistent with the absence of stable
divides. In the Spheres model, however, the appearance of more resistant
rocks at the surface often resulted in the migration of divides towards the
spheres (Fig. 5c, “Video supplement”; Scherler
and Schwanghart, 2020b). In this case, parts of the drainage divide network
were mobile, not stable, but they moved towards particularly stable portions
in the landscape. Therefore, the junction connectivity index (*C*_{J}) may
also be interpreted as an attractor or centrality index (Phillips et al.,
2015) that quantifies how strong a drainage divide network has been pulled
towards and anchored at a certain junction (Spotila, 2012).

5 Conclusions

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Based on landscape evolution model experiments in which we forced divides to migrate, we found that stable drainage divides strive to attain a constant hillslope relief and flow distance from the nearest stream, provided a sufficiently large divide distance to avoid confounding influences near the edges of the divide network. In our experiments this distance is ∼5 km from endpoints. Simple indicators of mobile divides are anomalously low hillslope relief or flow distance values, which could signal beheaded valleys or future capture events. Overall, drainage divides located high up in the network, i.e., at great distance from endpoints, are more vulnerable than divides closer to endpoints of the network and are more likely to record disturbance for a longer time period. In our comparison of different topographic metrics to assess drainage divide mobility, we found that across-divide differences in hillslope relief proved more useful for assessing divide migration than other tested metrics.

Code availability

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Code availability.

The divide algorithm developed in Scherler and Schwanghart (2020a) has been implemented in the TopoToolbox v2 (Schwanghart and Scherler, 2014). The codes will be made available with the next TopoToolbox release and shall be accessible at https://github.com/wschwanghart/topotoolbox (last access: 17 April 2020).

Video supplement

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Video supplement.

The “Video supplement” related to this article is available online at https://doi.org/10.5880/GFZ.3.3.2019.005 (Scherler and Schwanghart, 2020b).

Author contributions

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Author contributions.

DS conducted the modeling and led the writing of the paper. Both authors contributed to discussions, editing, and revising the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank two anonymous reviewers for constructive comments that helped improve the paper.

Financial support

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Financial support.

This research has been supported by the Deutsche Forschungsgemeinschaft (DFG; grant no. SCHE 1676/4-1).

The article processing charges for this open-access

publication were covered by a Research

Centre of the Helmholtz Association.

Review statement

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Review statement.

This paper was edited by Sebastien Castelltort and reviewed by two anonymous referees.

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Short summary

Drainage divides are believed to provide clues about divide migration and the instability of landscapes. Here, we present a novel approach to extract drainage divides from digital elevation models and to order them in a drainage divide network. We present our approach by studying natural and artificial landscapes generated with a landscape evolution model and disturbed to induce divide migration.

Drainage divides are believed to provide clues about divide migration and the instability of...

Earth Surface Dynamics

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