Laboratory-scale experiments of erosion have
demonstrated that landscapes have a natural (or intrinsic) response time to a
change in precipitation rate. In the last few decades there has been growth
in the development of numerical models that attempt to capture landscape
evolution over long timescales. However, there is still an uncertainty
regarding the
validity of the basic assumptions of mass transport that are made in deriving
these models. In this contribution we therefore return to a principal
assumption of sediment transport within the mass balance for surface
processes; we explore the sensitivity of the classic end-member landscape
evolution models and the sediment fluxes they produce to a change in
precipitation rates. One end-member model takes the mathematical form of a
kinetic wave equation and is known as the stream power model, in which sediment
is assumed to be transported immediately out of the model domain. The second
end-member model is the transport model and it takes the form of a diffusion
equation, assuming that the sediment flux is a function of the water flux and
slope. We find that both of these end-member models have a response time that
has a proportionality to the precipitation rate that follows a negative power
law. However, for the stream power model the exponent on the water flux term
must be less than one, and for the transport model the exponent must be
greater than one, in order to match the observed concavity of natural
systems. This difference in exponent means that the transport model generally
responds more rapidly to an increase in precipitation rates, on the order of

How river networks form and how landscapes erode remains a basic research
question despite more than a century of experimentation and study. At a
fundamental level, the root of the problem is a lack of an equation of motion
for erosion derived from first principles

The response of landscapes and sediment routing systems to a change in the
magnitude or timescale of precipitation rates is expected to depend on the
long-term erosion law implemented

It has been increasingly recognized over the last 2 decades that many basic
geomorphic measures of catchments, such as the scaling between channel slopes
and catchment drainage areas, are typically unable to distinguish the
erosional processes behind their formation

Non-uniqueness or equifinality is a common problem when comparing the
morphology generated from landscape evolution models

In this article we make a comparative study between the transport and stream power model to further explore the potential differences between these two end-member hypothetical landscape evolution models. We will focus on the transient period of adjustment to a perturbation in precipitation rates, and using end-member numerical models we attempt to evaluate how the response time varies as a function of the model forcing. To this end we aim to find the model parameters that generate similar landscape morphologies such that we can subsequently explore how the same end-member models respond to a change in precipitation rate. We believe that the results of this study have implications for understanding the responses of landscapes to past changes in climate and could potentially be compared with and tested against further laboratory experiments.

Diagram showing the conservation of mass within a 2-D domain where
mass enters the system through uplift,

We aim to understand the effects of the most basic assumptions of mass
transport in landscape evolution on the sediment flux record. In other words,
how do the response times vary for the advective stream power law and the
diffusive transport model? To this end we derive the two models from first
principles to demonstrate clearly how, from the same starting point, the
fundamental assumptions made about mass transport initially give rise to very
different model equations. We use this framework as a context for our
investigation of an eroding system responding to precipitation change. We
first define a one-dimensional system from which the basic equations can be
assembled. Following

It is important to realize that to solve Eq. (

One basic assumption to make is that there is always a supply of
transportable sediment; we can then follow through with the summation in
Eq. (

To solve this equation in one dimension we assume that the water flux is a
function of the precipitation transported down the river network. The water
collected is taken from the upstream drainage area,

However, returning to Eq. (

It is clearly plausible to suppose that erosion is primarily due to flowing
water, so the assumption of geologically instantaneous transport may well be
valid for mass that is transported as suspended load within the water column.
Such an assumption is less clear for bedload transport. We can assume that
the speed at which suspended loads travel down-system is a function of the
height achieved within each hop, which is a function of the water depth,
settling velocity, and flow velocity. For small grains

The percentage of mass transported in suspension may also be quite
significant. For a small Alpine braided river it was found that the majority
of mass was transported as suspended load

Assuming surface flow is the primary driver of landscape erosion and that
positive

In two dimensions the change in elevation is then given by

To solve Eq. (

We have demonstrated two different fundamental equations for change in
elevation in 2-D (Eqs.

If

To solve Eqs. (

We will explore how an idealized landscape evolves under uniform uplift at a
rate of 0.1 mm yr

The response time for the transport model scales by the effective
diffusivity and can be given by

It has been previously demonstrated that both end-member models can generate
convex-up long profiles

Six models have been run without a change in precipitation to find the steady-state topography. The models explored are first a set of three with varying

Sediment flux out of the model domain for the transport models in which

When the transport coefficient

We subsequently analyse the topography for the relationship between trunk
river slope and drainage area, as shown in Fig.

Slope–area relationship for trunk streams derived using

Sediment flux out of the model domain for the stream power models in which

In order to provide a comparison for the morphology of the transport model we
explore how the stream power model evolves to a steady state. The landscape
derived from the stream power model, as shown in Eq. (

Following the previous examples, we analyse the topography for the
relationship between trunk river slope and drainage area
(Fig.

