Hack's law

Hack's law is an empirical relationship between the length of streams and the area of their basins. If L is the length of the longest stream in a basin, and A is the area of the basin, then Hack's law may be written as

${\displaystyle L=CA^{h}\ }$

for some constant C where the exponent h is slightly less than 0.6 in most basins. h varies slightly from region to region and slightly decreases for larger basins (>8,000 mi², or 20,720 km²). In addition to the catchment-scales, Hack's law was observed on unchanneled small-scale surfaces when the morphology measured at high resolutions (Cheraghi et al., 2018).

The law is named after American geomorphologist John Tilton Hack.

References

• Hack, John T. (1957). Studies of Longitudinal Stream Profiles in Virginia and Maryland (PDF) (Report). Shorter Contributions to General Geology. pp. 45–97. doi:10.3133/pp294B. U.S. Geological Survey (USGS) Professional Paper 294-B. Archived from the original on 2024-07-11.
• Rigon, Riccardo; Rodriguez-Iturbe, Ignacio; Maritan, Amos; Giacometti, Achille; Tarboton, David G.; Rinaldo, Andrea (November 1996). "On Hack's law". Water Resources Research. 32 (11). American Geophysical Union: 3367–3374. Bibcode:1996WRR....32.3367R. doi:10.1029/96WR02397. ISSN 0043-1397 – via Wiley.
• Willemin, James H. (November 2000). "Hack's law: Sinuosity, convexity, elongation". Water Resources Research. 36 (11). American Geophysical Union: 3365–3374. Bibcode:2000WRR....36.3365W. doi:10.1029/2000WR900229. ISSN 0043-1397 – via Wiley.
• Cheraghi, Mohsen; Rinaldo, Andrea; Sander, Graham C.; Perona, Paolo; Barry, D. A. (28 September 2018). "Catchment drainage network scaling laws found experimentally in over-land flow morphologies". Geophysical Research Letters. 45 (18). American Geophysical Union: 9614–9622. Bibcode:2018GeoRL..45.9614C. doi:10.1029/2018GL078351. hdl:20.500.11820/2d9254dd-7ad9-40b0-a95f-3abbea834ec8.