CEE Seminar Self-similarity and vanishing diffusion in fluvial landscapes: theoretical and numerical challenges

With intricate ridge and valley networks, natural landscapes shaped by fluvial erosion often exhibit universal scaling laws and self-similar behavior. These properties are also displayed by the solutions of landscape evolution modes (LEMs), when fluvial erosion dominates over the smoothing tendency of diffusion (i.e., soil creep). 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. In the vanishing limit, diffusion remains crucial and localized in valleys and ridges where abrupt slope changes occur (as shock waves).
WE illustrate how focusing on the essential elements that distinguish landscape evolution, minimalist LEMs become amenable to dimensional analysis and other methods of nonlinear field equations, used for example in fluid mechanics and turbulence, offering fertile ground to unveil distinct dynamic regimes (e.g., unchannelized, from incipient valley formation, transitional and statistically self-similar fractal regime), and properly formulate questions related to the existence of steady state solution (as opposed to a situation of space time chaos, similar to a geomorphological turbulence). We also explore the parallelism between the landscape self-similarity and the self-similarity of fully developed turbulent flows and discuss challenges for evaluating numerical simulation and novel avenues for numerical methods, as well as ways to bridge between spatially discrete models (i.e., river networks) and continuous, partial-differential-equation models.

Porporato, A. (2022). Hydrology without dimensions. Hydrology and Earth System Sciences, 26(2), 355-374.
Anand, S. K., Bertagni, M. B., Drivas, T. D., & Porporato, A. (2023). Self-similarity and vanishing diffusion in fluvial landscapes. Proceedings of the National Academy of Sciences, 120(51), e2302401120.