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1. Introduction
The denudation of mountainous landscapes is central to many Earth processes. It modifies topography (e.g.,
Gilbert, 1877), accelerates soil production (e.g., Heimsath et al., 1997), affects soil nutrient supply (e.g., Y.
Lucas,2001), focuses rock exhumation (e.g., Willett,1999), perturbs sea level by unloading continental surfaces
(e.g., Dalca etal.,2013; Ferrier etal.,2017), and modulates Earth's climate by affecting weathering rates of sili-
cates and sulfides (e.g., Torres etal.,2014; Walker etal.,1981). Deciphering the controls on denudation rate is
therefore of wide interest across the geosciences.
Abstract Terrestrial cosmogenic nuclides (TCN) are widely employed to infer denudation rates in
mountainous landscapes. The calculation of an inferred denudation rate (Dinf) from TCN concentrations is
typically performed under the assumptions that denudation rates were steady during TCN accumulation and
that soil chemical weathering negligibly impacted soil mineral abundances. In many landscapes, however,
denudation rates were not steady and soil composition was significantly impacted by chemical weathering,
which complicates interpretation of TCN concentrations. We present a landscape evolution model that
computes transient changes in topography, soil thickness, soil mineralogy, and soil TCN concentrations. We
used this model to investigate TCN responses in transient landscapes by imposing idealized perturbations
in tectonically (rock uplift rate) and climatically sensitive parameters (soil production efficiency, hillslope
transport efficiency, and mineral dissolution rate) on initially steady-state landscapes. These experiments
revealed key insights about TCN responses in transient landscapes. (a) Accounting for soil chemical erosion
is necessary to accurately calculate Dinf. (b) Responses of Dinf to tectonic perturbations differ from those to
climatic perturbations, suggesting that spatial and temporal patterns in Dinf are signatures of perturbation type
and magnitude. (c) If soil chemical erosion is accounted for, basin-averaged Dinf inferred from TCN in stream
sediment closely tracks actual basin-averaged denudation rate, showing that Dinf is a reasonable proxy for actual
denudation rate, even in many transient landscapes. (d) Response times of Dinf to perturbations increase with
hillslope length, implying that response times should be sensitive to the climatic, biological, and lithologic
processes that control hillslope length.
Plain Language Summary In geomorphology, the rate at which mountains wear down is
commonly inferred from concentrations of cosmogenic nuclides in minerals in soil or river sand. Cosmogenic
nuclides are isotopes that build up in minerals during exposure to high energy particles from space,
accumulating at a rate that depends on the degree to which the soil is chemically altered and the mountain's
erosion rate itself. To explore how cosmogenic nuclide concentrations change in soil over time, we developed
a computer model that tracks cosmogenic nuclide concentrations in soil across a landscape. We used this
model to perform two model experiments: one driven by a change in the rate at which the bedrock is rising
(the so-called tectonic experiment), and the other driven by a change in rainfall (the climatic experiment).
Our experiments confirm that accounting for soil composition is necessary to accurately infer erosion rate.
They also show that cosmogenic nuclide concentrations respond differently to tectonic changes than climatic
changes, implying that patterns of cosmogenic nuclide concentrations may reflect perturbation type. The
time it takes cosmogenic nuclide concentrations to respond to a perturbation increases with the length of the
hills from the ridge to the valley, implying that response times should be influenced by climate, life, and rock
type. These results show how this model can be used to explore how mountainous topography, soils, and
cosmogenic nuclides change simultaneously. Our model will be a useful tool for improving field measurements
of cosmogenic nuclide concentrations in soil and in stream sediment.
REED ETAL.
© 2023. The Authors.
This is an open access article under
the terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.
Modeling Cosmogenic Nuclides in Transiently Evolving
Topography and Chemically Weathering Soils
Miles M. Reed1 , Ken L. Ferrier1 , and J. Taylor Perron2
1Department of Geoscience, University of Wisconsin-Madison, Madison, WI, USA, 2Department of Earth, Atmospheric, and
Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA, USA
Key Points:
• We developed a model to compute
terrestrial cosmogenic nuclide
(TCN) concentrations in transiently
evolving topography and chemically
weathering soils
• TCN-based denudation rates track
actual denudation rates more closely
during responses to changes in uplift
rate than to changes in climate
• Soil chemical weathering inf luences
modeled TCN concentrations,
confirming that this should be
accounted for in TCN-based
denudation rates
Supporting Information:
Supporting Information may be found in
the online version of this article.
Correspondence to:
M. M. Reed,
miles.reed@wisc.edu
Citation:
Reed, M. M., Ferrier, K. L., & Perron, J.
T. (2023). Modeling cosmogenic nuclides
in transiently evolving topography and
chemically weathering soils. Journal of
Geophysical Research: Earth Surface,
128, e2023JF007201. https://doi.
org/10.1029/2023JF007201
Received 12 APR 2023
Accepted 25 SEP 2023
10.1029/2023JF007201
RESEARCH ARTICLE
1 of 20
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At present, the most common tool for measuring millennial-scale denudation rates is terrestrial cosmogenic
nuclides (TCN). Concentrations of TCN in host minerals (e.g.,
10Be in quartz) can be used to infer denudation
rates averaged over a catchment drainage area (Bierman & Steig,1996; Brown etal.,1995; Granger etal.,1996)
or at a point on a hillslope (Dixon & von Blanckenburg,2012; Lal,1991; Riebe etal.,2003). Over the past few
decades, TCN-inferred estimates of denudation rate (which we refer to as Dinf) have been measured in thousands
of places (Codilean etal.,2022) and have been harnessed to investigate many things, including the influence of
climate and tectonics on denudation rate (e.g., Ferrier etal.,2012; Godard etal.,2020; Riebe etal.,2004; West
etal.,2005). The standard equations used to invert TCN concentrations for Dinf assume steady state—that is,
TCN-bearing minerals have been exhumed from depth to the surface at a steady rate and that the subsurface depth
profile of TCN concentrations is invariant in time (Lal,1991).
In landscapes that are out of steady state, TCN concentrations differ from what they would be in steady state.
If conventional expressions for steady-state Dinf were applied to TCN concentrations in transient landscapes,
then the resulting estimates of Dinf would differ from the actual denudation rates by an amount that depends on
the magnitude of the landscape's deviation from steady state (e.g., Bierman & Steig,1996; Mudd,2017; von
Blanckenburg,2006). Ferrier and Kirchner(2008) observed that Dinf is robust to minor deviations from steady
state, which suggests that Dinf is likely to be an accurate approximation of the true denudation rate in many land-
scapes. More practically, the Ferrier and Kirchner(2008) simulations imply that errors in Dinf associated with
transience are smaller than our ignorance of what denudation rates actually were over the past few thousand years
(an ignorance that is large in most landscapes, and in some landscapes is unbounded), given the absence of alter-
native methods for measuring millennial-scale denudation rates. This underscores the usefulness of measuring
Dinf with TCN, despite uncertainties associated with transient conditions. Nonetheless, if deviations from steady
state are not accounted for, they could lead to errors in Dinf of unknown size. It would be useful to be able to
interpret those sources of error as accurately as possible.
