Multifractality of flow distribution in the river-network model of Scheidegger

The multifractal structure of flow distribution is investigated in the river-network model of Scheidegger [Bull. IASH 12, 15 (1967)]. It is shown that the partition function Z(q)==${\mathit{tsum}}_{\mathit{i}}$${\mathit{I}}_{\mathit{i}}^{\mathit{q}}$ scales as Z(q)≊${\mathit{L}}^{\mathrm{\zeta }\left(\mathit{q}\right)}$ where ${\mathit{I}}_{\mathit{i}}$ is the flow of water passing over the bond i within the river network, the summation ranges over all bonds, and L is the size of the river network. In the limit of a sufficiently large q, ζ(q)/q gives the exponent of the drainage basin of a river. The exponent also equals to the fractal dimension ${\mathit{d}}_{\mathit{f}}$ of a single river. The f-α spectrum of the normalized flow distribution is calculated. It is found that the fractal dimension ${\mathit{d}}_{\mathit{f}}$ of a river is exactly given by ${\mathit{d}}_{\mathit{f}}$=2-α(∞). The flow distribution shows a characteristic multifractal structure for the river network. The river-width distribution also shows the multifractality if the width w of a river scales as w≊${\mathit{I}}^{\mathrm{\beta }}$.