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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.
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Dissipation enhancement by shear flows for generalized diffusion operators in high dimension
We establish conditions for shear flows on the d -dimensional torus that give enhanced dissipation for the associated linear advection-diffusion equation for well-prepared data. The diffusion operator can be of fractional or high order and does not need to have constant coefficients. We then construct flows that satisfy these assumptions and obtain a quantitative estimate on the dissipation enhancement. Our examples generalize known examples in two space dimensions to the high-dimensional setting, which is relevant in applications to sampling a distribution and in optimization.
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- PAR ID:
- 10648641
- Publisher / Repository:
- International Press
- Date Published:
- Journal Name:
- Communications in Mathematical Sciences
- Volume:
- 23
- Issue:
- 8
- ISSN:
- 1539-6746
- Page Range / eLocation ID:
- 2203 to 2213
- Subject(s) / Keyword(s):
- generalized diffusion advection dissipation enhancement shear flow high-dimensional flows
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.4310/CMS.250930031923
