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

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.