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

$L = C A^h\$

for some constant C where the exponent h is slightly less than 0.6 in most basins. Note that h varies slightly from region to region, and also slightly decreases for larger basins (>8,000 mi², or 20,720 km²). A theoretical value h = 4/7 ≈ 0.571 for the exponent has been derived (Birnir, 2008).

References

• Birnir, B., 2008, "Turbulent rivers", Quart. Appl. Math., 66, 3, pp. 565–594.
• 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.