A Vertical Sorting Model for Rivers with Non-uniform Sediment and Dunes

Abstract

Vertical sorting fluxes and the resulting organisation of grain size fractions over bed elevations, i.e. the vertical sorting profile, influence bedform dimensions and the bed surface composition. As a result, they will affect the bed roughness, the rate and composition of the sediment transport, the large-scale river morphology, and water levels during floods. An important problemof presentmorphologicalmodel systems for rivers with non-uniform sediment is their neglect of vertical sorting fluxes other than through net aggradation or degradation. The objective of the present study is to develop a method in order to better account for vertical sorting processes in the river bed. Net aggradation or degradation of the river bed results from divergences in the sediment transport rate, which is expressed by the mass balance equation. In rivers withnon-uniform sediment, a divergence in the transport rate of a specific size fraction will result in changes in the composition of the bed surface and net aggradation or degradation of the river bed. A further complication in the mass balance is the vertical sorting. A plane bed is usually covered with a coarse bed layer (mobile pavement or armour layer),whereas in bedform-dominated conditions the coarse size fractions are mainly found in the lower parts of the bedforms. In a morphological model system for non-uniform sediment, the above interaction is described in terms of sediment continuity models. Hirano (1971) was the first to develop such a model. Yet, the Hirano active layer model and most of its variants suffer from a number of shortcomings. Most bed layer models do not account for vertical sorting other than through net aggradation or degradation, whereas flume experiments have shown that vertical sorting also occurs through grain size-selective deposition down a bedformlee face and the variability in trough elevations. Hirano’s assumption that the bed elevations interacting with the flow constitute a distinct surface layer seems too limited to adequately account for these sorting processes. A final problem of these models with discrete bed layers is that in certain situations the set of equations becomes elliptic in parts of the space-time domain. This means that solving the equationswould require future time-boundaries,which is physically unrealistic. In the present study, two sets of flume experiments have been conducted, in order to obtain a better insight into vertical sorting processes and to obtain data on vertical sorting in bedforms under well-controlled laboratory conditions. Vertical sorting profiles were measured using newly-developed core sampling boxes. The experiments have shown that the sorting within migrating bedforms is dominated by (1) grain size-selective deposition down the avalanche lee face (resulting in a downward coarsening trend within the bedforms), (2) partial transport, and (3) the winnowing of fines from the trough surface and subsurface. The experiments point towards a close relation between the variability in trough elevations and the (time scale of) vertical sorting. This can be understood when considering that, through migration, a dune redistributes all bed material above its trough elevation. The author has developed a new continuum vertical sorting model for conditions dominated by bedforms 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-eveloped method to account for the variability in bedform trough elevations. Parker et al. (2000) developed a new type of framework for sediment continuity that no longer distinguishes discrete bed layers, thus enabling us to relate entrainment and deposition fluxes in the bed to the likelihood of a bed elevation being exposed to the flow. Parameters in the framework such as the grain size-specific and entrainment and deposition fluxes vary continuously over bed elevations. The present study comprehends the derivation of formulations for these fluxes for bedform-dominated conditions. For equilibrium conditions, the continuum sorting model is reduced to an equilibrium sorting model, which comprises two methods to solve for the equilibrium sorting profile. For non-equilibrium conditions, the continuum sorting model is reduced to a sorting evolution model. It solves for the time evolution of both the vertical sorting profile and the bed load transport composition fromthe following parameters: the initial sorting profile, the total bed load transport rate, and the probability density function (PDF) of relative trough elevations. Both reduced models describe sorting through grain size-selective deposition down the bedform lee face and the variability in trough elevations. In addition, methods are proposed to account for vertical sediment fluxes through both a change in time of the PDF of relative trough elevations and net aggradation or degradation. However, testing the continuumsorting model for situations with net aggradation or degradation was beyond the scope of the thesis. In a morphological model system, the equilibrium sorting model may be applied instantaneously in case the time scale of large-scale morphological changes is much larger than the time scales of vertical sorting, vertical dune dimensions, and dune migration. This constraint seems to be satisfied in situations with large-scale aggradation or degradation, such as long-term bed degradation after a river bend cut-off or long-term aggradation or degradation due to river training works. For situations with local small-scale aggradation or degradation, e.g., in river bends, it is likely that the time evolution of sorting needs to be accounted for when computing changes in morphology. In that case, the sorting evolution model must be applied. The equilibrium sorting model was calibrated and verified against flume experiments. Two constants in the lee sorting function were used as calibration constants. The sorting evolution model was verified by comparing the computed time evolution of both the vertical sorting profile and the bed load transport composition to measured data from flume experiments. No parameters were calibrated upon. The computations by both models agree reasonably well with the measured data. Note that the continuum sorting model computes the bed load transport composition by itself. This suggests that there is a strong interaction among (1) the vertical sorting profile, (2) the PDF of relative trough elevations, and (3) the bed load transport composition. It is believed that this strong interaction does not allow for one of the components to be modelled independently from one another. The results indicate that also the bed roughness is intimately connected to the vertical sorting profile and the PDF of trough elevations. The new continuum sorting model is deterministic in the computation of the vertical sorting profile, but probabilistic in terms of the bed surface. This probabilistic element arises from the fluctuations of the bed surface due to the presence of bedforms. The continuum sorting model is the first to incorporate sediment sorting fluxes through grain size-selective deposition down the bedform lee face, the variability in trough elevations, a change in time of the PDF of relative trough elevations, and net aggradation or degradation. Because of its improved description of the vertical sorting profile, the continuumsortingmodel yields better predictions of the bed surface composition than the discrete bed layermodels presently available. Since the bed surface composition plays a crucial role in the sediment transport, it is believed that themodelwill also lead to improved predictions of sediment transport rates and thus morphological changes, although testing the model as part of a morphological model system was beyond the scope of the present study. In order to use the new continuum sorting model in predictive applications, the model needs to be extended to plane-bed conditions and a sub-model for the PDF of relative trough elevations needs to be developed.