Self-similarity and vanishing diffusion in fluvial landscapes

Shashank Kumar Anand, Matteo B. Bertagni, Theodore D. Drivas, Amilcare Porporato

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article numbere2302401120
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number51
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • General


  • dimensional analysis
  • landscape evolution
  • ridge and valley patterns
  • self-similarity
  • vanishing diffusion


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