The stream power and transport models can both fit the observed slope–area
relationships of the present day landscape morphology, for example

Response of the transport and stream power model to a reduction in
precipitation rate.

When the model is perturbed by a change in precipitation rate the sediment
flux output will first change as the erosive power changes (e.g.
Fig.

Response to a change in precipitation rate;

The response of the stream power model to an identical reduction in
precipitation at a model time of 10 Myr takes a similar form, with an
initial decrease in sediment flux out followed by a gradual recovery
(Fig.

The magnitude of the response for all the runs is greater for the transport
model when compared to the stream power model (Fig.

Transport model evolution due to a reduction in precipitation.

Stream power model evolution due to a reduction in
precipitation.

To explore how the difference in response time and magnitude is expressed in
the landscape, we extract the river profiles of the main trunk systems for
models in which

The response time of the transport model is known to be a function of the
transport coefficient and the magnitude of the precipitation rate

Log–log plot of response time to a change to a precipitation rate
of

The stream power model likewise has a response time that is a function of
precipitation rate (Fig.

Both models display a response time that is a function of the precipitation
rate (Figs.

In contrast, the response time of the stream power model is not as strongly
inversely dependent on the precipitation rate (Fig.

The position of the critical point at which the stream power model responds more
rapidly than the transport model is a function of the water flux and the
collection of coefficients. In the model comparison developed here, we have
compared two model catchments that have a similar slope–area exponent,

The stream power model is insensitive to the size of the model domain because
of the particular choice of

A final key difference between the transient sediment flux responses of the
two models is that the peak magnitude of system response to a change in
precipitation rate is systematically larger for the transport model
(Fig.

Up to this point we have compared how the models respond to a precipitation
rate change when the solutions are linear. However, there is reasonable
debate as to the value of the slope exponent

Log–log plot of response time for different slope exponents and
uplift rates to a change to a precipitation rate from an initial value of

We find that for both the transport and stream power model, when the slope
exponent is greater than one, the model response time is a function of uplift
rate. The faster the rate of uplift, the faster the system responds to a
change in precipitation rate. If the response time for a system recovery to
steady state by 50 or 10 % is plotted on a log–log plot against uplift rate
we find that the response time is proportional to the uplift rate raised to a
negative power (Fig.

Up until this point we have only explored how the numerical models respond to
an increase or decrease in precipitation rate by keeping the initial
precipitation rate fixed at

Log–log plots for the transport model and the stream power model in
1-D for a step change in precipitation rate; the initial precipitation
rate,

For the linear and non-linear transport model we find that if the initial
precipitation is less than the final precipitation (

In the case of the linear and non-linear stream power model, the response
time has a no dependence on the initial precipitation rate and is only a
function of the final precipitation rate (Fig.

In deriving the two end-member models to describe landscape evolution, we
showed that if the rate of transport of sediment were assumed to be
instantaneous (i.e. all sediment is transported out of the model domain) then
the stream power model would be appropriate to describe catchment erosion.
However, if it is instead assumed that the rate of sediment transport is not
instantaneous, then we arrive at a model in which erosion scales with the
rate of change of sediment flux, which itself is dependent on both linear and
potentially non-linear slope-dependent terms. These two end-members can
produce similar steady-state landscapes, as noted by a number of previous
studies

It is also important to stress that the catchment responses and the
predicted sediment fluxes out of these two model domains might be variously relevant
to different erosional and depositional domains

Under certain parameter sets it is relatively straightforward to generate two
landscapes eroded by the transport or stream power model that have similar
elevation, slope, and area metrics (Figs.

We have demonstrated that models limited by their ability to transport
sediment tend to have shorter response times to an increase in rainfall rate
and thus re-achieve pre-perturbation sediment flux values more rapidly
compared to stream-power-dominated systems, particularly when catchment
length scales are small (e.g.

