Variational analysis of landscape elevation and drainage networks

Milad Hooshyar; Shashank Anand; Amilcare Porporato

doi:10.1098/rspa.2019.0775

Variational analysis of landscape elevation and drainage networks

Proc Math Phys Eng Sci. 2020 Jul;476(2239):20190775. doi: 10.1098/rspa.2019.0775. Epub 2020 Jul 1.

Authors

Milad Hooshyar¹, Shashank Anand², Amilcare Porporato³

Affiliations

¹ Princeton Environmental Institute and Princeton Institute for International and Regional Studies, Princeton University, Princeton, NJ, USA.
² Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA.
³ Princeton Environmental Institute and Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ, USA.

Abstract

Landscapes evolve towards surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here, we analyse variational principles giving rise to minimalist models of landscape evolution as a system of partial differential equations that capture the essential dynamics of sediment and water balances. Our results show that in the absence of diffusive soil transport the steady-state surface extremizes the average domain elevation. Depending on the exponent m of the specific drainage area in the erosion term, the critical surfaces are either minima (0 < m < 1) or maxima (m > 1), with m = 1 corresponding to a saddle point. We establish a connection between landscape evolution models and optimal channel networks and elucidate the role of diffusion in the governing variational principles.

Keywords: landscape; network; optimality.