The Quasi‐Equilibrium Longitudinal Profile in Backwater Reaches of the Engineered Alluvial River: A Space‐Marching Method

Abstract

An engineered alluvial river (i.e., a fixed‐width channel) has constrained planform but is free to adjust channel slope and bed surface texture. These features are subject to controls: the hydrograph, sediment flux, and downstream base level. If the controls are sustained (or change slowly relative to the timescale of channel response), the channel ultimately achieves an equilibrium (or quasi‐equilibrium) state. For brevity, we use the term “quasi‐equilibrium” as a shorthand for both states. This quasi‐equilibrium state is characterized by quasi‐static and dynamic components, which define the characteristic timescale at which the dynamics of bed level average out. Although analytical models of quasi‐equilibrium channel geometry in quasi‐normal flow segments exist, rapid methods for determining the quasi‐equilibrium geometry in backwater‐dominated segments are still lacking. We show that, irrespective of its dynamics, the bed slope of a backwater or quasi‐normal flow segment can be approximated as quasi‐static (i.e., the static slope approximation). This approximation enables us to derive a rapid numerical space‐marching solution of the quasi‐static component for quasi‐equilibrium channel geometry in both backwater and quasi‐normal flow segments. A space‐marching method means that the solution is found by stepping through space without the necessity of computing the transient phase. An additional numerical time stepping model describes the dynamic component of the quasi‐equilibrium channel geometry. Tests of the two models against a backwater‐Exner model confirm their validity. Our analysis validates previous studies in showing that the flow duration curve determines the quasi‐static equilibrium profile, whereas the flow rate sequence governs the dynamic fluctuations.

River Response to Anthropogenic Modification: Channel Steepening and Gravel Front Fading in an Incising River

“…This effect results in a reduction of the net annual sediment mobility (sediment is transported less effectively), and the channel slope increases to compensate for this. This mechanism is similar to the reason for the stream‐wise decrease of net sediment mobility in a backwater segment, which is compensated for by a stream‐wise increase of channel slope (Arkesteijn et al., 2019). The slope increase in the upstream part of our alluvial domain competes with the effect of narrowing that tends to reduce the equilibrium slope (Figure 3c3).…”

A Rapid Method for Modeling Transient River Response Under Stochastic Controls With Applications to Sea Level Rise and Sediment Nourishment

“…In a BWS, the flow depth at the downstream end of the reach (under subcritical flow conditions) differs from the normal flow depth. This leads to an alternation of M1 or M2 backwater curves depending on the flow rate (Arkesteijn et al, 2019;Blom, Arkesteijn, et al, 2017;Chow, 1959;Lamb et al, 2012;Lane, 1957;Nittrouer et al, 2011Nittrouer et al, , 2012 and alternating channel bed aggradation and incision (Chatanantavet & Lamb, 2014;Ganti et al, 2016). A BWS forms upstream of, for instance, the river mouth, a confluence, bifurcation, or change in channel width or bed friction.…”

“…If the statistics of the hydrograph, sedigraph, and base level are constant in time, the river reach may tend to a dynamic equilibrium state where the bed level and bed texture vary around stable mean values (Arkesteijn et al, 2019;Lane, 1957;Parker, 2004). This dynamic equilibrium state is characterized by a static component (i.e., the mean or time-averaged longitudinal profile) and a dynamic component (i.e., fluctuation of the bed level and surface texture around a mean state).…”

Alluvial Morphodynamics of Low‐Slope Bedrock Reaches Transporting Nonuniform Bed Material

“…This is associated with a streamwise reduction of equilibrium alluvial cover and an alluvial‐bedrock transition may form. Throughout this paper, sediment load and transport capacity refer to temporal averages over intervals that are long compared to the time scales of change in bed elevation due to the variability of the flow rate (peak flow events), bedform migration, and sediment transport (Arkesteijn et al, 2019; Parker et al, 2000).…”

“…In alluvial rivers, equilibrium is a condition in which bed elevation averaged over time scales that are long compared to the time scales of bedform migration (Blom et al, 2006) and bedload transport (Wong et al, 2007) is constant in time (Anderson et al, 1975;Blom et al, 2017). If base level, formative discharge, and sediment supply are constant in time and abrasion is not accounted for, equilibrium sediment load of each grain size is everywhere equal to sediment supply of that grain size (Arkesteijn et al, 2019;Blom et al, 2016;Parker, 2004, and references therein). This equilibrium state does not depend on initial conditions; it only depends on water discharge, sediment supply, and caliber (Blom et al, 2016(Blom et al, , 2017Parker & Wilcock, 1993).…”