Astrid Blom

Quantification of variability in bedform geometry

We analyze the variability in bedform geometry in laboratory and field studies. Even under controlled steady flow conditions in laboratory flumes, bedforms are irregular in size, shape, and spacing, also in case of well-sorted sediment. Our purpose is to quantify the variability in bedform geometry. We use a bedform tracking tool to determine the geometric variables of the bedforms from measured bed elevation profiles. For each flume and field data set, we analyze variability in (1) bedform height, (2) bedform length, (3) crest elevation, (4) trough elevation, and (5) slope of the bedform lee face. Each of these stochastic variables is best described by a positively skewed probability density function such as the Weibull distribution. We find that, except for the lee face slope, the standard deviation of the geometric variable scales with its mean value as long as the ratio of width to hydraulic radius is sufficiently large. If the ratio of width to hydraulic radius is smaller than about ten, variability in bedform geometry is reduced. An exponential function is then proposed for the coefficients of variation of the five variables to get an estimate of variability in bedform geometry. We show that mean lee face slopes in flumes are significantly steeper than those in the field. The 95% and 98% values of the geometric variables appear to scale with their standard deviation. The above described simple relationships enable us to integrate variability in bedform geometry into engineering studies and models in a convenient way.

Vertical sorting in bed forms: Flume experiments with a natural and a trimodal sediment mixture

Two sets of flume experiments were conducted to examine grain size selective transport and vertical sorting in conditions with migrating bed forms and bed load transport. In the two sets of experiments we used a sediment mixture from the river Rhine and a trimodal mixture, respectively. The vertical sorting profiles showed a downward coarsening trend within the bed forms, and in some experiments an essentially immobile coarse bed layer formed underneath the migrating bed forms. Three mechanisms contribute to the formation of such a coarse bed layer: (1) the avalanching process at the lee face, (2) conditions of partial transport in which a relatively large amount of coarse material does not participate in the transport process, and (3) the winnowing of fines from the trough surface and subsurface. The experiments show that vertical sorting fluxes not only occur through net degradation or aggradation but also through the migration of bed forms and through the variability in trough elevations. This is contradictory to the way vertical sorting processes are modeled in most existing sediment continuity models for nonuniform sediment. The present study is therefore also a plea for modifying existing sediment continuity models to account for vertical sorting processes other than through net aggradation or degradation.

Vertical sorting and the morphodynamics of bed form–dominated rivers: A modeling framework

Uncertainties and errors in predictions by morphodynamic models of rivers with nonuniform sediment are usually attributed to shortcomings pertaining to the submodel of sediment transport. This mistakenly neglects shortcomings in the submodel of sediment continuity, which describes the vertical sorting and the bed surface composition, whereas the latter is one of the main input parameters for calculating sediment transport and thus morphological changes. Hirano [1971 , 1972] was the first to develop a sediment continuity model for nonuniform sediment. Since this commonly used Hirano active layer model and its variants suffer from a number of shortcomings, the authors have developed a new type of sediment continuity model that describes the bed composition and vertical sorting fluxes without distinguishing discrete bed layers. This continuum sorting model is aimed at conditions dominated by bed forms and bed load transport. It is based on (1) the Parker-Paola-Leclair framework for sediment continuity, (2) the Einstein step length formulation, (3) a newly developed lee sorting function, and (4) a newly developed method to account for the variability in bed form trough elevations. The resulting model is deterministic in the computation of the vertical sorting profile and is probabilistic in terms of the riverbed surface due to the presence of dunes.

Different approaches to handling vertical and streamwise sorting in modeling river morphodynamics

This paper presents an assessment of the strengths and weaknesses of four sediment continuity models for nonuniform sediment by applying these models to an aggradational flume experiment that is dominated by nonuniform sediment and dunes. The author makes simulations of the flume experiment using four numerical morphodynamic model systems to which the following sediment continuity models are applied: the commonly applied active layer model (A), a two-layer model (B), a sorting evolution model while assuming bed form size to be regular (C1), and a sorting evolution model while taking into account the variability in bed form geometry (C2). The model systems that incorporate the variability of bed form geometry, i.e., the two-layer model and the sorting evolution model with irregular dunes, show an improved prediction of the adaptation timescale of the composition of both the bed surface and the transported sediment, as well as the vertical sorting profile. This is because including the variability of bed form geometry enables the model system to account for sediment being stored (temporarily) at elevations that are exposed to the flow less frequently. Future application of both models to field cases is difficult, however, as the two-layer model is not sufficiently generic and may lead to ellipticity of its set of equations, whereas the sorting evolution model requires a very small numerical time step.

