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From Turbulence to Landscapes: Universality of Logarithmic Mean Profiles in Bounded Complex Systems

The logarithmic mean-velocity profile is a key experimental and theoretical result in wall-bounded turbulence. Similarly, here we show that the topographic surface emerging between parallel zero-elevation boundaries presents an intermediate region with a logarithmic mean-elevation profile. We use model simulations, which account for growth, erosion, and smoothing processes and give rise to complex topography with channel branching and fractal river networks, as well as data from a physical landscape-evolution experiment. Dimensional and self-similarity arguments are used to corroborate this finding. Our results suggest a universality of the logarithmic scaling in bounded complex systems out of equilibrium, of which landscape topography and turbulence are quintessential examples.