Basin-averaged denudation rates may locally exhibit a wide dispersion, even in areas where the topographic steady state is supposedly achieved regionally. This dispersion is often attributed to the accuracy of the data or to some degree of natural variability of local erosion rates which can be related to stochastic processes such as landsliding. Another physical explanation of this dispersion is local and transient disequilibrium between tectonic forcing and erosion at the scale of catchments. Recent studies have shown that basin divide migration can potentially induce such perturbations, and they propose metrics to assess divide mobility based on cross-divide contrasts in headwater topographic features. Here, we use a set of landscape evolution models assuming spatially uniform uplift, rock strength and rainfall to assess the effect of divide mobility on basin-wide denudation rates. We propose using basin-averaged aggressivity metrics based on cross-divide contrasts (1) in channel

Topographic steady state, in which average topography is constant over time, is one of the key concepts of modern geomorphology (e.g. Gilbert, 1877; Hack, 1960; Montgomery, 2001). Though simple, this paradigm provides a useful framework to study landscape evolution related to tectonic and/or climatic forcing (e.g. Willett et al., 2001; Reinhart and Ellis, 2015), to spatial variations in rock strength (Perne et al., 2017) or to the geometry of active crustal structures (Lavé and Avouac, 2001; Stolar et al., 2007; Scherler et al., 2014; Le Roux-Mallouf et al., 2015). To define topographic steady state, the temporal and spatial scales of the processes involved are essential parameters. Compared to large-scale geodynamic processes operating over 1–100 Myr timescales, river incision and sediment transport are rapid processes driving landscapes to stable forms over this long timescale, whereas rapid climatic fluctuations during the Quaternary may prevent the occurrence of steady-state conditions in modern landscapes (Whipple, 2001).

The timescale of divide migration has received increasing attention in recent years. Although rivers exhibit a rapid adjustment to tectonic or
climatic changes to maintain their profiles, Whipple et al. (2017) show that
divides continue to migrate over time periods of 10

As an example, in the Great Smoky Mountains in the southern Appalachians,
uplift and erosion rates integrated over varying time periods from 10 kyr
to 100 Myr give a similar average magnitude of ca. 0.03 mm yr

Basin-wide denudation rate variability as a function of drainage
area in the Great Smoky Mountains. Original dataset from Matmon et al. (2003a, b); denudation rates reprocessed by Portenga and Bierman (2011).
Dashed black line shows the estimated background uplift rate for the region
of 0.03 mm yr

More recently, to characterize divide migrations, Forte and Whipple (2018)
introduced other metrics, referred to as “Gilbert metrics” (Gilbert, 1877),
based on the cross-divide contrast in channel local gradient and
height. This last study indeed focused on cross-divide contrasts in
headwater basin shape. Here, we propose extending these approaches by
modelling divide migration and by developing new metrics to assess divide
stability at the scale of the entire watershed, which are an expansion of
the aggressivity metric initially suggested by Willett et al. (2014). We use
these metrics to assess the effect of persistent divide mobility on
basin-averaged erosion rates at a timescale of 10

We use TTLEM (TopoToolbox Landscape Evolution Model) (Campforts et al.,
2017), a landscape evolution model based on the MATLAB function library
TopoToolbox 2 (Schwanghart and Scherler, 2014). This LEM uses a finite
volume method (Campforts and Govers, 2015) to solve the following equation
of mass conservation for rock or regolith subject to uplift and denudation:

Since the computation is performed using a discretized land surface, smaller
mesh sizes lead to detailed topography but lengthen the computation time and
memory requirements. Hereinafter, we consider a reference square landscape
model of 50 km side with a grid resolution of 90 m, which is a good
compromise between computation time (3–5 h on a PC workstation) and the
total amount of basins that can be studied (

In order to isolate the effect of divide migrations on the variability of
basin-wide denudation rates, we explore simple models with constant and
spatially uniform uplift and precipitation rates, and we assume no horizontal
advection,

Firstly, we consider a reference model with parameters commonly used for
moderately active orogens: an uplift rate

Secondly, all other parameters held constant, we investigate the specific
impact of uplift rate, erodibility and hillslope processes in other models
by varying

In order to better constrain the variability of our results under similar conditions, we ran for each model five simulations using the same parameters but with different initial random topographies.

