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, J., 1957, "Studies of longitudinal stream profiles in Virginia and Maryland", U.S. Geological Survey Professional Paper, 294-B.
• Rigon, R., et al., 1996, "On Hack's law" Water Resources Research, 32, 11, pp. 3367–3374.
• Willemin, J.H., 2000, "Hack’s law: Sinuosity, convexity, elongation". Water Resources Research, 36, 11, pp. 3365–3374.
• Cheraghi, M.; Rinaldo, A.; Sander, G. C.; Perona, P.; Barry, D. A. (2018). "Catchment drainage network scaling laws found experimentally in over-land flow morphologies". Geophysical Research Letters. 45 (18): 9614–9622. Bibcode:2018GeoRL..45.9614C. doi:10.1029/2018GL078351. hdl:20.500.11820/2d9254dd-7ad9-40b0-a95f-3abbea834ec8.

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