At-a-Station Hydraulic Geometry: Mean Depth from Discharge

At-a-station hydraulic geometry describes one gauging station as the flow rises and falls. It is an empirical power law, not a derived one: the coefficient $c$ and the exponent $f$ come from the station's own fit of mean depth against discharge. Mean depth $d$ is the cross-section area over the surface width at that discharge, and $Q$ is the discharge. Taking logarithms turns the law into a straight line, which is how the exponent is solved for: $f = \ln(d/c)/\ln Q$.

The coefficient $c$ has a direct meaning: it is the mean depth at a discharge of 1 m³/s, since $Q^{f}$ is then 1. The exponent says how fast depth responds. It is typically 0.2 to 0.6, with a mean near 0.40 in Leopold and Maddock's original study, and it is larger than the width exponent: as a river rises it deepens faster than it widens. This is the second of the three laws, for width, depth and velocity, that together carry the flow. Because width times depth times velocity must give the discharge, the exponents of the three add to one.

For the worked example take $c = 0.5$, $f = 0.40$ and $Q = 50$ m³/s. Then $d = 0.5 \times 50^{0.40}$ is 2.391 m. Run backward, a mean depth of 2.391 m at 50 m³/s with $c = 0.5$ returns the exponent 0.40. Keep $Q$ in m³/s and $d$ in metres, because $c$ is defined at 1 m³/s and is entered as a plain number.

The limits come from where the numbers come from. The law holds for one station and for the range of flows its fit covers, and $c$ and $f$ belong to that station, so values from one river should not be carried to another. The typical range of the exponent is only a guide to what to expect. The sources are Knighton's account of at-a-station hydraulic geometry in Fluvial Forms and Processes and Dingman's Fluvial Hydraulics. The width law and velocity law at the same station complete the set, and the Froude number and specific energy pages use the depth found here.