Concerns about transients are common because transient landscapes are common. Stream capture (Beeson
etal.,2017), exhumation of rocks of different strengths (Forte etal.,2016; Zondervan et al.,2020), climatic
shifts (Marshall etal.,2017), and changes in tectonic movements (Hurst etal.,2019) can all give rise to transients
in topography and soil composition (Ferrier & West,2017). Moreover, detecting landscape transience can be
difficult at the soil profile or hillslope scale (Heimsath etal.,2002; Hippe etal.,2021). Recent work with paired
nuclides has presented a method for detecting the magnitude but not the timing of transient conditions (Skov
etal.,2019), but quantifying the history of deviations from steady state remains challenging.
The recognition that landscapes are rarely in perfect steady state has inspired efforts to investigate transient
effects on TCN through numerical models. Small etal.(1999) developed the first hillslope-profile model of
TCN conservation that included process representations of hillslope sediment transport and soil production.
Anderson(2015) extended the hillslope-profile framework to investigate the influence of soil mixing on particle
transit times and TCN concentrations. Heimsath(2006) modeled the response of TCN depth profiles to changes in
physical erosion rate and showed that TCN-inferred erosion rates lag behind actual rates. Mudd(2017) modeled
TCN responses in a 2D landscape to changes in uplift rate, hillslope transport, and fluvial incision efficiency and
showed that Dinf responses to climate-related parameters closely matched the forcing while Dinf responses to uplift
rate were damped and lagged. These studies and many others have advanced our understanding of the effects of
transients on TCN concentrations.
To date, no study has investigated the sensitivity of TCN to transient conditions in a full 2D landscape evolution
model that includes explicit treatment of eroding and chemically weathering soil. That is our goal here. In this
study, we present a model to compute TCN concentrations in soil and the underlying rock, and we show how to
apply this model across landscapes undergoing simultaneous changes in topography, soil production rate, physi-
cal erosion rate, chemical erosion rate, soil thickness, and soil composition. This extends the model of Ferrier and
Perron(2020), which accounted for production, transport, and chemical erosion of soils in transiently evolving
topography but did not include TCN.
A key aspect of this model is that it is able to account for the effects of soil chemical erosion on TCN. These
effects can be significant. Small etal.(1999) and Riebe etal.(2001) recognized that chemical weathering of solu-
ble minerals increases the exposure times of dissolution-resistant host minerals such as quartz to cosmic rays, and
hence increases TCN concentrations in quartz beyond what they would be in the absence of chemical weathering.
If this additional exposure were neglected, this would yield erroneously low estimates of TCN-based denudation
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rate. Riebe and Granger(2013) extended this work to consider chemical erosion in deep saprolite, showing that
Dinf can be in error by tens of percent if chemical erosion is neglected. Because our simulations calculate soil
chemical depletion directly, they are able to confirm the importance of accounting for soil chemical erosion in
TCN-based estimates of denudation rate. More generally, our simulations show how this model can be used to
investigate the influence of tectonic and climatic perturbations on transiently evolving hillslopes and illustrate
how to identify transient conditions from multiple TCN measurements across a landscape.
2. Methods
2.1. Model Description
We designed our model to capture simultaneous responses of TCN, topography, soil thickness, and soil compo-
sition to changes in boundary conditions (e.g., bedrock uplift rate) and rates of soil production, transport, and
chemical weathering. The model does not treat bare-bedrock hillslopes, so application of this model is restricted
to soil-mantled landscapes. We incorporated nonlinear hillslope soil transport and TCN conservation into the
coupled landscape evolution-soil mineralogy model of Ferrier and Perron(2020). The model tracks and conserves
TCN concentrations in the soil, at the soil-bedrock interface, and in deeper bedrock (Figure1). Our aim was to
construct and test a model that simulates cosmogenic nuclide production and transport in soil-mantled uplands,
a common type of landscape.
In our model, stream channels are restricted to vertical movement relative to the hillslopes and set the local
base-level of the hillslopes (Mudd & Furbish,2007). We do have the capacity to specify a rate law for stream
channel incision (Text S4 in Supporting Information S1), but, in this work, we wish to focus solely on the
distributed hillslope response for simplicity. During transient conditions, rates of hillslope soil transport, soil
production, and mineral dissolution vary across the landscape. This results in temporal and spatial variations in
soil thickness, soil mineral concentrations, and TCN concentrations. In this section, we detail the governing equa-
tions, numerical implementation, and several methods for solving for inferred denudation rate using the modeled
TCN concentrations.
2.2. Governing Equations for Topography, Soil Thickness, and Soil Composition
The elevation of the bedrock-soil boundary, zb, increases at a bedrock uplift rate, U (L T
−1), and decreases at a soil
production rate, ϵ (L T
−1). The change in zb is expressed as
Figure 1. Schematic of model framework (Section2.1). At each grid point within a synthetic landscape (panel a), soil mass (panel b) and cosmogenic nuclides (panel
c) are conserved. (a) Stream channels indicated in blue, soil-mantled hillslopes in shaded relief with elevation contours. (b) Soil thickness (H) is governed by physical
erosion rate (E), chemical erosion rate (W), and soil production rate (ϵ). Elevation of the bedrock-soil boundary (zb) is governed by bedrock uplift rate (U) and ϵ.
Soil mineral concentrations (CXs) are determined by mineral supply from ϵ and losses through E and W. Soil and bedrock densities (ρs and ρr) are held constant. (c)
Soil terrestrial cosmogenic nuclides (TCN) concentrations (Ns) are determined by supply from bedrock, hillslope soil transport, and cosmogenic nuclide production
and decay (star symbol). In the soil, Ns (
10Be in this case) is uniform with depth, consistent with a vertically well-mixed soil. The TCN concentration at soil-bedrock
boundary (Nr,zb) arises through the calculation of a bedrock TCN profile Nr at a depth ξ below the soil-bedrock interface through time.
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=−0−∕0
.
(1)
In Equation1, the second term is the exponential soil production function (Heimsath etal.,1997,1999), where
H (L) is soil thickness, H0 (L) is the characteristic length that regulates soil production rate with regard to H, and
ϵ0 (L T
−1) is the maximum soil production rate at zero H. U is imposed upon the landscape, and a static channel
network incises at a rate equal to U. H changes via soil production rate, the divergence of downslope soil trans-
port, and chemical erosion rate:
=
0−∕0−∇⋅−
𝑊
(2)
where ρr and ρs are bedrock and soil density (M L
−3), respectively. In our model, ρr and ρs are time-invariant.
The downslope volumetric flux of soil per unit contour width, qs (L
2 T
−1), increases nonlinearly with respect to
hillslope gradient of the model surface elevation, (zs=zb+H), and is defined as
𝑞
𝑠=
−𝐾nl∇𝑧
𝑠
1−(|∇𝑧𝑠|∕𝑆𝑐)
2
.
