Mathematical Modeling of Landforms: Optimality and Steady-State Solutions

The purpose of this paper is to first show how an Euler-Lagrange theorem can be applied to the steady-state fluvial landscape model, thereby allowing the geomorphic solution surfaces to be understood as those that globally optimize the difference between kinetic and potential energy dissipation (Hamilton’s Principle) while conserving mass. It is then shown how this variational formulation of the steady-state problem makes it possible to exploit the well-known Ritz-Galerkin method for finding finite-element numerical solutions to partial differential equations. A “local equation” is derived that shows how elevations at a lattice node are related to the elevations of neighboring nodes. A variety of special solutions to this local equation are given that exhibit many of the ata-point geometric features that characterize real fluvial landscapes, including peaks, ridges, saddles, hillslopes and forks. It is also shown that pits cannot occur as a solution to the local equation, so that numerical solutions will always be hydrologically sound.