  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 1D analytical solutions and 2D
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.
