The River Profile

This is the authors' abstract. We don't add key points for this paper.

The long profile of a representative natural stream is composed of segments, each of which is exponential, or nearly so. Each segment tends to plot on semilog paper (distance vs. log elevation) as a straight line, with a faint upward convexity near the upper end of the segment, and a faint upward concavity near the lower end of the segment. These small, but real, departures from the straight-line plot require that we undertake a refinement of the classical profile equation, dA/dz = kA. Several modifications are proposed, and are explored mathematically. One attempts to separate dynamic sorting of bed load from attrition effects. Another (expressed initially as Lagrange's linear equation) treats the development of the two-dimensional profile with time-varying uplift. A third is the Leopold-Langbein random-walk model. A fourth is a modified random-walk model which specifically includes the upstream convexity and which can be altered to include the downstream concavity. A fifth is Devdariani's heat-flow analogy, which seems to be too general, but which can be forced, arbitrarily, to describe (without explaining) many anomalies of curvature. Three more are nonlog profiles based on reasoning associated with precipitation or discharge effects. No one of the attempts is completely successful, but each sheds a little light on the overall problem.