The branching patterns in landscape evolution has features reminiscent of other complex systems including fluid turbulence. We consider the temporal dynamics of land surface elevation in detachment-limited conditions, modeled by a system of partial differential equations for the surface elevation and specific contributing area. We show that for parallel boundary conditions of constant elevation, the steady-state surface exhibits features typical of wall-bounded turbulent flow. In particular, the mean-elevation profile develops a logarithmic region in the intermediate distance from the boundary, a behavior which resembles the well-known turbulence logarithmic velocity profile. The deviation of the surface from its mean profile is linked to the (negative) correlation between the fluctuations of surface elevation and specific catchment area by equations similar to the Reynolds-averaged Navier-Stokes equations. Solutions in conditions of negligible diffusion show that while the global mean elevation approaches steady-state fast and monotonically, the variance evolves toward its steady state more slowly and with possible overshooting.
Porporato, A.M., M. Hooshyar, S.K. Anand (2019): Scaling and dynamics of surface elevation in terrestrial landscape evolution. American Geophysical Union Fall Meeting, San Francisco, CA, December 9-13, 2019.
This Paper/Book acknowledges NSF CZO grant support.