# Fluvial systems : a re-evaluation of Horton's laws

Dissertation

1972

## Keywords

Fluvial geomorphology, Bodies of water, Mathematical models.

## Degree Name

Doctor of Philosophy (PhD)

## Department

Geological Sciences

Donald R. Coates

Nicolay P. Timofeeff

Marie E. Morisawa

## Series

Science and Mathematics

## Abstract

The number of different order streams entering a channel of known order can be approximated by a geometric progression similar to the Law of Stream Numbers. Hence, the frequency and order of different size tributaries along any channel can be described by a single parameter. Similarly, the number of specified order tributaries to different Strahler order streams also decreases with order following a geometric progression. The ratios of these functions are related to the bifurcation ratio by a simple algebraic equation. The length ratio can be directly related to the rate of change in the number of tributaries for different order streams.

The average vertical properties of the channel network can be related to planimetric dimensions of the basin. The slope ratio is related to the area ratio by a power function with exponent z equal to the rate of change in slope with discharge. Theoretical calculation by Leopold and Langbein of z agrees with the solution obtained here and support the hypothesis that z represents a central tendency in its value for natural channel systems. It was found that the variability in the vertical properties of the drainage network as expressed by the stream gradient standard deviation is unaffected by changes in geology, basin maturity, or hydrologic characteristics, but varies directly with the average stream gradient in the form of a power relation; The stream variance is therefore directly dependent on the variability of the basin geometry, discharge, and z. The distribution of stream gradient can also be expressed in terms of the distributions and correlation between stream length and relief. Because these parameters are related by a power function, the value of the exponent will partially control the variance of stream fall and will cause the stream gradient distribution to be stratified with respect to channel length and relief. Stream fall and gradient for given order streams take various distribution ranging from normal to right and left skewed. These distributions, result from the grouping of stream data from more than one population which can be derived in terms of the link fall distribution, magnitude, and rate of change in slope or relief downstream. Strong concavity in the stream gradient will cause the stream gradient distribution to be asymmetrical while relatively straight channels will tend to have a more normal distribution.

Temporal changes in the basin size, total stream length, and number of streams were studied through experimental development of a small drainage network. Initially changing at an exponential rate, these parameters attain a steady state as the basin becomes aerially fixed. The ratio of eroded area to stream length is constant during basin evolution. Similarly, the exponential channel profile maintained equilibrium in spite of lowering and extension of the stream in time. The rate of decrease in the hypsometric integral markedly decreased as the basin attained a steady state in its growth.

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