Variational analysis of landscape elevation and drainage networks

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.