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 convex profiles. Towards this aim, the conical convergent and divergent surfaces are approximated by a trapezoidal shape, and the overland flow is assumed to be always one-dimensional. A simple ‘shape factor’ accounting for both planform geometry and profile shape was introduced: for each planform geometry, a brachistochrone profile was obtained by minimizing a functional 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 of overland flow increases with the degree of divergence; for a fixed planform geometry, however, the overland flow associated with convex profiles is more rapid than that associated with concave profiles. An extended analytical solution is also proposed for the instantaneous response function.