Modeling river erosion and knickpoint propagation using a lagrangian description: numerical benefits and implications for river profiles
Résumé
Landscape evolution models generally solve for the time variation of elevation due to erosional processes on a
fixed grid. This Eulerian description offers a tractable approach, despite leading to unnecessary dependency to
slope, and in turn to numerical diffusion, when solving for the specific stream power incision model, e.g. with a
slope exponent equal to 1. Indeed, under this condition, the stream power model reduces to a linear kinematic
wave equation with a wave celerity independent of the local slope (Whitham, 1974; Whipple & Tucker, 1999).
Using a Lagrangian description and solving for the horizontal advection of the topography due to erosion enables
to develop a simple finite difference numerical scheme fully independent of local slope. This scheme presents
numerous benefits over Eulerian descriptions, including the absence of numerical diffusion, unconditional stability
and the possibility to track geomorphological features and slope instabilities such as knickpoints. We exploit the
latter to track co-seismic knickpoints in simulations where uplift history is computed from a statistical model
of earthquakes (Turcotte et al., 2007; Leonard 2010). This model results in spatial and temporal distributions
of earthquakes that are consistent with Gutenberg-Richter frequency-magnitude scaling, a modified Båth’s law,
Omori’s law in space and time (Turcotte et al., 2007) and earthquake dimension scaling laws (Leonard 2010).
Motivated by recent results from sandbox experiments (Baynes, 2016), we then introduce a dependency of
knickpoint celerity to knickpoint height, that is initially here equals to co-seismic displacement. This model
authorizes the merging of successive and colliding knickpoints due to different celerities. We use this model to
investigate the role of knickpoint height and temporal distribution and of the dependency of their celerity to height
on the shape of river profiles. This model leads to river profiles with concavities deviating from predicted ones and
to particular morphologies, including river profile characterized by a succession of knickzones separated by flat
zones. We then discuss these implications in the light of natural river profiles.