Power law distribution of discharge in ideal networks

Data for several river networks show algebraically decaying distribution functions for the mean annual discharge and the dissipation rate (Rodríguez‐Iturbe et al., 1992). We derive a relation between the exponent a in the integrated mean annual discharge distribution P ( Q ≥ q ) ∼ q −α and the topological dimension D t of the network, α = 1 − 1/ D t . Using the experimentally determined value D t ≈ 1.8 (Tarboton et al., 1988) we find α ≈ 0.45, in good agreement with the data of (Rodríguez‐Iturbe et al., 1992). For the random model (Shreve, 1967, 1969; Smart and Werner, 1974) we find an exponent of 1/2.