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

A river speeds up as it rises, but not in proportion to the flow. At a single gauging station the mean velocity $v$ follows a power law of the discharge $Q$: $v = kQ^{m}$. This is an empirical fit, not a derived law. The coefficient $k$ and the exponent $m$ come from the station's own measurements, so the fitted relation describes that station and no other. It is one of the at-a-station hydraulic geometry relations described by Knighton (1998) and Dingman (2009).

The coefficient $k$ is the mean velocity at a discharge of 1 m³/s, because $Q^{m}$ is 1 there. The exponent $m$ is typically 0.1 to 0.5, with a mean near 0.34 in Leopold and Maddock's original study. Since $m$ is less than one, velocity grows more slowly than the flow does. The rest of the extra discharge has to go somewhere: width times depth times velocity is the discharge, so the channel also gets wider and deeper, and the exponents of the width, depth and velocity laws add to one. Taking logarithms turns the power law into a straight line of $\ln v$ against $\ln Q$, so the exponent is the slope on a log-log plot.

Take $k = 0.5$, $m = 0.34$ and $Q = 50$ m³/s. The relation gives 1.8907 m/s. By continuity that is also $Q/(w\,d)$, the discharge over the cross-section area, which is what the three laws are built to agree on. Run backwards, a velocity of 1.8907 m/s at 50 m³/s with $k = 0.5$ gives an exponent of 0.34.

The fit belongs to one station and to the range of flows its record covers, so stretching it far beyond them is extrapolation. Manning's equation is a second route to the velocity, from the channel's roughness and slope instead of from the station's fit.