FEASIBLE OPTIMALITY IMPLIES HACKS LAW

We analyze the elongation (the scaling properties of drainage area wit h mainstream length) in optimal channel networks (OCNs) obtained throu gh different algorithms searching for the minimum of a functional comp uting the total energy dissipation of the drainage system. The algorit hms have different capabilities to overcome the imprinting of initial and boundary conditions, and thus they have different chances of attai ning the global optimum. We find that suboptimal shapes, i.e., dynamic ally accessible states characterized by locally stationary total poten tial energy, show the robust type of elongation that is consistently o bserved in nature. This suggestive and directly measurable property is not found in the so-called ground state, i.e., the global minimum, wh ose features, including elongation, are known exactly. The global mini mum is shown to be too regular and symmetric to be dynamically accessi ble in nature, owing to features and constraints of erosional processe s. Thus Hack's law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the syst em as a whole explains the dynamic origin of fractal forms in nature.