(3)
Here, Knl is the soil transport coefficient (L
2 T
−1), ∇zs is the local hillslope gradient, and Sc is the critical gradi-
ent (Roering etal., 1999,2001). Our framework is flexible enough to adopt other formulations for qs, such as
those that depend on soil thickness (Braun etal., 2001). For simplicity, the simulations we show in this study
are governed by Equation3. This translates into a physical erosion rate E (L T
−1) by taking the divergence of qs
(Roering etal.,1999):
𝐸
=−𝐾nl
∇2𝑧𝑠
1−(
∇𝑧𝑠
∕𝑆𝑐)2+2
𝜕𝑧𝑠
𝜕𝑥
2
𝜕2𝑧𝑠
𝜕𝑥2
+
𝜕𝑧𝑠
𝜕𝑦
2
𝜕2𝑧𝑠
𝜕𝑦2
+2
𝜕𝑧𝑠
𝜕𝑥
𝜕𝑧𝑠
𝜕𝑦
𝜕2𝑧𝑠
𝜕𝑥𝜕𝑦
𝑆2
𝑐
1−(
∇𝑧𝑠
∕𝑆𝑐)2
2
,
(4)
where x and y are the two axes of a regular grid. As implemented, E is limited by the thickness of the explicitly
modeled soil layer, H, such that, for a given timestep, erosion can never be greater than H (Text S2 in Supporting
InformationS1). This parameterization for soil transport is most applicable to nonlinear creep and is not directly
applicable to large discrete events such as deep-seated landslides. Chemical erosion rate, W (L T
−1), is calculated
following Ferrier and Kirchner (2008) as the difference between mineral dissolution and secondary mineral
formation rates summed over n mineral species throughout the soil:
=
∑
=1 ( −
)
,
(5)
where kj (mol L
−2T
−1), Aj (L
2 mol
−1), Cjs (M M
−1), sj (mol L
−3T
−1), and wj (M mol
−1) are the mineral dissolution
constant, specific surface area, soil mineral concentration, secondary mineral production rate, and molar mass of
the jth mineral from the set of n soil minerals, respectively.
The concentration of an individual soil mineral X, CXs (M M
−1), evolves through the competition between disso-
lution and mineral supply. We assume a well-mixed soil such that mineral concentration is independent of soil
depth (Ferrier & West,2017). Soil production, hillslope soil transport, mineral dissolution, secondary mineral
formation, and the mass losses of the other mineral phases combine to modify a mineral's concentration:
Xs
=
0−∕0(Xr −Xs)−
⋅∇Xs −Xs +
+Xs
∑
=1
(
−
).
(6)
Here, CXr (M M
−1) is the concentration of mineral X in the bedrock. This formulation differs from that in Ferrier
and Perron(2020) only in the transport term (second term), where qs is represented as a nonlinear function of
hillslope gradient (Equation3) rather than a linear function of it (Yoo etal.,2007). Combining Equations1 and2,
the rate of change of zs is expressed as
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=−+(
−1
)0−∕0−
∑
=1
( −
)
.
(7)
Model outputs can exhibit nonlinear behavior during transient conditions through the interacting changes in
topography, mineral supply from soil production, soil thickness, and chemical erosion (Ferrier & Perron,2020).
2.3. Governing Equations for Cosmogenic Nuclides
We extended the model of Ferrier and Perron(2020) to track concentrations of cosmogenic nuclides in soil and
the underlying rock (Riebe & Granger, 2013). First, we implemented TCN production in the soil and bedrock
such that production rates evolve with topography. Here, we outline how the processes described above modify
TCN concentrations in soil and bedrock.
To compute changes in TCN concentrations in the soil, we account for TCN production in the soil, addition
of TCN from soil production, changes in TCN through the divergence of soil transport, and losses of TCN to
radioactive decay and host mineral dissolution. The full expression for the change in soil TCN concentration, Ns
(atoms
M−1
host
), is
=1
∑
=1(0)Λ(1−−⁄Λ)−
host,
host,
0−⁄0(−,)
−
⋅∇(host,)−−host host
.
(8)
Here, i is the ith of n cosmogenic production pathways, Pi (atoms
M−1
host
T
−1) is the cosmogenic production rate
of the ith production pathway (at zero depth), and Λi (M L
−2) is the attenuation length associated with the ith
production pathway. Chost,r and Chost,s (M M
−1) are the concentrations of the TCN host mineral in the bedrock and
soil, respectively. Nr,zb is the concentration of TCN at the soil-bedrock interface, and λ (T
−1) is the decay constant
of the TCN. The last term in Equation8 represents the loss of TCN concentration by chemical erosion, which
can be ignored if the host mineral is inert or nearly so. A more complete derivation of Equation8 can be found in
Text S2 of Supporting InformationS1.
In this study, we used n=3 production pathways representing spallation, negative muon capture, and fast muogenic
production, which is appropriate for several TCNs, including
10Be,
26Al, and
14C in quartz (Dunai,2010). For all
production pathways, we employed the Lifton-Sato-Dunai scaling scheme (Lifton etal.,2014) within CRONUS-
calc (Marrero etal.,2016) using a sea-level, high-latitude
10Be production rate of 3.92 atoms
g−1
qtz
yr
−1 (Borchers
et al., 2016). We used Madison, Wisconsin, as a reference location to scale production rates. We obtained a
uniform spallation pathway attenuation length (Λs) from CRONUScalc using the mean elevation of the reference
model topography. To parameterize muogenic production rates and attenuation lengths, we used the codes of
Balco(2017) to create 1,000 synthetic muogenic production profiles for both negative muon capture and fast
production pathways across the entire range of elevations comprising our modeling runs. In keeping with previ-
ous work (Braucher etal.,2013), profiles were fitted with a single-term exponential function to arrive at a surface
production rate and an attenuation length for each pathway. Similar to spallation, we used a mean attenuation
length for each muogenic pathway as the range was small. We then created piecewise cubic Hermite interpolants
(Fritsch & Carlson,1980) for each production pathway using the corresponding surface production rates and
elevations, allowing the model to update surface production rates as elevations change.
Calculating Nr,zb in Equation8 requires determining the rate of change in bedrock TCN concentration, Nr, at an
arbitrary depth below the soil-bedrock interface, ξ (L), as
𝜕𝑁
𝑟(𝜉)
𝜕𝑡
=
3
∑
𝑖
𝑃𝑖(0)𝑒−𝜌𝑠𝐻∕Λ𝑖𝑒−𝜌𝑟𝜉∕Λ𝑖−𝜆𝑁𝑟(𝜉)−𝜖0𝑒−𝐻∕𝐻0𝜕𝑁𝑟
𝜕𝜉 ||||𝜉
.
(9)
This equation represents the sum of nuclide production in bedrock and decay while recognizing that the soil
production rate history influences the bedrock nuclide concentration profile. By including deep production,
periods of transient denudation rate can induce changes in the bedrock TCN concentration profile (Knudsen
etal.,2019), which can alter soil TCN (Skov etal.,2019). The change in Nr,zb is then evaluated at the soil-bedrock
interface (i.e., ξ=0) as
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𝜕𝑁
𝑟,𝑧𝑏
𝜕𝑡
=
3
∑
𝑖
𝑃𝑖(0)𝑒−𝜌𝑠𝐻∕Λ𝑖−𝜆𝑁𝑟(𝜉)−𝜖0𝑒−𝐻∕𝐻0𝜕𝑁𝑟
𝜕𝜉 ||||𝜉=0
.
(10)
We solved an approximation of these two equations with a semi-Lagrangian advection scheme (Blackburn
etal.,2018; Spiegelman & Katz,2006) described in Section2.4.
2.4. Inferring Denudation Rates From Cosmogenic Nuclides
To fully exploit a landscape evolution model that conserves TCN, we must be able to efficiently infer denudation rates
from TCN concentrations. Throughout this study, we applied Equation11 to infer denudation rates (Dinf) from Ns.