Given that the response time is a function of the water flux exponent (

For such conditions, the stream power model predicts a landscape response
time to a change in precipitation of the order of

Finally, it is worth noting that the model response time has implications for
the inverse modelling of river profiles. When river-long profiles are inverted
for uplift, erosion is typically assumed to be captured by the stream power
model

To what extent do these model results, which start from similar steady-state
topographies, help us to understand whether stratigraphic records of sediment
accumulation through time do or do not reflect the effects of climatic
change on sediment routing systems governed by differing long-term erosional
dynamics? One motivation for this study has come from the increasing number
of field and stratigraphic investigations of terrestrial sedimentary
deposits, apparently contemporaneous with (and taken to record) known past
climate perturbations, such as the Palaeocene–Eocene thermal maximum (PETM),
a hyperthermal event that occurred around 56 Ma. Stratigraphers often
correlate changing stratigraphic characteristics with changing environmental
boundary conditions in a qualitative way

To compare our model predictions with observations it is clear that we have
to use the depositional record. Therefore, there is an implicit assumption
that stratigraphy is a faithful recorder of erosion. It is, however, possible
that climatic change will also alter processes that control sediment
deposition, for example by altering how sediment partitions from transport
into stratigraphy. By using estimates of the total volume of sediment
deposited within the Escanillia Eocene sedimentary system in the Spanish
Pyrenees, it has been demonstrated that climatic change can recreate observed
changes in grain size deposition

The PETM is arguably the most rapid global warming event of the Cenozoic,
with a rise in global surface temperatures by 5 to 9

The initial warming associated with the PETM event occurred at ca. 55.5 Ma
and may have been as abrupt as 20 kyr, with a duration of 100 to 200 kyr
based on the synthesis of

A clear response to the PETM is recorded within both the Spanish Pyrenees and
the western US; however, the responses are arguably not
the same. At the onset of the PETM there is strong evidence for the
contemporaneous increase in precipitation rates and the deposition of coarse
gravels known as the Claret Conglomerate

The Claret Conglomerate was likely deposited rapidly, representing a fast
response to climate change. If we assume a constant rate of deposition, then
the Claret Conglomerate accounts for roughly 30 % of the total duration of
deposition for the CIE (170 kyr;

Erosional source catchment areas were likely

The time-equivalent sections in the Bighorn and Piceance Creek basins of the
western US also provide clear evidence of anomalous sedimentation at the
PETM; however, in this case the duration of deposition is somewhat longer,

In contrast, the Bighorn Basin boundary sandstone sediments are contained
within the PETM time period and indicate uniform flow depths and widths
during this time, while also being coarser than the underlying horizons.
Moreover, proxy data suggest a net decrease rather than an increase in
precipitation

In this section we consider the implications of our model
outputs, both generally for interpreting sediment routing system response to
boundary condition change and specifically in the context of the well-studied
PETM event. While the sediment flux response of the models to a change in
precipitation are at a first-order level broadly similar, there are four key
differences to highlight. First, starting from the same initial conditions,
the sediment transport model appears to be more sensitive to precipitation
change than the equivalent stream power model. It is therefore a good
candidate for which rapid catchment-wide responses are recorded to, for example, a climate
change event, as we argued for the PETM Claret Conglomerate in the Spanish
Pyrenees. Second, we note that in both model cases there is a quicker response
to a wetting than a drying event, something which has not been well established
or demonstrated from field observations. Nevertheless we argue that field
data sets, including PETM studies, may already have recorded this asymmetry,
although it may not have been recognized as such. Third, the sediment
transport model has a greater magnitude of peak sediment flux and is
particularly sensitive to catchment size. Finally, we note that response time
in both models is a function of uplift rate for

However, it is important to recognize that in deriving these two classic
end-member models we have simplified landscape evolution considerably. We
acknowledge that change in the model parameters,

Deterministic numerical models of landscape evolution rest on fundamental
assumptions on how sediment is transported down-system. The stream power law
is based on the assumption that all sediment generated is transported
instantaneously out of the landscape. Transport models assume that there is
an endless supply of sediment to be transported. The existence of knickpoints
within river-long profiles, assumed to be produced by a system perturbation
such as a base level, has been used to provide evidence in support of the
stream power law in upland areas

Both models suggest that the response time of landscape to a change in
precipitation rate has a proportionality of the form of a negative power law
(Eqs.

While this study does not address whether or not these sediment flux signals
will be preserved in the stratigraphic record, a problem that fundamentally
rests on the availability of accommodation to capture the eroded sediment

The 1-D solution to the transport model is available from
John Armitage (armitage@ipgp.fr). The 1-D solution to the stream power model
is available from Benjamin Campforts (benjamin.compforts@kuleuven.be).
Fastscape is available from Jean Braun (GFZ Potsdam) by request. The 2-D
solution to the transport model was developed by Guy Simpson (University of
Geneva) and is available as part of

The solution to the one-dimensional stream power law (Eq.

For the transport model (Eq.

The steady-state solutions are plotted in the case that

The range of gradients found for river catchments for this type of slope–area
analysis, usually referred to as concavity, generally lies within the range

Gradient,

Clearly there is a combination of

The positive slope–area relationship for the transport model for small
catchment areas (see Figs.

The authors declare that they have no conflict of interest.

We would like to thank Guy Simpson (University of Geneva) for sharing his
numerical model that solves the