Stability of Alluvial Channels

By making a slight extension to E. W. Lane's tractive-force theory of stable channel design, and by combining it with Strickler's formula, equations are deduced similar inform to G. Lacey's “regime” equations. In particular it is shown that Lacey's width-discharge relation, P ∞ Q, is true for the narrow channel developed in Lane 's theory. It is pointed out that the similarity is of limited significance because the regime condition implies that the bed is live, whereas the tractive force criterion assumes that the bed is only on the thresh old of motion. The latter criterion requires that the shear stress, and hence RS, is constant at all points along the channel. Lacey 's observations on the other hand were to the effect that R2S is constant along the channel, and it is shown by using the Einstein bed load function that this condition is fulfilled in a stable channel with a live bed. These results are then applied to the consideration of longitudinal profiles of stable rivers and canals. It is pointed out that only two stable channel equations can be obtained from consideration of bed conditions and flow resistance, and that a third equation, such as Lacey 's P∞ can be true only if there is a certain slope-discharge relationship, that is a certain longitudinal profile. The nature of this relationship is determined both for fixed (threshold) and live bed conditions, and the former is found to fit well to some data of L. B. Leopold, F. ASCE, and M. G. Wolman on natural rivers in coarse alluvium. Hence, it is argued that in such rivers the relations obtaining at the threshold of motion substantially determine the processes of natural river formation, both in cross section and in longitudinal profile, and that the results obtained herein may form a basis for a general attack on the problem.