=
3
∑
(0)
+inf ∕Λ
[
host,
host,
(
1−−∕Λ
)
+−∕Λ
]
(11)
Equation 11 is nearly identical to an analogous expression from Riebe and Granger (2013) for a two-layer
bedrock-soil system similar to our model; the only difference is that it includes radioactive decay. It was derived
under assumptions of steady state, with constant Dinf and H in a well-mixed, chemically weathered soil produced
from an underlying parent rock in which negligible chemical erosion occurred. Equation 11 does not have an
analytical solution for Dinf, so in practice it must be solved numerically. Our goal is to investigate the extent to
which denudation rate estimates inferred from Equation11 would deviate from actual denudation rates during
landscape transience. At all soil-mantled cells, we solved for Dinf in Equation11 and others presented in Section3
using local values of Pi, Ns, H, and Chost,s/Chost,r, all of which are measurable in the field or in the lab.
Calculators such as CRONUScalc or CRONUS solve Dinf from Ns numerically (Balco et al., 2008; Marrero
etal.,2016). Similarly, we used model outputs of Ns, Chost,s, and H to solve Dinf in Equation11 as a post-processing
step as it is too computationally expensive to solve at each timestep.Using a Newton-Raphson method implemented
within MATLAB's variable-precision solver for Equation11 under steady-state conditions at ridge locations not
subject to downslope soil transport, Dinf deviates by only ∼0.18% from the modeled, actual denudation rate, Dact.
For basin-averaged inferred denudation rate (Dinf,basin), we solved Equation11 using a mean H and Chost,s/Chost,r,
and hypsometrically averaged Pi(0) for the entire basin. We did this as a post-processing step on a delineated basin
through time using outputs of Ns, Chost, and E. We constructed a virtual stream sediment TCN concentration using
these variables for all stream-side grid points. This concentration is calculated as
𝑁
𝑠,stream =
∑𝑁
𝑠,
stream-side𝐸stream-side 𝜌
𝑠
𝐶
ℎ𝑜𝑠𝑡,𝑠,
stream-sideΔ𝑥Δ𝑦Δ𝑡
∑𝐸stream-side𝜌𝑠𝐶ℎ𝑜𝑠𝑡,𝑠,stream-side Δ𝑥Δ𝑦Δ𝑡.
(12)
Here, ∆x and ∆y are the grid point lengths in the x and y directions, and ∆t is the duration of the timestep.Thus
far, few studies have accounted for enrichment or depletion of the host mineral in cosmogenic nuclide-based
estimates of basin-averaged denudation rate, but recent work in carbonate landscapes underscores the importance
of doing so (Ott etal., 2022,2023). In Section 3, we show the implications of neglecting chemical erosion in
point-based and basin-averaged estimates of denudation rate.
2.5. Numerical Implementation
Here, we describe some details associated with the model domain, boundary conditions, and solutions to the
governing equations. In all modeling scenarios, we used a 150× 150 regular grid with a resolution of 10 m.
The boundaries of the model domain were pinned to zero elevation, allowing the boundaries to act as a regional
base-level. Mass transfers to the channel network do not change the channel elevation (Ferrier & Perron,2020).
We used a timestep of 10years to maintain an acceptable agreement between Dact and Dinf at steady state. The
governing equations are solved using explicit finite difference methods. The supplement contains a detailed
description of the implementation of nonlinear hillslope soil transport based on Perron(2011).
In order to solve Equations9 and10, we used a semi-Lagrangian advection scheme to account for deep cosmo-
genic production under variable denudation rates (Figure S1 in Supporting InformationS1). The scheme can
solve for Nr(ξ) at regularly spaced depths of an evolving bedrock concentration profile (Knudsen etal.,2019).
The soil production rate at a grid cell provides a vertical velocity from which we find a departure depth via cubic
interpolation (Fletcher,2019). During each timestep, TCN production rate is calculated using a depth obtained by
the averaging the depth of evaluation (i.e., ξ) and the departure depth. Nr,zb is then filled with the value at ξ=0,
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which is the bedrock-soil interface. The lowermost value of the profile is filled with a steady-state TCN concen-
tration corresponding to a running average of bedrock exhumation velocities. This mimics a pre-perturbation
history of TCN accumulation below the profile, which cannot be accounted for analytically. The number and
spacing of evaluation points below the surface affect the accuracy of the TCN concentration relative to an analyt-
ical steady-state value due to interpolation errors (T. R. Lucas,1974). We observed a depth of 10m and point
spacing of 10mm to be a good compromise between accuracy and computational speed. In the supplement, we
detail an alternative, speedier method for calculating Nr,zb that does not directly track TCN profiles (Figure S2 in
Supporting InformationS1).
2.6. Attaining Steady-State Topography
To prepare for simulations under transient conditions (Section4), we ran two successive simulations to obtain
steady-state initial conditions. First, we generated a steady-state topography with the Tadpole landscape evolution
model (Perron etal.,2008,2009,2012; Richardson etal.,2020). This model differs from that in Section2 in that
it includes not only soil transport but also stream incision, and in that it does not track changes in soil thickness,
soil composition, or chemical erosion rate. In this model, under a threshold stream-power incision law, the change
in elevation with time due to fluvial erosion is expressed as
=
⎧
⎪
⎨
⎪
⎩
0≤
−(−)>
,
(13)
where Kf (L
1–2m T
−1) is an effective erodibility, A (L
2) is drainage area, S is slope (L L
−1), m and n are unit-
less constants, and θc (L
2m) is an incision threshold (e.g., Pelletier, 2012; Theodoratos & Kirchner, 2020).
Starting from a surface generated by red noise, we ran the model until the topography reached steady state
using the same boundary conditions and model parameters used in the transient simulations (U = 50 mm
kyr
−1, Knl=0.0032m
2yr
−1 and Sc=1.2; Section4.1) and the following values for stream incision parameters:
Kf=0.0001year
−1, m=0.5, n=1, and θc=8m. We extracted the channel network from the resulting steady-state
topography using a threshold drainage area of 5,000m
2.
We then used the topography output from this first simulation as the initial condition for a second simulation with
the model in Sections2.1 and2.2. We ran this simulation to steady state to obtain steady state values of zs, H, CXs,
and TCN concentrations. In this simulation, the channel network was fixed in the geometry it was in at the end
of the Tadpole simulation. The median difference in elevation between the initial topography and the steady-state
topography is ∼3.5m and is due to the inclusion of soil chemical erosion, soil production, bedrock lithology, and
the fixed stream channel network.
3. Results: Steady-State TCN-Inferred Denudation Rates Are Sensitive to Soil
Chemical Erosion
How do inferred denudation rates computed with Equation11 compare to denudation rate estimates inferred from
Ns via some other common formulations? Since we solved Equation11 numerically as a post-processing step,
we tracked and outputed Dinf by ignoring radioactive decay. If decay is neglected, Equation11 would simplify to
inf,CEF =
3
∑
(0)Λ
[host,
host,
(
1−∕Λ
)
+−∕Λ]
,
(14)
which is identical to that in Riebe and Granger(2013). Most inferred denudation rates in the literature do not correct
for chemical erosion (Riebe & Granger,2013). If chemical erosion is neglected, Equation11 would simplify to
=
3
∑
(0)
+inf,Lal ∕Λ
,
(15)
which is similar to that originally derived in Lal(1991) and identical to those used in denudation rate calculators
when evaluating surface samples (Marrero etal., 2016) or stream sediment samples (Mudd etal.,2016). We
solved Equation15 via the same method as Equation11.