The total duration of simulations is 10 Myr. The implicit scheme used to
simulate linear hillslope processes provides stable solutions regardless of
the time step. In contrast, the explicit scheme used to model fluvial
incision requires a time step that satisfies the Courant–Friedrich–Lewy
criterion. Hereinafter, we choose a time step

We derive basins from the synthetic DEMs (digital elevation models) using an
accumulation map computed with a single flow direction algorithm implemented
in TopoToolbox (Schwanghart and Scherler, 2014). Next, we calculate for each
basin the variation in average elevation over a time interval of 10 kyr. The
drainage network migrates during the simulation, so we only survey the
basins that keep the same outlet location during this time interval.
Furthermore, due to divide mobility, the geometry of watersheds can also
change. Hence, we measure the average difference in elevation inside the
basin perimeter after 10 kyr. Here we only assess the surface uplift

Most recent studies have focused on the relationship between drainage divide
mobility and headwater cross-divide contrast in either

First, we assess

Conceptual relationship between cross-divide contrast in

Then, we calculate the difference in metrics (

A detailed analysis of the DEM suggests that during the initial phase, the flat initial surface (Fig. 3a) is progressively uplifted to form a plateau. At the same time the edges of this plateau are gradually regressively eroded by drainage networks that spread from the base level toward the centre of the model (Fig. 3b and c). This transient landscape is completely dissected after 2 Myr. From this time and until the end of the simulation, landscape changes are mainly due to competition between watersheds, resulting in continuous divide migrations with decreasing intensity as the model is moving toward a total topographic equilibrium (Fig. 3d–f; video no. 1 in the Supplement).

Map view of the temporal evolution of the reference model. Colour bar gives the model elevation. Black lines show the evolution – and the migration – over time of drainage divides for five drainage basins. Red
circles in

To define the time period of regional steady state, we measure the average elevation, the maximum elevation and the average denudation rate over the entire model for each time step (Fig. 4a). We identify two distinct stages during the evolution of our reference simulation. During the first million years, due to long wavelength topographic building, the calculated landscapes are far from steady state. This leads to a major increase in the mean elevation from ca. 25 to ca. 75 m. In a second stage, this trend reverses and the mean elevation decreases asymptotically toward ca. 60 m until the end of the simulation.

Evolution of the reference model over time.

The evolution of the maximum elevation follows the same pattern but can be affected by temporal changes in the location and altitude of highest peaks. The maximum elevation increases between ca. 50 and ca. 250 m over the first 3 Myr (Fig. 4a) then decreases progressively to remain at ca. 200 m during the rest of the simulation.

We compute the average denudation rate from the rock uplift rate and from
average elevation change over the entire model between two time steps:

Based on these results, we will consider that a regional topographic
steady state is reached between 1.5 and 2 Ma, when the plateau relict
topography is totally eroded and

Variability of denudation rates over time for a compilation of five simulations of the reference model with different initial noised DEM.

We calculate basin-wide denudation rates

The spatial variability of the denudation rates is neither homogeneous nor
randomly distributed (Fig. 6a). The location of drainage basins with
denudation rates far from the equilibrium value of 0.1 mm yr

Denudation rates and cross-divide contrast metrics obtained for the reference model after 2.5 Myr of simulation. Drainage network is extracted from a minimal drainage area of 1 km

Willett et al. (2014) showed that the basin-averaged cross-divide contrast
in

We here assess the relationship between the

Denudation rates normalized by uplift rate as a function of aggressivity metrics and parameters influencing data dispersion.

In natural settings, the stage of evolution of landscapes cannot be easily
defined and the total amount of basins with a specific size may be limited.
The large dataset from our modelling can provide further insights by
gathering the results obtained every 0.5 Myr for seven classes of basin
areas expanding geometrically with a multiplying factor of 2 from 1–2 to
64–128 km

Sensitivity to uplift rates. Colour scale indicates basin area. Basins of variable sizes are sorted into seven area classes expanding geometrically with a multiplying factor of 2 from 1–2 to 64–128 km

The reference model involves various parameters related to uplift, fluvial
incision and hillslope denudation. A systematic analysis of trade-offs
between all parameters is out of the scope of this article. In this
section, we assess the sensitivity of the results to both tectonic and
erosion processes by studying the specific impact of uplift

We test rock uplift rates of 0.01, 0.1 (hereafter called reference model) and 1 mm yr

Maximum variability of

Effect of erodibility. Colour scale indicates basin area. Basins of
variable sizes are sorted into seven area classes expanding geometrically with a
multiplying factor of 2 from 1–2 to 64–128 km

Fluvial erosion is proportional to the erodibility coefficient

Hillslope denudation is proportional to the diffusivity coefficient

Effect of diffusivity. Colour scale indicates basin area. Basins of variable sizes are sorted into seven area classes expanding geometrically with
a multiplying factor of 2 from 1–2 to 64–128 km

Effect of critical slope. Colour scale indicates basin area. Basins of variable sizes are sorted into seven area classes expanding geometrically with a multiplying factor of 2 from 1–2 to 64–128 km

Altogether, these sensitivity tests demonstrate the robustness of our findings. Regardless of the tested parameter values, we observe a relationship between aggressivity metrics and deviation of denudation rates from uplift rates. Thus, aggressivity metrics are, to the 1st-order, reliable metrics to assess the effect of divide mobility on basin-wide denudation rates inferred from simulations. In the following section, we apply this approach to field observations and discuss the consequences for sampling and interpretation.

Over the last decades, measurements of cosmogenic radionuclide (CRN) concentrations in alluvial sediments (see Granger et al., 2013, and references therein), of suspended sediments (Gabet et al., 2008) and of detrital thermochronology (Huntington and Hodges, 2006) have become common practices to assess basin-wide denudation rates. However, their interpretation remains debated, even in settings where topographic steady state is supposedly achieved regionally.

