Numerical modelling of landscape and sediment flux response to precipitation rate change
Authors: Armitage John J. , Whittaker Alexander C. , Zakari Mustapha , Campforts Benjamin 
Published: February 2018

Abstract. 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
105 years for post-perturbation sediment fluxes to return to within
50 % of their initial values, for theoretical landscapes with a scale of
100×100 km. Additionally from the same starting conditions, the
amplitude of the sediment flux perturbation in the transport model is
greater, with much larger sensitivity to catchment size. An important finding
is that both models respond more quickly to a wetting event than a drying
event, and we argue that this asymmetry in response time has significant
implications for depositional stratigraphies. Finally, we evaluate the extent
to which these constraints on response times and sediment fluxes from simple
models help us understand the geological record of landscape response to
rapid environmental changes in the past, such as the Paleocene–Eocene thermal
maximum (PETM). In the Spanish Pyrenees, for instance, a relatively rapid (10
to 50 kyr) duration of the deposition of gravel is observed for a climatic shift
that is thought to be towards increased precipitation rates. We suggest that the
rapid response observed is more easily explained through a diffusive
transport model because (1) the model has a faster response time, which is consistent
with the documented stratigraphic data, (2) there is a high-amplitude spike
in sediment flux, and (3) the assumption of instantaneous transport is
difficult to justify for the transport of large grain sizes as an alluvial
bedload. Consequently, while these end-member models do not reproduce all
the complexity of processes seen in real landscapes, we argue that variations
in long-term erosional dynamics within source catchments can fundamentally
control when, how, and where sedimentary archives can record past
environmental change.