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We computed Dinf with Equations11, 14, and 15 in five steady-state landscapes that differ in their bedrock
concentration of the soluble phase plagioclase (5%, 20%, 35%, 50%, and 65%) and the complementary concen-
tration of the bedrock's only other mineral phases, quartz, which is treated as insoluble (95%, 80%, 65%, 50%, and
35%, respectively). All the landscapes had a U of 50mm kyr
−1. Figure2 shows Dinf/Dact for these five landscapes
using Equations11, 14 and15 at a ridge location not subject to downslope soil transport (left panel) and a small
basin (right panel). Comparing Dinf/Dact for Equations11 and14 in Figure2 shows that ignoring decay results in
a small error in Dinf (∼2.5%) in these simulations. By contrast, comparing Dinf/Dact for Equations11 and15 shows
that neglecting the enrichment of Chost,s by chemical erosion leads to larger errors in these simulations, up to 29%
in the simulation with the highest W/D (and plagioclase concentration). These errors increase with increasing
W/D, consistent with the observations in Small etal.(1999), Riebe etal.(2001), and Riebe and Granger(2013).
Figure2 also shows that the ratio diverges from 1 with increasing W/D for basin-averaged Dinf. In the model, the
effect of W on Dinf is indirectly affected by U, to the extent that U affects properties like soil thickness and soluble
mineral concentrations (Text S5 in Supporting InformationS1).
To summarize, Figure2 implies that calculating Dinf requires accounting for soil chemical depletion if the cosmogenic
host mineral has a different soil residence time than the average soil material (e.g., quartz, which stays in the soil while
other minerals are lost to dissolution). Our study is not the first to make this point—Small etal.(1999) and Riebe
etal.(2001) did this more than 20years ago—but our model is well-equipped to show the importance of this effect.
4. Results: Transient Landscapes
4.1. Responses to Tectonic and Climatic Perturbations
To investigate the transient response of TCN to perturbations, we performed two numerical experiments. In the
first experiment, we simulated a simple tectonic perturbation in which initially steady-state landscape experi-
ences an instantaneous doubling of rock uplift rate (or equivalently a doubling of the lowering rate of the domain
boundaries and the channels) from 50mm kyr
−1 to 100mm kyr
−1. In the second experiment, we simulated a
Figure 2. Ratio of inferred denudation rate (Dinf, computed with Equations11, 14 and15) to actual denudation rate,
Dact, in steady-state landscapes at a ridge (left panel) and averaged over a small basin (right panel). Markers represent
simulations with a range of ratios of chemical erosion rate to denudation rate (W/D), resulting from a range of bedrock
plagioclase concentrations (5%–65%). Values of Dinf/Dact close to 1 for Equations11 and14 show that accounting for soil
chemical erosion yields denudation rate estimates close to actual denudation rates. Values of Dinf/Dact far from 1 computed
with Equation15, which neglects chemical erosion, show that Dinf grows increasingly farther from Dact with increasing
W/D. This underscores the importance of accounting for soil chemical erosion in TCN-based estimates of denudation rate.
Basin-averaged rates are nearly identical to those on ridges because these simulations are steady-state landscapes.
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simple climatic perturbation in which the maximum soil production rate, hillslope transport efficiency, and the
dissolution rate constant of plagioclase were simultaneously increased by 50%. This tests the response of TCN to
a “top-down” perturbation where the entirety of the hillslope area is affected at the same time (Mudd,2017). In
both experiments, we restricted our attention to the responses of the hillslopes by fixing the lateral position of the
stream network in place during the simulation (Hurst etal.,2012; Mudd & Furbish,2007). In each simulation,
we ran the model until CXs, H, and Ns reached new steady-state values, which occurred at 3 and 3.1 Myr of model
run time in the tectonic and climatic experiments, respectively.
To quantify the difference between the actual instantaneous denudation rate Dact and the TCN-based estimate
of denudation rate Dinf (Equation11), we introduce the metric Ddiff=Dinf−Dact. This reveals the magnitude of
the error in the TCN-inferred denudation rate and is a measure of the deviation from steady-state conditions in
a non-steady landscape. This can be calculated at any point on the landscape, and therefore can reveal spatial
patterns in deviations from steady state. It can also be calculated for a drainage basin as the difference between
the basin-averaged inferred denudation rate, Dinf, basin, and the actual basin-averaged denudation rate, Dact, basin.
In both experiments, the initial U is 50mm kyr
−1. All model parameters used in the perturbation experiments
are listed in Table1. We use values of Knl, Sc, and ϵ0 from the Oregon Coast Range as it is one of the few land-
scapes where these parameters have been calibrated from field measurements (Heimsath etal.,2001; Roering
etal.,1999). The kplag parameter comes from the field measurements of Clow and Drever(1996). Both scenarios
are assigned the same bedrock mineralogy (50% plagioclase and 50% quartz) to simplify the interpretation of soil
composition in terms of one easily weathered phase and one insoluble phase.
The responses of landscape-wide Dinf and Dact differed spatially and temporally between the two experiments. In the
tectonic scenario, the areas around channels responded first to the increase in uplift rate, steepening and increasing
Dact. This was followed by a wave of increased physical erosion rate that gradually propagated up the hillslopes to
the ridges (Figure3). Although Dact exhibited notable change around channels by 100 kyr, Dinf showed little change
(Figure3). This is due to downslope soil transport from upslope areas that have not yet reached the new steady-state
conditions, which buffers the response in convergent hollows and footslopes. Both Dact and Dinf along ridges equili-
brated slowly to the perturbation. Dinf stayed close to local steady state (Dinf≈Dact) on ridges throughout the experi-
ment, such that the minimum Ddiff stayed above −10mm kyr
−1. By comparison, Ddiff on footslopes and hollows was
larger and persisted due to the buffering effect of hillslope transport (Figure3). Across the entire experiment, Dinf
deviated from Dact by at most 64% during the experiment (Ddiff=−46mm kyr
−1), which occurred at 71 kyr in hollows
directly above channel heads and along channels below long hillslopes. Positive Ddiff values were low (Ddiff=3mm
kyr
−1) during the entirety of the experiment, implying that Dinf did not exceed Dact by much at any time or place.
Model parameter Tectonic Climatic
∆t—model timestep 10yr 10yr
∆x, ∆y—grid resolution 10m 10m
U—bedrock uplift rate 0.05 to 0.1mm yr
−1 0.05mm yr
−1
Knl—hillslope transport efficiency (nonlinear) 0.0032m
2 yr
−1 a 0.0032–0.0048m
2 yr
−1
SC—critical hillslope gradient 1.2m m
−1 b 1.2m m
−1
ε0—maximum soil production rate 0.268mm yr
−1 c 0.268–0.402mm yr
−1
H0—soil production rate scaling length 1/3m c 1/3m
ρs—soil density 1,325kg m
−3 1,325kg m
−3
ρr—bedrock density 2,650kg m
−3 2,650kg m
−3
kplag—plagioclase dissolution rate constant 0.000005mol m
−2yr
−1 d 0.000005–0.0000075mol m
−2yr
−1
Aplag—plagioclase mineral specific surface area 117m
2 mol
−1 d 117m
2 mol
−1
ξmax—max depth of bedrock concentration profile 10m 10m
ξ spacing—distance between profile evaluation points 0.01m 0.01m
# averaging timesteps for cosmogenic profile infilling 20,000 20,000
aRoering etal.(1999).
bRoering etal.(2001).
cHeimsath etal.(2001).
dFerrier and Kirchner(2008) and Clow and Drever(1996).