Vertical sorting and the morphodynamics of bed form-dominated rivers: a sorting evolution model

Existing sediment continuity models for nonuniform sediment suffer from a number of shortcomings, as they fail to describe vertical sorting fluxes other than through net aggradation or degradation of the bed and are based on a discrete representation of the bed material interacting with the flow. We present a new type of sediment continuity model that is based on a stochastic description of the bed surface rather than discrete bed layers. The model is aimed at conditions dominated by bed load transport wherein the river bed is fully covered by river dunes. Application of the model should be limited to spatial scales covering a significant number of bed forms. The resulting model, i.e., the sorting evolution model, is suitable for unsteady conditions, as it takes into account the time evolution of, for instance, vertical sorting in modeling net aggradation or degradation of the river bed. The present paper lists the various submodels of a morphodynamic model system to which the sorting evolution model is applied. We compare the results of the morphodynamic model system to measured data from two flume experiments. The new model shows reasonable results for the predicted time evolution of the vertical sorting profile, as well as the time evolution of the grain size distribution of the bed load transport. Yet the model does not properly include sorting mechanisms associated with partial transport and the winnowing of fines from the trough surface and subsurface. The model serves as a basis for a future simplification of the model into a new stochastics-based bed layer type sediment continuity model in which vertical sediment fluxes are included in a parameterized way.

Vertical sorting and the morphodynamics of bed-form-dominated rivers: an equilibrium sorting model

A modeling framework is developed for taking into account the effects of sediment sorting in the morphodynamic modeling of bed-form-dominated rivers for the case of equilibrium or stationary conditions dominated by bed load transport. To this end, the Blom and Parker (2004) framework for sediment continuity is reduced to an equilibrium sorting model. The predicted equilibrium sorting profile is mainly determined by the probability density function (PDF) of bed form trough elevations and by a lee sorting function. The PDF of trough elevations needs to be known from either model predictions or measurements. A simple formulation for the lee sorting function is suggested, yet data on the avalanche mechanism down lee faces of dunes is required so as to improve the function and make it generic. The equilibrium sorting model is calibrated and verified using data from flume experiments. The agreement between the predicted and measured equilibrium sorting profiles is reasonable, although the model does not reproduce an observed coarse top layer. In a hydraulic-morphodynamic model this equilibrium sorting model may be applied instantaneously if the timescale of large-scale morphological changes is much larger than the ones of changes in vertical sorting and dune dimensions.

Mathematical analysis of the Saint-Venant-Hirano model for mixed-sediment morphodynamics

Sediment of different size are transported in rivers under the action of flow. The first and still most popular sediment continuity model able to deal with mixed sediment is the so-called active layer model proposed by Hirano (1971, 1972). In this paper, we consider the one-dimensional hydromorphodynamic model given by the Saint-Venant equations for free-surface flow coupled with the active layer model. We perform a mathematical analysis of this model, extending the previous analysis by Ribberink (1987), including full unsteadiness and grainsize selectivity of the transported load by explicitly considering multiple sediment fractions. The presence of multiple fractions gives rise to distinct waves traveling in the downstream direction, for which we provide an analytical approximation of propagation velocity under any Froude regime. We finally investigate the role of different waves in advecting morphodynamic changes through the domain. To this aim, we implement an analytical linearized solver to analyze the propagation of small-amplitude perturbations of the bed elevation and grainsize distribution of the active layer as described by the system of governing equations. We find that initial gradients in the grainsize distribution of the active layer are able to trigger significant bed variations, which propagate in the downstream direction at faster pace than the “bed” wave arising from the unisize-sediment Saint-Venant-Exner model. We also verify that multiple “sorting” waves carry multiple associated bed perturbations, traveling at different speeds.

The graded alluvial river: profile concavity and downstream fining

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.

The equilibrium alluvial river under variable flow and its channel-forming discharge

When the water discharge, sediment supply, and base level vary around stable values, an alluvial river evolves toward a mean equilibrium or graded state with small fluctuations around this mean state (i.e., a dynamic or statistical equilibrium state). Here we present analytical relations describing the mean equilibrium geometry of an alluvial river under variable flow by linking channel slope, width, and bed surface texture. The solution holds in river normal flow zones (or outside both the hydrograph boundary layer and the backwater zone) and accounts for grain size selective transport and particle abrasion. We consider the variable flow rate as a series of continuously changing yet steady water discharges (here termed an alternating steady discharge). The analysis also provides a solution to the channel-forming water discharge, which is here defined as the steady water discharge that, given the mean sediment supply, provides the same equilibrium channel slope as the natural long-term hydrograph. The channel-forming water discharge for the gravel load is larger than the one associated with the sand load. The analysis illustrates how the load is distributed over the range of water discharge in the river normal flow zone, which we term the “normal flow load distribution”. The fact that the distribution of the (imposed) sediment supply spatially adapts to this normal flow load distribution is the origin of the hydrograph boundary layer. The results quantify the findings by Wolman and Miller (1960) regarding the relevance of both magnitude and frequency of the flow rate with respect to channel geometry.

Sediment Continuity for Rivers with Non-Uniform Sediment, Dunes, and Bed Load Transport

Models describing the interaction among grain size-selective sediment transport, the vertical sorting profile within the bed, and bed level changes constitute a critical component of morphological model systems for rivers with non-uniform sediment. This interaction is described in terms of sediment continuity models. Hirano (1971) was the first to develop such a sediment continuity model. In the Hirano active layer model, the bed is divided into a homogeneous top layer, i.e. the active layer, and an inactive substrate. Only sediment in the active layer interacts with the flow and participates in the transport process. Sediment fluxes between the active layer and the substrate occur only when there is a change in the average bed level. In the last few decades, a number of variants to the Hirano active layer model have been proposed in order to overcome several shortcomings and obtain a better description of vertical sediment sorting mechanisms within the bed.