Hack's law: Sinuosity, convexity, elongation

Abstract

Hack's law, an empirical, power law relationship between drainage basin area and the length of the main stream channel, has long been taken to imply that drainage basins become more elongate (relatively longer and narrower) with increasing basin size. A study of the geometry of 38 basins from three distinct geomorphic settings shows that this geometric interpretation of Hack's law is only occasionally true: Even though Hack's power law relationship holds between basin area and main channel length, these basins do not necessarily become more elongate with increasing size. Rather, Hack's law is an expression of a balance between changes in basin shape and changes in channel planform geometry. For the basins in this study, changes in channel sinuosity play the most important role in this balance; changes in basin shape are far less regular. Local conditions appear to determine the partitioning of importance between changes in basin shape and channel sinuosity.