Deterministic Tree Networks for River Drainage Basins

This paper shows that the dendritic patterns formed by low-resistance channels in a river drainage basin are reproducible and can be deduced from a single principle that acts at every step in the development of the pattern: the constrained minimization of global resistance in area-to-point flow. The river basin is modeled as a two-dimensional territory with Darcy flow through a saturated heterogeneous porous medium with uniform flow addition per unit area. From one step to the next, small elements of the porous medium are dislodged and removed in ways that minimize the global flow resistance. The removed elements are replaced by channels with lower flow resistance. The channels form a dendritic pattern that is deterministic, not random. The finest details of this structure are sensitive to internal properties and external forcing, i.e. variations in the local properties of the flow medium, and the manner in which the total area-to-point flow rate varies as the structure develops. Remarkably insensitive to such effects are the basic type and rough size of the flow structure (channels versus no channels, dendrite, number of branches) and the minimized global resistance to flow.