A notable drawback in mixed-size sediment morphodynamic modeling is the fact that the most commonly used mathematical model in this field (i.e., the active layer model Hirano, 1971) can be ill-posed under certain circumstances. Under these conditions the model loses its predictive capabilities, as negligible perturbations in the initial or boundary conditions produce significant differences in the solution. In this paper we propose a preconditioning method that regularizes the model to recover well-posedness by altering the time scale of the sediment mixing processes. We compare results of the regularized model to data from four new laboratory experiments conducted under conditions in which the active layer model is ill-posed. The regularized active layer model captures the change of bed elevation and surface texture averaged over the passage of several bedforms. Neither the active layer model nor the regularized one account for small scale changes due to individual bedforms.@en

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.

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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.

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A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.@en

Throughout the last two centuries engineers have intervened large rivers for the sake of, among others, improving navigability and preventing flooding (Lonnquest et al., 2014). These interventions have induced morphodynamic changes that we still face nowadays. One if the consequences of past interventions is the ongoing bed degradation occurring in several large rivers including the Rhine River in the Netherlands (Blom, 2016). Measures aimed at preventing further degradation require predicting the effects of such interventions using morphodynamic models. A problem arises when using a morphodynamic model to predict degradational conditions of a river: the model may become ill-posed. An ill-posed model becomes useless in practice, as unphysical oscillations appear in the results. The origin of the oscillations lays in the unsuitability of the model to represent the physical processes under consideration and for this reason the emergence of oscillations is independent of the numerical solver. The most striking feature of an ill-posed model is that the result does not converge when the numerical grid is refined. Worded differently, when using smaller grid cells unphysical oscillations grow faster leading to ever changing results. In this paper we present two alternative strategies to prevent ill-posedness in morphodynamic modelling. @en

The mixed-size character of sediment is a necessary property to ex- plain physical phenomena such as downstream fining or the presence of armor layers. The active layer model was developed to model mixed-size sediment in river morphodynamics. This model assumes that the topmost part of the bed, the active layer, has no vertical stratification and interacts with the flow. The substrate, below the active layer, only interacts with the active layer in case of aggradation or degradation. The active layer model has been used in morphody- namic modelling for more than four decades but under certain conditions it may become mathematically ill-posed. When a model becomes ill-posed, the solu- tion presents unphysical oscillations and its predictive capabilities are lost. We present two alternatives to the active layer model. The first one retains the basic concepts and guarantees well-posedness by means of an additional parameter controlling the celerity of mixed-size sediment processes. The second solution yields a well-posed model by means of considering the sediment transport rate as a stochastic process rather than to adapt instantaneously to the flow. Both models provide reasonable results when compared to measured data from a lab- oratory experiment conducted under conditions in which the active layer model is ill-posed.@en

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.

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Two-dimensional fl ow models are widely used and necessary to predict phenomena such as the dynamics of bars. In these problems the effect of the bed slope in the direction of the sediment transport rate needs to be accounted for to obtain physically realistic results. This effect is included using an empirical closure relation. Even including bed slope effects, 2D models do not capture 3D phenomena such as secondary fl ow which occurs when the fl ow curvature is large. This process is accounted for in a parameterized manner including one equation to model the advection and diffusion of the secondary fl ow intensity.@en

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.

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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.

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A gravel-sand transition (GST) seems to be the result of a gravel wedge. Such a wedge can prograde, halt, and even retreat. It is typical of an ungraded or transient reach (Blom et al., 2016), where profile concavity and downstream fining can be much stronger than in a graded or equilibrium river reach. The GST migration speed decreases in time and is likely of the order of a few meters or less per year. The effect of abrasion on GST migration is limited to mainly affecting the sand and silt load transported into the sand-bed reach and so its slope. Our simple GST migration model describes GST dynamics fairly well. It provides an estimate of the time scale of GST migration and a criterion for the onset for a GST to halt or retreat. We recommend analysis of temporal change in bed elevation and bed surface texture in GST zones, and the development of new techniques to measure the bed surface texture to assess GST dynamics in the field @en

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.  @en

The active layer model (Hirano, 1971) is the most commonly used model to account for mixed-size sediment processes in modeling morphodynamics of rivers, coasts, and estuaries. In this model, only the sediment in the topmost part of the bed (the active layer, characterized by a certain thickness, and assumed to be fully mixed) interacts with the flow. The sediment in the active layer can be entrained and the transported sediment can be deposited in the active layer. The grain size distribution of the sediment below the active layer, the substrate, typically varies with elevation. There is a net flux of sediment between the active layer and the substrate if the bed aggrades or degrades. Due to the highly schematized treatment of the bed processes, the active layer model may present elliptic (rather than hyperbolic) behavior (Ribberink, 1987). A system of equations that models changes in time cannot be of an elliptic type. This is because in that case future conditions influence the present, which is physically unrealistic. Such a model is mathematically ill-posed. The solution of an ill-posed problem is unstable to short wave perturbations. Another example of an ill-posed problem is the twofluid model. Zanotti et al. (2007) developed a regularization strategy to restore the hyperbolic character when it becomes ill-posed. Our objective is to apply a similar concept to guarantee the hyperbolic character of the active layer model. @en

We present a new image analysis technique for measuring the grain size distribution (texture) of the bed surface during flow in a laboratory experiment. A camera and a floating device are connected to a carriage used to take images of the bed surface over the entire flume length. The image analysis technique, which is based on color segmentation, provides detailed data on spatial and temporal changes of the areal fraction content of each grain size at the bed surface. The technique was applied in a laboratory experiment conducted to examine a degradational reach composed of a well sorted two-fraction mixture of sand and gravel. The initial bed consisted of an upstream reach that was characterized by an imposed stepwise fining pattern (the bimodal reach) and a downstream sand reach. A lack of sediment supply and partial transport conditions led to the formation of a static armor in the bimodal reach, which resulted in a more abrupt spatial transition in the bed surface mean grain size. The associated spatial transition in slope led to a backwater effect over the bimodal reach, a streamwise reduction in sand mobility, and so a static armor that was governed by a downstream fining pattern. Although a morphodynamic equilibrium state under steady flow is generally characterized by normal flow, here the partial transport regime prevented the bed from adjusting toward normal flow conditions and the morphodynamic steady state was governed by a backwater. We applied a numerical morphodynamic sand-gravel model to reproduce the laboratory experiment. The numerical model captured the hydrodynamic and morphodynamic adjustment and the static armor well, yet the armoring occurred too slowly. Although the final configuration of the experiment shows features of a gravel-sand transition (i.e., a sudden transition in slope and mean grain size), we are hesitant to claim similarities between our results and the physical mechanisms governing a gravel-sand transition in the field.
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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.

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Available online 9-04-2014@en