Overland flow generation on hillslopes of complex topography: Analytical Solutions

The analytical solution of the overland flow equations developed by Agnese et al. (2001; Hydrological Processes 15:3225–3238) for rectangular straight hillslopes was extended to convergent and divergent surfaces and to concave and convexprofiles. Towards this aim, the conical convergent and divergent surfaces are approximated by a trapezoidal shape, and theoverland flow is assumed to be always one-dimensional. A simple ‘shape factor’ accounting for both planform geometryand profile shape was introduced: for each planform geometry, a brachistochrone profile was obtained by minimizing afunctional containing a slope function of the profile. Minima shape factors are associated with brachistochrones; interestingly,brachistochrones associated with rectangular surfaces have a simple power-law form. For a fixed profile shape, the rapidness ofoverland flow increases with the degree of divergence; for a fixed planform geometry, however, the overland flow associatedwith convex profiles is more rapid than that associated with concave profiles. An extended analytical solution is also proposedfor the instantaneous response function.