  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.
