Hack’s law

Three questions that the literature leaves open. Does h exceed 0.5 consistently, or does the value depend on how basins and their lengths are sampled? What controls its variation from basin to basin and from scale to scale? And what makes the departure from 0.5 — are basins elongated, or are their channels sinuous? The length is Hack’s own: the flow path from the outlet to the farthest point of the divide.

The study samples the pairs of length and area in five ways — along the main stem, at tributary junctions, and at independent outlets, pooled over the world and separately per basin — so the same grid answers whether the sampling paradigm, and not the landscape, is what moves the exponent. A geometric decomposition and four shape diagnostics separate elongation from sinuosity, and a random-forest analysis asks which of terrain, substrate and drainage regime explains what is left over.

FullBasin on MERIT Hydro · 90 m, version 1.0

The 340,991 basins of at least 1 km² and, for each, its longest flow path (Hack’s L), its drainage area (A) and the terrain of its upstream land; every pair of L and A in the study is read off this grid. The geometric decomposition and the attribution use the 26,970 basins of at least 100 km².

MERIT Hydro (Yamazaki et al., 2019)

The flow directions and the terrain underneath FullBasin, at 3 arc-seconds.

This is what the versioned addresses are for: a reader of the paper can open the same release of the same collection and find the same basins, numbered the same way.

Submitted to Water Resources Research in April 2026; in revision after review, September 2026. The results, the figures and the data and scripts behind every figure will be linked from this page when the paper is published; until then, nothing of the findings is reported here.

Questions about the study: [email protected].