Variational Analysis of Landscape Elevation and Drainage Networks

TL;DR

This paper develops a variational framework for landscape evolution, linking models of erosion and drainage networks, and identifies conditions under which landscapes reach extremal surface states.

Contribution

It introduces a variational approach to landscape evolution models, connecting them with optimal channel networks and analyzing the role of diffusion.

Findings

Steady-state surfaces extremize average elevation without diffusion.

Critical surfaces are minima or maxima depending on the erosion exponent m.

Diffusion influences the variational principles governing landscape shapes.

Abstract

Landscapes evolve toward surfaces with complex networks of channels and ridges in response to climatic and tectonic forcing. Here we analyze 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 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. Our results establish a connection between Landscape Evolution Models (LEMs) and Optimal Channel Networks (OCNs) and elucidate the role of diffusion in the governing variational principles.