Analytical and numerical study of optimal channel networks

We analyze the optimal channel network model for river networks Using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected ill the power-law behavior of various quantities characterizing the morphology of the basin, In the context of a finite-size scaling ansatz, the exponents describing the power-law behavior are calculated exactly and show mean-field behavior, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modified versions of the model, incorporating quenched disorder, are considered: the first simulates heterogeneities in the local properties of the soil and the second considers the effects of a nonuniform rainfall. In the region of mean-field behavior, the model is shown to be robust for both kinds of perturbations. In the two limiting cases the random rainfall is still irrelevant, whereas the heterogeneity in the soil properties leads to different universality classes. Results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum. The statistics of the local minima are found to resemble more strongly observational data on real rivers.