Table 1
Model Parameters for Transient Perturbation Experiments
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Closer investigation of changes in H, Ns, and CXs at different hillslope positions through time showed how large
Ddiff developed in hollows and footslopes through the transport of Ns and CXs (Figure 5b–5e). In the hollows,
the decrease in H far outpaced the slower response of Ns and CXs. This suggests that estimates of Dinf inferred
from samples on ridges (Ferrier etal.,2012; Larsen etal.,2014) should be good indicators of current Dact during
tectonic-style perturbations like the one imposed here. At the ridge, larger magnitude changes in U induced larger
Ddiff that equilibrated faster due to hillslope steepening and soil thinning as the landscape approached SC (Figure
S3 in Supporting InformationS1).
Across the landscape in the climatic experiment, Dact increased rapidly after the initial perturbation, and Dinf
increased nearly as rapidly. This resulted in negative Ddiff values at 10 kyr (Figure4). Dinf and Dact attained a mean
maximum difference of around 44% (Ddiff=−21mm kyr
−1) across all hillslope positions at 2 kyr. By 100 kyr,
stream channel incision rate started to dominate the response, and a wave of decreased Dact migrated upslope,
which generated positive Ddiff (Figure4). This was again due to the buffering capacity of soil transported from
hillslope positions at different stages of transient evolution. When Ns, CXs, and H from different hillslope posi-
tions are considered, the perturbation to ϵ0, soil production efficiency, rapidly increased H at all positions, which
resulted in a decrease in Ns. The relatively fast change in these variables explains the quick recovery of Ddiff to
near zero (Figures5g and 5h). As the channels began to dominate, the hillslope positions displayed differing
responses. Soil transport delayed the recovery to steady-state conditions by supplying Ns and CXs from ridges to
footslopes and hollows, leading to prolonged positive Ddiff in these locations (Figures5g and 5h). Given these
results, perturbations that synchronously disrupt the entire landscape may cause inaccuracies in real-world esti-
mates of Dinf on ridges directly after the perturbation. These inaccuracies would grow and be prolonged if the
magnitude of the perturbation change were increased (Figure S4 in Supporting InformationS1).
Figure 3. Landscape distributions of actual and inferred denudation rates Dact and Dinf (Equation11), and the difference Ddiff=Dinf−Dact for the tectonic experiment
at 10, 100, and 1,000 kyr after the perturbation. Topography shown at 2x vertical exaggeration. The model domain is 1.5 km by 1.5 km. Topographic contours are every
15m. Maximum elevation is 180m at 1,000 kyr. A movie of the experiment is available as a supplement (MovieS1). Ridges not subject to hillslope soil transport
remain near local steady state (Dinf≈Dact) throughout the simulation.
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Figure6 shows Ddiff, basin, the difference between the basin-averaged TCN-inferred denudation rate, Dinf, basin, and
the basin-averaged instantaneous denudation rate, Dact, basin, for the drainage basin outlined in Figure6. Ddiff, basin
was relatively small throughout most of each simulation, yet the time series of Ddiff, basin differed in style between
the tectonic and climatic experiments. In the tectonic experiment, Ddiff, basin progressed to −7mm kyr
−1 (10%
difference) at ∼158 kyr and then declined gradually to nearly zero over ∼1 Myr (Figure6b). In contrast, in the
climatic experiment, Ddiff, basin evolved as a sharp initial pulse to a maximum of −23mm kyr
−1 (37% difference)
at 2 kyr and then returned to near zero within 83 kyr (Figure6c), a much faster response than that in the tectonic
experiment. This was followed by a longer, lower-amplitude response in which Ddiff, basin grew as large as 2mm
kyr
−1 and then decayed to nearly zero over ∼1 Myr, similar to the duration of the response in the tectonic exper-
iment (Figure6b). Larger magnitude changes in the perturbed parameters for each scenario generated greater
Ddiff, basin which equilibrated either more rapidly (tectonic) or slowly (climatic) due to steeper (tectonic) or gentler
(climatic) gradients (Figures S5 in Supporting InformationS1).
The results show that these transient changes in bedrock uplift rate, soil production rate, dissolution rate, and soil trans-
port rate do not generate large errors in estimates of Dinf, basin from TCN in stream sediment. This implies that large
errors in Dinf, basin in practice are more likely to arise from processes not represented in the model, such as deep-seated
landslides (Schide etal.,2022; Yanites etal.,2009) or sediment storage along or within the stream network (Grischott
etal.,2017). The inclusion of dynamic channels would increase the response time in the tectonic scenario since the
response would need to transit the channel network before propagating through the hillslopes (Hurst etal.,2012).
Figure 4. Landscape distributions of actual and inferred denudation rates Dact and Dinf (Equation11), and the difference Ddiff=Dinf−Dact for the climatic experiment
at 0, 10, 100, and 1,000 kyr after the perturbation. Topography shown with 2x vertical exaggeration. The model domain is 1.5km by 1.5km. Contours are every
15m. The maximum elevation is 135m at 1,000 kyr. A movie of the experiment is available as a supplement (MovieS2). The initial surfaces (t=0 kyr) illustrate the
abrupt initial change in all variables during this experiment. For this type of perturbation, Ddiff varies between negative and positive values depending on the time since
perturbation and hillslope position.
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4.2. Response Times
The time it takes a landscape to respond to external forcings is a fundamental property of the landscape. This
response time is useful because it can be compared to time scales of climatic or tectonic forcings and used to iden-
tify whether a landscape is likely to be close to steady state (e.g., Roering etal.,2001). Here we use the tectonic
and climatic experiments in Figures3 and4 to calculate response times of several characteristics of the landscape
(soil TCN concentrations, soil thickness, soil mineralogy, and denudation rates). To do this, we applied a modi-
fied version of the response time calculation in Ferrier and Perron(2020), adapted to accommodate variables with
Figure 5. (a) Map view of the initial topography for the tectonic and climatic perturbation experiments. Channels marked in blue. Numbers 1–4 indicate the hillslope
positions (1: ridge; 2: channel head; 3: sideslope; and 4: near-channel) that are tracked in panels (b–i). (b–e) Time-series plots of Ns, H, Chost,s, and Ddiff from tectonic
perturbation experiment. (f–i) Time series plots from climatic perturbation. Initial variability in Ns is due to hillslope soil transport. The variable responses show that
hillslope position determines how the underlying variables used to obtain Dinf evolve.
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similar initial and final values (e.g., for Dact and Dinf in the climatic experiment in Figure5). For instance, for Dinf,
we define the response time as the time it takes Dinf to get 95% of the way to its new steady-state value relative to
the maximum deviation of Dinf from its new steady-state value. In mathematical terms, this is the time at which
Dinf first satisfies the condition
|inf −inf, final|<max(0.05|inf, −inf, final |)
and continues to satisfy it for the
remainder of the simulation. Here, Dinf,t is the entire time series of Dinf for the modeling run, Dinf, final is the final
steady-state value of Dinf, and 0.05 is a 5% threshold level. The response time is defined this way for every point
in the landscape. We used this definition to calculate the response time, τ, for Ns, H, Chost,s, Dact, and Dinf, which
we denote τNs, τH, τChost,s, τDact, and τDinf, respectively.
