Influence of Valley Type on the Scaling Properties of River Planforms

Abstract

Scaling properties of 44 individual river planforms from the Cascade and Olympic Mountains of Washington State were defined using the divider method. Analysis of the standardized residuals for least squares linear regression of Richardson plots reveals systematic deviations from simple self‐similarity that correlate with the geomorphological context defined by valley type. A single fractal dimension describes rivers flowing through bedrock valleys. Those flowing in inherited glacial valleys exhibit two distinct fractal dimensions, with a larger fractal dimension at small scales. Rivers flowing in alluvial valleys are also described by two fractal dimensions, but with a larger dimension at large scales. We further find that the wavelength of the largest meander defines an upper limit to the scaling domain characterized by fractal geometry. These results relate scaling properties of river planforms to the geomorphological processes governing valley floor morphology.

Hack's Law: Sinuosity, convexity, elongation

“…The window length is taken to be slightly greater than the upper scale limit of the meander pattern, generally 2500 m [cf. Beauvais and Montgomery, 1996] (Table 2) Values for exponents and coefficients in the power law relationships presented below were obtained from least squares regression on the logarithms of each parameter against the logarithms of basin area. The correlation coefficient R 2 is taken as a goodness-of-fit measure.…”

“…The slope and scale boundaries of these segments for the larger streams in each area are summarized in Table 2. It should be noted that this analysis carries the assignment of slope values to much larger scales than those advocated by Beauvais and Montgomery [1996]; this is justified by the low scatter of residuals at the larger scales. A possible reason for the reduced scatter at large scales lies in the closer estimate of self-similar character provided by an assiduous search for a minimum stream length at each step, a search precluded by the data-gathering method of Beauvais and Montgomery [1996].…”

“…Since the Hausdorff-Besicovitch dimension is defined as the limit of Nr/r as r approaches zero and Nr is the minimum number of disks of radius r needed to cover the channel planform [e.g., Schroeder, 1991], the minimum number of divider steps is the appropriate measure. The resulting Richardson plots were analyzed using the method of residuals [Andrle, 1992;Beauvais and Montgomery, 1996]. Each plot appears to consist of several straight line segments.…”

Form and growth of bars in a wandering gravel‐bed river

“…Much earlier, Nikora et al (1993) discovered that the geometry of the channel itself (a reflection of bar morphology) along single thread channels exhibits selfsimilar behaviour at small scales (which presumably are not constrained by geological controls such as valley dimension), and self-affine behaviour at large scales. Beauvais and Montgomery (1996) pointed out that the transition from self-similar to self-affine behaviour occurred at the scale of valley width, in meandered rivers corresponding with maximum meander amplitude (or the threshold of confinement).…”

Fractal characteristics of the main channel of Yellow River and its relation to regional tectonic evolution

“…However, they appeared to be smooth, Euclidian shapes when viewed at scales approaching those of channel width or at the whole length of the stream segment, which implies that there are upper and lower limits for the application of fractal analysis to channel planform geometry. Similarly, Beauvais and Montgomery (1996) used the divider method to reveal that river planforms in the Cascade and Olympic Mountains of Washington State exhibit fractal scaling properties over scaling ranges bounded by the channel width and the largest meander wavelength and explore their relation to the geomorphological context defined by valley type.…”