Feasible optimality implies Hack's law

We analyze the elongation (the scaling properties of drainage area with mainstream length) in optimal channel networks (OCNs) obtained through different algorithms searching for the minimum of a functional computing the total energy dissipation of the drainage system. The algorithms have different capabilities to overcome the imprinting of initial and boundary conditions, and thus they have different chances of attaining the global optimum. We find that suboptimal shapes, i.e., dynamically accessible states characterized by locally stationary total potential energy, show the robust type of elongation that is consistently observed in nature. This suggestive and directly measurable property is not found in the so-called ground state, i.e., the global minimum, whose features, including elongation, are known exactly. The global minimum is shown to be too regular and symmetric to be dynamically accessible in nature, owing to features and constraints of erosional processes. Thus Hack's law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dynamic origin of fractal forms in nature.