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.

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.

In this paper we analyze the Hirano active layer model used in mixed sediment river morphodynamics concerning its ill-posedness. Ill-posedness causes the solution to be unstable to short-wave perturbations. This implies that the solution presents spurious oscillations, the amplitude of which depends on the domain discretization. Ill-posedness not only produces physically unrealistic results but may also cause failure of numerical simulations. By considering a two-fraction sediment mixture we obtain analytical expressions for the mathematical characterization of the model. Using these we show that the ill-posed domain is larger than what was found in previous analyses, not only comprising cases of bed degradation into a substrate finer than the active layer but also in aggradational cases. Furthermore, by analyzing a three-fraction model we observe ill-posedness under conditions of bed degradation into a coarse substrate. We observe that oscillations in the numerical solution of ill-posed simulations grow until the model becomes well-posed, as the spurious mixing of the active layer sediment and substrate sediment acts as a regularization mechanism. Finally we conduct an eigenstructure analysis of a simplified vertically continuous model for mixed sediment for which we show that ill-posedness occurs in a wider range of conditions than the active layer model.

Laboratory experiments were conducted on a sand-gravel Gilbert delta to gain insight on its dynamics under varying base level. Base level rise results in intensified aggradation over the topset, as well as a decrease in topset slope and topset surface coarsening, the signals of which migrate in an upstream direction. Preferential deposition of coarse sediment in the topset results in a finer load at the topset-foreset break, which creates a fine signature in the foreset deposit. Base level fall has the opposite effects. Entrainment of the topset mobile armor causes a coarsening of the load at the topset-foreset break and so a coarse signature in the foreset deposit. The entrainment of the topset substrate and fine top part of the foreset may follow, which causes a fining of the load and a fine signature in the foreset deposit. The fact that the upstream sediment supply requires a certain slope and bed surface texture to be transported downstream under quasi-equilibrium conditions counteracts the effects of base level change. This information travels in the downstream direction. In nature base level change is likely so slow that the upstream sediment load maintains the topset slope and bed surface texture and so keeps the topset in a quasi-equilibrium state. Base level change is therefore not expected to leave a clear signal in a mixed-sediment Gilbert delta other than a change in elevation of the topset-foreset interface.

Downstream fining of bed sediment in alluvial rivers is usually gradual, but often an abrupt decrease in characteristic grain size occurs from about 10 to 1 mm, i.e., a gravel-sand transition (GST) or gravel front. Here we present an analytical model of GST migration that explicitly accounts for gravel and sand transport and deposition in the gravel reach, sea level change, subsidence, and delta progradation. The model shows that even a limited gravel supply to a sand bed reach induces progradation of a gravel wedge and predicts the circumstances required for the gravel front to advance, retreat, and halt. Predicted modern GST migration rates agree well with measured data at Allt Dubhaig and the Fraser River, and the model qualitatively captures the behavior of other documented gravel fronts. The analysis shows that sea level change, subsidence, and delta progradation have a significant impact on the GST position in lowland rivers.

The set of equations used in modelling river  morphodynamics needs to be (at least) wellposed  to be representative of the real natural  phenomenon. As we deal with a time dependent  process the solution needs to be wave-like to be  well-posed. In other words, the solution must  have a domain of dependence and of influence.  Otherwise, the future river state influences the  present solution, which is physically unrealistic.  Based on an analysis of the system of  equations to model one-dimensional river  morphodynamics with unisize sediment and a  Chezy-based friction term, Cordier (2011)  concluded that the system is always well-posed.  Stecca (2014) extended the analysis to a mixture  of sediment with 2 size fractions and concluded  that under degradational conditions the system  may become ill-posed. This result supported the  first analysis that found ill-posedness in mixedsize  sediment morphodynamics conducted by  RIbberink (1987) assuming a simpler model.  Here we extend these analyses by adding the  effects of flow curvature which creates an  intrinsically 3D flow referred to as secondary or  spiral flow (Van Bendegom, 1947). In this study  the flow is assumed bi-dimensional which implies  that the secondary flow needs to be  parameterized. 

There has been quite some debate on the relative importance of particle abrasion and grain size selective transport regarding the river profile form and the associated grain size trends in a graded alluvial stream. Here we present new theoretical equations for the graded alluvial river profile that account for the effects of particle abrasion and grain size selective transport in the absence of subsidence, uplift, and sea level change. Under graded conditions we find that abrasion results in a mild profile concavity and downstream fining, whereas under aggradational conditions grain size selective transport can lead to large spatial changes in channel slope and bed surface mean grain size.