Stream gradient as a function of order, magnitude, and discharge

TLDR

The three approaches to deriving stream‑gradient change are mathematically equivalent, implying comparable slope rates. Stream‑gradient change is modeled as a power function of area or discharge using order data, link slope, and hydraulic‑geometry equations. The derived gradient‑change rates are nearly identical across methods, averaging about –0.6, consistent with published values and suggesting a quasi‑equilibrium tendency for fluvial systems.

Abstract

The downstream change in the average channel gradient derived from order data can be expressed in terms of either area or discharge in the form of a power function. Similarly, the average channel profile based on link slope can be related to link magnitude or discharge in the form of a power equation. Finally, from the downstream hydraulic geometry equation the change in stream gradient can be expressed in terms of discharge or area as a power function. Because these relations are identical in form and in their independent parameters, rates of change in slope obtained by all three approaches should be equivalent. The rates of change in stream gradient derived from the power functions above yield almost identical averages for entire channel networks. The order data give a rate of −0.63, whereas link slope exponents average −0.60. These values are well within the range of variation for published data obtained for the hydraulic geometry equation (averages between −0.49 and 0.95) and may represent a quasi‐equilibrium tendency for entire fluvial systems.