When response times are mapped across the landscape, they reveal spatial patterns in the propagation of signals
throughout the domain. Figure7 shows that response times varied between the tectonic and climatic experiments
and among variables within each experiment. In the tectonic experiment, the response time of Dinf was greater
than the response times of all other variables. τDinf varied in space across the landscape from 368 kyr to 1.735
Myr and had a mean (± s.d.) of 1.304 Myr± 223 kyr (Figure7f). For the variables not associated with cosmo-
genic nuclides, areas near channels responded fastest (e.g., τH and τDact in Figures7b and7e). However, τDinf was
high along channels where hillslope sediment transport from ridges carried down parcels of soil with higher Ns
(Figure7f). Ridges responded slower than other parts of the landscape for every characteristic (Figure7a–7f),
mimicking the way the perturbation in bedrock uplift is translated upslope (Mudd & Furbish,2007). When this
experiment is viewed holistically, the redistribution of relict Ns downslope controls the response time of Dinf.
Response times in the climatic experiment were longer than those in the tectonic experiment for all variables
(Figure7g–7l). This differs from a similar experiment in Ferrier and Perron(2020) and is likely due to the intro-
duction of more complex topography and different parameter values. The mean (± s.d.) of τDinf in the climatic
experiment was 1.604 Myr± 276 kyr and varied from 784 kyr to 2.238 Myr. High τDinf in top-central area of
the modeling domain is indicative of topographic control on τDinf (Figure7l). This zone underwent the most
topographic adjustment during the perturbation. These response times show that relatively small perturbations
in the efficiencies of hillslope processes could lead to transient states lasting much longer than the timescale of
climatic fluctuations. Although this is the case, as shown in Figures5 and6, Ddiff on ridges is quite low for the vast
duration of modeling time. This means that even though τDinf is high for perturbations affecting climate-related
hillslope processes, Dinf values should reflect Dact at almost any point in time except for directly after the onset
of a step-change.
5. Discussion
5.1. Sensitivity of Response Times for Maximum Ddiff and Dinf to Hillslope Length
Hillslope length (Lh) is a fundamental landscape property and is defined as the distance from ridges to stream
channels (Grieve etal., 2016). Numerical modeling and experimental work have shown that Lh is set by the
competition between advective fluvial processes and diffusive hillslope processes (Perron etal.,2009; Sweeney
Figure 6. (a) Initial topography with selected basin outlined in black. (b) Difference between inferred and actual
basin-averaged denudation rates, Ddiff, basin, for the initial 2,000 kyr of modeling time in the tectonic experiment. During this
simulation, errors in Dinf, basin were no larger than 7mm kyr
−1, damped by the range of cosmogenic nuclide production rates
within the basin and through the collection of an integrated sediment sample. (c) The climatic experiment shows that Ddiff, basin
could be large directly after a perturbation but recover relatively rapidly given a constant tectonic regime.
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etal.,2015). In numerical simulations in Ferrier and Perron(2020), chemical erosion rates took longer to respond
to perturbations in landscapes with longer hillslopes. This matches previous modeling of topographic response
times of a single hillslope for both linear and nonlinear hillslope sediment transport to changes in bedrock uplift
(Fernandes & Dietrich,1997; Roering etal.,2001). These studies suggest that Lh should influence cosmogenic
nuclide behavior as well, including response times and the magnitude of Ddiff.
To test the sensitivity of the maximum Ddiff (Ddiff,max) and Dinf response time (τDinf) to Lh in our model, we gener-
ated four synthetic landscapes with a range of Lh values. We generated these using the previously mentioned
stream power incision model and applied the same parameters as in the numerical experiments in Figures3–5
except for Kf (effective erodibility) and the drainage area threshold for channel extraction. This yielded four land-
scapes that had average Lh values ranging from 34.5 to 93.2m. With these landscapes, we explored the sensitivity
of Ddiff,max and τDinf to Lh by applying the same tectonic-style perturbation to each of them, which involved a
step-change in bedrock uplift rate from 50mm kyr
−1 to 100 kyr
−1, as shown in Figure4. To demonstrate spatial
variations in the responses within each landscape, we computed Ddiff,max at multiple hillslope positions (ridge,
sideslope, channel head) and as a basin-averaged quantity in a small basin in each simulation.
The landscape position at which Dinf deviated the least from Dact was on ridges. Figure8a shows that this was the
case at all average hillslope lengths and that the magnitude of the deviation of Dinf from Dact decreased with Lh.
In contrast, the largest deviations of Dinf from Dact occurred at channel heads, where the largest deviations were
temporarily greater than the imposed change in uplift rate at the beginning of the simulation.
The patterns of Ddiff,max are instructive. Because ridges respond last to base-level perturbations, they are the parts
of the landscape where Dinf most closely tracks Dact, making them ideal sampling locations for measuring physical
and chemical erosion rates (Ferrier etal.,2016). This is particularly so above long hillslopes, where the differ-
ence between Dinf and Dact approaches zero (Figure8a). However, in landscapes with ridge-visiting landslides
(Campforts etal.,2022; Dahlquist etal.,2018), this agreement between Dinf and Dact could be violated. Similar
to the ridges, basin-averaged Dinf better depicts basin-averaged Dact as Lh increases, as the method considers
nuclide production rates throughout the basin, including slowly responding areas (Granger etal.,1996; Mudd
etal.,2016). Channel heads and sideslopes produced greater Ddiff,max with increasing Lh due to enhanced buffering
of Dinf by hillslope transport of relatively high Ns soil from the slowly responding ridges.
τDinf increased with Lh at all topographic positions and was largest at ridges (Figure8b). Landscapes with longer
hillslopes exhibited shorter basin-averaged τDinf relative to the sideslope and channel head. This is due to lower
final steady-state Dinf from downslope transport from ridges to these positions, which makes meeting the response
Figure 7. (a–l) Response times (τ) of soil cosmogenic nuclide concentration (Ns), soil thickness (H), cosmogenic nuclide host mineral concentration (Chost,s), actual
denudation rate (Dact), and inferred denudation rate (Dinf) for the tectonic (a–f) and climatic experiments (g–l) (Figures3 and4). The response time of Dinf is longest on
ridges due to the combined response times of all the underlying variables, but, as the text details, the deviation of Dinf from Dact is smallest on ridges.
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criterion more demanding. At the longest Lh, the channel head τDinf is lower than the sideslope because Dinf at the
channel head goes down by ∼5mm kyr
−1, easing the satisfaction of the response criterion. Response times were
also sensitive to U, with faster responses at higher U (Figure S3 in Supporting InformationS1).
5.2. Detection of Transience in a Small Catchment Directly With One Cosmogenic Nuclide
Recognition of ongoing topographic change at the scale of headwater streams can be challenging. Testing theoret-
ical concepts in fluvial and hillslope geomorphology often begins by assuming a condition of topographic steady
state (Dietrich etal.,2003). A test of transience using cosmogenic nuclides could eliminate concerns over this
confounding factor when formulating geomorphic transport laws relating mass fluxes to landscape properties.
