Recent analysis of equilibrium and quasi-equilibrium channel geometry in engineered (fixed-width) rivers has successfully shown that two temporal scales can be distinguished, with quasi-static (long-term) and dynamic (short-term) components. This distinction is based on the fact that channel slope cannot keep pace with short-term fluctuations of the controls. Here we exploit the distinction between the two temporal scales to model the transient (so time-dependent) phase of channel response, which is the phase wherein the channel approaches its new equilibrium. We show that: (a) besides channel slope, also the bed surface texture cannot keep pace with short-term fluctuations of the controls, and (b) mean transient channel response is determined by the probability distributions of the controls (e.g., flow duration curve rather than flow rate sequence). These findings allow us to set up a rapid numerical method that determines the mean transient channel response under stochastic controls. The method is based on distinguishing modes (i.e., sets of controls) and takes the probability density of each mode into account. At each time step, we compute the mode-specific flow, sediment transport rate, and corresponding change in bed level and surface texture. The net change within the time step is computed by weighting the mode-specific changes in bed level and surface texture with the probability density of each mode. The resulting mean transient channel response is a deterministic one, despite the controls being stochastic variables. We show that the proposed method provides a rapid alternative to Monte Carlo analysis regarding the mean time-dependent channel response.

The active layer model (Hirano, 1971) is frequently used for modeling mixed-size sediment river morphodynamic processes. It assumes that all the dynamics of the bed surface are captured by a homogeneous top layer that interacts with the flow. Although successful in reproducing a wide range of phenomena, it has two problems: (1) It may become mathematically ill-posed, which causes the model to lose its predictive capabilities, and (2) it does not capture dispersion of tracer sediment. We extend the active layer model by accounting for conservation of the sediment in transport and obtain a new model that overcomes the two problems. We analytically assess the model properties and discover an instability mechanism associated with the formation of waves under conditions in which the active layer model is ill-posed. Numerical simulations using the new model show that it is capable of reproducing two laboratory experiments conducted under conditions in which the active layer model is ill-posed. The new model captures the formation of waves and mixing due to an increase in active layer thickness. Simulations of tracer dispersion show that the model reproduces reasonably well a laboratory experiment under conditions in which the effect of temporary burial of sediment due to bedforms is negligible. Simulations of a field experiment illustrate that the model does not capture the effect of temporary burial of sediment by bedforms.

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.