Controls on the hydraulic geometry of alluvial channels: bank stability to gravitational failure, the critical-flow hypothesis, and conservation of mass and energy

Jon D Pelletier

doi:10.5194/esurf-9-379-2021

Abstract

Abstract. The bank-full depths, widths, depth-averaged water velocities, and along-channel slopes of alluvial channels are approximately power-law functions of bank-full discharge across many orders of magnitude. What mechanisms give rise to these patterns is one of the central questions of fluvial geomorphology. Here it is proposed that the bank-full depths of alluvial channels are partially controlled by the maximum heights of gravitationally stable channel banks, which depend on bank material cohesion and hence on clay content. The bank-full depths predicted by a bank-stability model correlate with observed bank-full depths estimated from the bends in the stage–discharge rating curves of 387 U.S. Geological Survey gaging stations in the Mississippi River basin. It is further proposed that depth-averaged water velocities scale with bank-full depths as a result of a self-regulatory feedback among water flow, relative roughness, and channel-bed morphology that limits depth-averaged water velocities to within a relatively narrow range associated with Froude numbers that have a weak inverse relationship to bank-full discharge. Given these constraints on channel depths and water velocities, bank-full widths and along-channel slopes consistent with observations follow by conservation of mass and energy of water flow.

Highlights

The bank-full depths, h, widths, w, depth-averaged water velocities, v, and along-channel slopes, S, of alluvial channels exhibit power-law relationships with bank-full discharge, Q:h ∝ Qk, w ∝ Qb, v ∝ Qm, S ∝ Qz, (1)where k ≈ 0.4, b ≈ 0.5, m ≈ 0.1, and z ≈ −0.4 (Leopold and Maddock, 1953)
Further tests of the model of this paper may require a better understanding of how channel depths and/or bank angles adjust to spatial variations in bank material cohesion through a channel network
This paper proposed that the bank-full depths of alluvial channels may be partially controlled by the maximum heights of gravitationally stable channel banks, which depend on bank material cohesion and on clay content

Summary

Introduction

The bank-full depths, h, widths, w, depth-averaged water velocities, v, and along-channel slopes, S, of alluvial channels exhibit power-law relationships with bank-full discharge, Q:h ∝ Qk, w ∝ Qb, v ∝ Qm, S ∝ Qz, (1)where k ≈ 0.4, b ≈ 0.5, m ≈ 0.1, and z ≈ −0.4 (Leopold and Maddock, 1953). The bank-full depths, h, widths, w, depth-averaged water velocities, v, and along-channel slopes, S, of alluvial channels exhibit power-law relationships with bank-full discharge, Q:. Bank retreat is driven by gravitational failure (ASCE, 1999), a process that limits bank heights to values that depend on bank material cohesion and on clay content. The gravitational failure of channel banks may partially control bank-full depths via a self-regulatory mechanism in which channel incision and/or floodplain deposition tend to increase bank height, triggering bank failure when a critical bank height, dependent on bank material cohesion, is exceeded (Andrews, 1982), introducing new sediment into the channel bed that, as it is redistributed by fluvial processes, tends to reduce the channel depth back towards a critical value.

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