Short communication: Analytical models for 2D landscape evolution

Abstract

Abstract. Numerical modelling offers a unique approach to understand how tectonics, climate and surface processes govern landscape dynamics. However, the efficiency and accuracy of current landscape evolution models remain a certain limitation. Here, I develop a new modelling strategy that relies on the use of 1D analytical solutions to the linear stream power equation to compute the dynamics of landscapes in 2D. This strategy uses the 1D ordering, by a directed acyclic graph, of model nodes based on their location along the water flow path to propagate topographic changes in 2D. This analytical model can be used to compute in a single time step, with an iterative procedure, the steady-state topography of landscapes subjected to river, colluvial and hillslope erosion. This model can also be adapted to compute the dynamic evolution of landscapes under either heterogeneous or time-variable uplift rate. This new model leads to slope–area relationships exactly consistent with predictions and to the exact preservation of knickpoint shape throughout their migration. Moreover, the absence of numerical diffusion or of an upper bound for the time step offers significant advantages compared to numerical models. The main drawback of this novel approach is that it does not guarantee the time continuity of the topography through successive time steps, despite practically having little impact on model behaviour.

Mountain Rivers Reveal the Earthquake Hazard of Geologic Faults in Silicon Valley

“…Independent geologic mapping and geophysical studies reduce epistemic uncertainty by identifying fault geometry and lithologically‐induced variations in channel steepness. We also acknowledge that the predictive power of our elastic, quasi‐static, fixed‐geometry mechanical, and constant‐uplift geomorphic models can be improved, for instance, by accounting for: (a) long‐term planform modifications of river networks (Castelltort et al., 2012), (b) time variations of uplift‐rate (Goren et al., 2014; Steer, 2021), and (c) material yielding and block advection (Baden et al., 2022), particularly in tectonic environments dominated by lateral motion. Nevertheless, our inversion approach might be used as a starting point in areas where sparse geodetic and geologic information may leave geoscientists with few other tools for constraining seismic hazard.…”

Physically‐based analytical erosion for fast terrain generation

“…Compared to previous simulation-based methods that required iterating over time, this solution directly expresses the elevation of the terrain from the uplift and drainage area. This is why we follow geology literature [Ste21] to call this an analytical solution (with respect to time), even if we need to resort to a numerical evaluation of the integral over space.…”

“…We follow [Ste21] and order the computation along the river network. This network consists of a set of trees that covers the terrain and represents the progressive merging of high-altitude small streams down to the larger rivers.…”

“…A solution of Eqn. 3 was proposed in Earth sciences [Ste21] with the assumption that u is constant in space and varies in time, which is important for geomorphologists who study the erosional response to tectonic perturbations. We prefer an orthogonal approach where the uplift varies in space but not in time, as we expect the uplift to be easier to control for the user as a function of space alone [SPF*23].…”

Understanding tectonics from fluvial topography by using the stochastic‐threshold incision model: Theory and application to the Dadu River basin, eastern Tibetan Plateau

“…Knickpoints move upstream and reshape their downstream channel profile, modulating the erosion rate pattern to attend an incision–uplift equilibrium state. An assumption of linear dependence of erosion rate on channel gradients makes it easy to derive the analytical solution to the SPIM, which has been used to forward topographic evolution (Steer, 2021) and to invert tectonic uplift history from river long profiles (Fox et al, 2015; Goren et al, 2022; Goren, Fox, & Willett, 2014; Rudge et al, 2015).…”