Hack's law of debris-flow basins

Hack's law was originally derived from basin statistics for varied spatial scales and regions. The exponent value of the law has been shown to vary between 0.47 and 0.70, causing uncertainty in its application. This paper focuses on the emergence of Hack's law from debris-flow basins in China. Over 5,000 debris-flow basins in different regions of China with drainage areas less than 100km² are included in this study. Basins in the different regions are found to present similar distributions. Hack's law is derived from maximum probability and conditional distributions, suggesting that the law should describe some critical state of basin evolution. Results suggest the exponent value is approximately 0.5. Further analysis indicates that Hack's law is related to other scaling laws underlying the evolution of a basin and that the exponent is not dependent on basin shape but rather on the evolutionary stage. A case study of a well known debris-flow basin further confirms Hack's law and its implications in basin evolution.