Variation of mainstream length with basin area in river networks

Mainstream length in river networks from various parts of the world varies statistically in proportion to basin area raised to a power that decreases from about 0.6 for small to medium basins (1–103 km2) to near 0.5 for the largest in the world (nearly 107 km2). This relationship is predicted by the statistical theory of channel networks, which is founded on the basic postulates that (1) natural channel networks in the absence of strong geologic controls are very nearly topologically random and (2) interior and exterior link lengths and associated areas in basins with homogeneous climate and geology have separate statistical distributions that are approximately independent of location. The calculation was carried out by a Monte Carlo method, which produced a random sample of networks from the postulated population, and was checked by analytical results for networks up to magnitude 500. The necessary empirical data on link lengths and associated areas were obtained by measurement on maps of an 80-km2 basin in relatively flatlying coal-bearing sandstones in eastern Kentucky. Agreement with observation is excellent for the small to medium basins, but the ratio of predicted to observed mainstream length progressively decreases to about 0.7 for the largest basins. This discrepancy can be accounted for by moderate downstream increase in channel sinuosity and decrease in drainage density. The particular data on link lengths and associated areas used in the calculation have only slight influence because mean link length varies statistically directly as the square root of mean associated area, so that taking data from different areas merely displaces the predicted points nearly parallel to their trend.