Keeping the edge: A numerical method that avoids knickpoint smearing when solving the stream power law

Abstract

The stream power equation is commonly used to model river incision into bedrock. Although specific conditions allow an analytical approach, finite difference methods (FDMs) are most frequently used to solve this equation. FDMs inevitably suffer from numerical smearing which may affect their suitability for transient river incision modeling. We propose the use of a finite volume method (FVM) which is total variation diminishing (TVD) to simulate river incision in a more accurate way. The TVD_FVM is designed to simulate sharp discontinuities, making it very suitable to simulate river incision pulses. We show that the TVD_FVM is much better capable of preserving propagating knickpoints than FDMs, using Niagara Falls as an example. Comparison of numerical results obtained using the TVD_FVM with analytical solutions shows a very good agreement. Furthermore, the uncertainty associated with parameter calibration is dramatically reduced when the TVD_FVM is applied. The high accuracy of the TVD_FDM allows correct simulation of transient incision waves as a consequence of older uplift pulses. This implies that the TVD_FVM is much more suitable than FDMs to reconstruct regional uplift histories from current river profile morphology and to simulate river incision processes in general.

Vegetation cover and topography rather than human disturbance control gully density and sediment production on the Chinese Loess Plateau

“…We used a value of 0.75 for a and a value of 0.45 for b. These values are similar to those reported for incising rivers (Campforts and Govers, 2015) as well as to the optimal values relating rill erosion to slope gradient and length . Also a recent study on gully headcut retreat rates worldwide indicates that gully erosion rates are more or less proportional to the square root of the contributing area (Vanmaercke et al, 2016).…”

Numerical modelling of landscape and sediment flux response to precipitation rate change

“…If n = 1 and if γ = 1 the equations become non-linear. In this case the numerical solutions can become unstable for simple explicit schemes and may suffer from too much numerical diffusion for implicit schemes, unless the size of the time step is limited by the appropriate Courant-Friedrichs-Lewy (CFL) condition (Campforts and Covers, 2015). Given the short time steps required to obtain an accurate solution, we explore the nonlinear solutions for erosion down a river-long profile in 1-D. We solve for the stream power model (Eq.…”

“…Given the short time steps required to obtain an accurate solution, we explore the nonlinear solutions for erosion down a river-long profile in 1-D. We solve for the stream power model (Eq. 12) using an explicit total variation diminishing scheme with the appropriate CFL condition (Campforts and Covers, 2015). For the transport model (Eq.…”

“…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 (Castelltort and Van Den Dreissche, 2003;Armitage et al, 2011Armitage et al, , 2013. Some numerical modelling studies based on treating erosion as a length-dependent diffusive problem suggest that landscape responses to a change in rainfall are also on the order of 10 5 to 10 6 years, similar to tectonic perturbations, although they produce diagnostically different stratigraphic signatures from the latter (e.g.…”

Numerical modelling landscape and sediment flux response to precipitation rate change

“…In this case the numerical solutions 5 can become unstable for simple explicit schemes, and may suffer from too much numerical diffusion for implicit schemes, unless the size of the time step is limited by the appropriate Courant-Friedrichs-Lewy (CFL) condition (Campforts and Covers, 2015). Given the short time steps required to obtain an accurate solution, we explore the non-linear solutions for erosion down a river long profile in 1-D. We solve for the stream power model (equation 8) using an explicit total variation diminishing scheme with the appropriate CFL condition (Campforts and Covers, 2015). For the transport model (equation 12) we use an 10 explicit finite element model with quadratic elements and the appropriate CFL condition to find a stable solution.…”