The direction of landscape erosion

Abstract. The rate of erosion a landscape depends largely on local gradient and material fluxes. Since both quantities are functions the shape catchment surface, this dependence constitutes mathematical straitjacket, in sense that – subject to simplifying assumptions about process, absent variations external forcing erodibility change surface geometry is solely function geometry. Here we demonstrate how use geometric self-constraint convert gradient-dependent model into its equivalent Hamiltonian, explore implications having Hamiltonian description process. To achieve conversion, recognize defines velocity motion orthogonal direction, express reciprocal form as surface-normal slowness. By rewriting tilt terms normal slowness components deploying substitution developed mechanics, extract what known fundamental metric phase space; square Hamiltonian. Such provides several new ways solve for evolution an surface: here it derive Hamilton's ray-tracing equations, which describe point at point. In context, involves two distinct directions: (i) points subvertically downwards, (ii) ray upstream generally small angle horizontal with sign controlled by scaling slope. If scales faster than linearly gradient, rays obliquely upwards, but if sublinearly downwards. This erosional anisotropy explains why, previous studies have shown, knickpoints behave depending exponent. Analysis shows carry boundary-condition information upstream, they geodesics, meaning takes path least time. Correspondingly, time changes propagate set these rays. also reveals surfaces critical tilt, given simple exponent, ray-propagation behaviour changes. Channel profiles generated from non-dimensionalized entirely determined exponents dimensionless expressed downstream boundary.