Normalized denudation rates in the Great Smoky Mountains as a function of aggressivity metrics. Original dataset from Matmon et al. (2003b) with

As previously mentioned (Matmon et al., 2003a, b), while the Great Smoky
Mountains in the southern Appalachians are expected to be in a
quasi-topographic steady state, basin-wide denudation rates show a strong
dispersion up to a factor of 2 in comparison to the estimated uplift rate
(ca. 0.03 mm yr

Based on both our simulations and this field dataset, we propose favouring
the use of

Topographic steady state is a very convenient assumption and concept to deduce the uplift pattern in mountains ranges from denudation rates, and thus to obtain significant information on the geometry of active structures and on orogen dynamics (Lavé and Avouac, 2001; Godard et al., 2014; Scherler et al., 2014; Le Roux-Mallouf et al., 2015). However, this assumption is seldom verified at the scale of sampled watersheds.

On the basis of our modelling, we show that the competition between low-order basins has a significant impact on basin-wide denudation rates. The proposed approach provides a new tool to assess the potential deviation from topographic steady state based on aggressivity metrics and drainage area, which can both be inferred from a simple DEM: the closer to zero the aggressivity metrics and the lower the standard deviation of cross-divide metrics, the more representative of uplift rate the measured denudation rates.

Deviation from steady state due to drainage migration as a function of basin size. Colour lines show the maximum dispersion of denudation rates (0.5 and 99.5 percentiles) due to divide mobility. Green lines indicate the reference model. Seven sets of basin size are considered: 1–2, 2–4, 4–8, 8–16, 16–32, 32–64 and 64–128 km

Basin-wide denudation rates obtained from CRN concentration measurements, suspended sediments or detrital thermochronology depend on many parameters including lithology, ice cover, rainfall, landslide activity or tectonic uplift (Vance et al., 2003; Bierman and Nichols, 2004; Wittmann et al., 2007; Yanites et al., 2009; Norton et al., 2010; Godard et al., 2012; Whipp and Elhers, 2019). Hence, to unravel the influence of tectonics from other processes, a specific sampling strategy is usually recommended: (1) to sample catchments with homogeneous lithologies to limit the effect of spatial variations in the abundance of target minerals in bedrock formations; (2) to select catchments with no ice cover (past or present) because the input of glacier-derived sediments can significantly complicate the interpretation of CRN concentrations; (3) to choose areas with spatially uniform rainfall distribution; and (4) to consider watersheds where the relative contribution of landslides to long-term landscape evolution is low. Unfortunately, these different criteria imply selection of watersheds with variable sizes. The first three criteria favour the sampling of small catchments, whereas the last one requires basins large enough to be less affected by landslides.

Our approach suggests the need to pre-assess targeted basins for their potential divide mobility before sampling for CRN concentration measurements. If the objective is to quantify the background uplift rate, one should sample basins that satisfy the conditions we previously described in the current section and also display an aggressivity close to zero and with the smallest associated standard deviation. Conversely, to quantify the specific denudation rate associated with the migration of drainage divides, small aggressor or victim basins should be favoured.

Based on our simulations, a relationship between the maximum of erosion
variability (0.5 and 99.5 percentiles, respectively) due to divide mobility

Calculations from a landscape evolution model assuming spatially uniform uplift, rock strength and rainfall confirm that the concept of topographic steady state is relevant at the scale of entire mountain belts, but this represents an oversimplification at the scale of individual watersheds. Our simulations underline the role of divide mobility on deviations from equilibrium, which can lead to significant differences between tectonic uplift rate and basin-wide denudation rates even if an overall topographic steady state is achieved at large scale.

To better assess these deviations, we propose new basin-averaged
aggressivity metrics –

For the sake of simplicity our models involve spatially homogenous and time
invariant parameters. Additional simulations are now needed to test this
approach in more complex settings, including spatial and temporal
variability in climate and tectonic forcing or parameters like stream power
equation exponents

The data that support the findings of this study are available from the corresponding author on request.

The supplement related to this article is available online at:

RC and MF initiated this study. TSS performed the simulations and topographic analyses. All authors contributed to the writing of the paper.

The authors declare that they have no conflict of interest.

We are greatly indebted to referees Fiona Clubb and Adam Forte for providing constructive reviews that significantly improved the quality of the article. We thank Wolfgang Schwanghart and Benjamin Campforts for providing the TopoToolbox and TTLEM codes to analyse and to simulate landscape evolution. Through the work of Martine Simoes, this study contributes to the IdEx Université de Paris ANR-18-IDEX-0001 and is IPGP contribution no. 4082.

Timothée Sassolas-Serrayet's PhD is supported by a fellowship from the French Ministry for Higher Education. This research has been supported by the Agence Nationale de la Recherche (project ANR-18-CE01-0017 (Topo-Extreme)).

This paper was edited by Simon Mudd and reviewed by Adam Forte and Fiona Clubb.