Recently, Mudd(2017) highlighted that at the scale of a soil profile (experiencing no soil transport from upslope)
one cannot detect instantaneous removals of overlying material or step-changes in denudation rate using single
cosmogenic nuclide depth-profile sampling. This condition can be overcome by using a radionuclide pair such
as in situ
14C/
10Be, where one nuclide decays much more rapidly (Hippe,2017). Using this technique, changes
in the magnitude and timing of denudation rate can be identified for perturbations occurring in the last several
thousand years (e.g., anthropogenic change) (Hippe etal.,2019). Due to its fast decay rate, in situ-produced
14C
is only moderately responsive to changes in Dact in the 30mm kyr
−1 to 100mm kyr
−1 range (Skov etal.,2019),
which is a common range in soil-mantled uplands (Dixon & von Blanckenburg,2012).
Our results show that detection of basin-scale transience due to changes in U could be feasible through systematic
sampling using one nuclide. Divergent hillslopes close to the channel respond rapidly to the change (Figure9a).
If a low ridge and a higher ridge above it were sampled in the same basin, measured Dinf values on these ridges
would differ markedly during transience, potentially beyond commonly reported analytical uncertainties (10%).
In Figure9a, the difference between these Dinf curves is labeled as a “transience detection window.” The differ-
ence in Dinf values corresponds to the current magnitude of transience, but both the future steady-state Dact and the
time since the perturbation would be indeterminate. The maximum magnitude of the difference and the duration
of this transience detection window would scale with the length of the hillslope below the sampled ridge. This
suggests that measuring Ns on low and high ridges within a given basin may provide a constraint on the current
magnitude of basin-scale landscape transience. These could be paired with topographic analysis of knickpoints
(Neely etal.,2017) and stream network disequilibrium between neighboring basins (Willett etal.,2014) to obtain
a more complete picture of a landscape's deviation from steady state.
In contrast, no analogous transience detection window appeared in the climatic perturbation experiment due to
the synchronous response across the landscape (Figure9b). Instead, values of Dinf on different ridges agreed with
Figure 8. (a) Maximum difference between the cosmogenic
10Be-inferred and actual denudation rates (Ddiff,max) during
simulations in which rock uplift rate instantaneously increases. (b) Response times of inferred denudation rate (τDinf) to a
step-change in bedrock uplift rate from 50 to 100mm kyr
−1 on landscapes with different average hillslope lengths (Lh). The
ridge, sideslope, and channel head points are specific, manually selected hillslope positions similar to those in Figure5. The
basin-averaged points correspond to small basins similar to that in Figure6.
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one another within uncertainty throughout this simulation. This suggests that detecting long-lived changes in Dact
arising from changes in hillslope soil transport efficiency may require sedimentary record-based paleo-denudation
rates (Marshall etal.,2017).
6. Conclusions
We used our newly introduced landscape evolution model to compute changes in soil cosmogenic nuclide concen-
trations alongside simultaneous changes in topography, soil thickness, and soil composition stemming from ideal-
ized tectonic and climatic perturbations. Our simulations highlighted several key conclusions.
First, accounting for the effects of chemical erosion on soil quartz enrichment is necessary for accurately infer-
ring denudation rates in weathered soils (Figure2). This supports previous work by Small etal.(1999), Riebe
etal.(2001), and Riebe and Granger(2013). It also suggests that soil chemical erosion may explain some of
the scatter in compilations of TCN-based denudation rates (e.g., Portenga & Bierman, 2011) and thus may
help resolve climatic, lithologic, tectonic, and topographic controls on landscape evolution (e.g., Kirchner &
Ferrier,2013; Larsen etal.,2014; Perron,2017; Willenbring etal.,2013).
Second, tectonic perturbations produce spatiotemporal variations in Dinf that differ from those induced by climatic
perturbations (Figures3–5). This suggests that spatial and temporal patterns in TCN-based denudation rate esti-
mates are useful indicators of the drivers of past changes in topography and sediment fluxes.
Third, estimates of basin-averaged Dinf closely track actual basin-averaged denudation rates at most times during
these simulations, with Dinf differing from Dact by less than ∼10% for the entire tectonic experiment (Figure6).
The exception to this is a brief excursion after the climatic perturbation in Figure5, when Dact changed immedi-
ately everywhere across the landscape, while Dinf took ∼15 kyr to adjust. This implies that TCN-based estimates
of basin-averaged denudation rate should be a reliable reflection of actual denudation rates in transient land-
scapes, except shortly after perturbations that affect the entire landscape at once. This supports early studies that
validated the use of TCN to infer basin-averaged denudation rates (Bierman & Steig,1996; Brown etal.,1995;
Granger etal.,1996), which noted that mixing of stream sediment should strongly damp within-basin variations
in TCN, and therefore that well-mixed stream sediment should have TCN concentrations that closely match the
basin-average TCN concentration.
Fourth, the response times of Dinf to climatic and tectonic perturbations are long (∼10
5–10
6years) and increase
with hillslope length (Figures7 and8). This implies that the response times of Dinf should be influenced by the
Figure 9. (a) Cosmogenic
10Be-inferred denudation rate Dinf at two hillslope positions—one at a high-elevation ridge, the
other at a low-elevation ridge—in a simulation undergoing a step change in rock uplift rate from 50 to 100mm kyr
−1 at t=0
kyr. The uncertainty bounds are ±10% on Ns and are meant to show how Dinf values could be distinguished from one another
in practice. The transience detection window exists for around ∼700 kyr in this case. As both the high-ridge and low-ridge
Dinf are near Dact during the entire perturbation, the gap between the low ridge and high ridge would indicate the current
magnitude of transience but not the final Dact at steady state. (b) Same but for a step change in climatic parameters, which
does not exhibit a detection window.
Journal of Geophysical Research: Earth Surface
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climatic, biological, and lithologic processes that control the efficiencies of river incision and soil transport and
hence hillslope length (Perron etal.,2009,2012; Richardson etal.,2019). This in turn raises the possibility that
the primary influences of life, climate, and rock type on denudation rate may not be on the magnitude of denuda-
tion rate, but rather on the duration and spatial pattern of its responses to perturbations.
Lastly, in landscapes that are still adjusting to a past perturbation, it may be possible to quantify the degree to
which the landscape deviates from steady state. For example, measuring a single TCN in two ridgetop soils within
a single basin can yield the current magnitude of topographic transience after a simple tectonic perturbation
(Figure9). Since ridges tend to have Dinf that most closely track Dact, sampling ridges rather than other landscape
positions may yield the clearest picture of this transient evolution.
Together, these results illustrate the ways in which this model can be used to investigate the coupled evolution of
topography, soil chemistry, and cosmogenic nuclide concentrations. The model will be a useful tool for exploring
the controls on TCN in soil and stream sediment in transient landscapes, and this in turn can help improve inter-
pretations of denudation rate estimates derived from field measurements of TCN.
Data Availability Statement
MATLAB code to run the model and initial conditions for featured model runs are available from the Zenodo
repository (Reed etal.,2023, https://doi.org/10.5281/zenodo.8256787). Model output files for the two numerical
experiments are large (>5GB) but are available upon request from the corresponding author.
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Acknowledgments
We would like to thank Simon Mudd and
Sean Gallen for their reviews that greatly
improved the manuscript. We would
also like to thank Associate Editor Jon
Pelletier for his insightful comments. This
work was supported by NSF Grant EAR
2045433 to KLF.
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