Self-similarity and vanishing diffusion in fluvial landscapes

Significance

With intricate ridge and valley networks, natural landscapes shaped by fluvial erosion exhibit universal scaling laws and self-similar behavior. Here, we show that these properties are also displayed by the solutions of a landscape evolution model when fluvial erosion dominates over the smoothing tendency of soil diffusion. Under such conditions, an invariant self-similar regime is reached where the average landscape properties become independent of the balance between fluvial erosion and soil diffusion. Soil diffusion remains crucial and localized in valleys and ridges where abrupt slope changes occur. We also explore the parallelism between the landscape self-similarity and the self-similarity of fully developed turbulent flows.

Abstract

Complex topographies exhibit universal properties when fluvial erosion dominates landscape evolution over other geomorphological processes. Similarly, we show that the solutions of a minimalist landscape evolution model display invariant behavior as the impact of soil diffusion diminishes compared to fluvial erosion at the landscape scale, yielding complete self-similarity with respect to a dimensionless channelization index. Approaching its zero limit, soil diffusion becomes confined to a region of vanishing area and large concavity or convexity, corresponding to the locus of the ridge and valley network. We demonstrate these results using one dimensional analytical solutions and two dimensional numerical simulations, supported by real-world topographic observations. Our findings on the landscape self-similarity and the localized diffusion resemble the self-similarity of turbulent flows and the role of viscous dissipation. Topographic singularities in the vanishing diffusion limit are suggestive of shock waves and singularities observed in nonlinear complex systems.