CC BY
69
Erik Mosselman,
Deltares & Delft University of Technology,
The Netherlands
RIVER MORPHOLOGY AND RIVER ENGINEERING AT DELTARES
МОРФОЛОГИЯ РЕКИ И СТРОИТЕЛЬСТВО РЕЧНЫХ ВОДНЫХ ПУТЕЙ
В ДЕЛЬТАХ
В статье представлена концепция морфологии реки и строительства речных водных путей, разработанная Независимым институтом технологии дельт Deltares.
The article presents the conception of river morphology and river engineering at deltas , elaborated by the independent institute for delta technology Deltares.
Ключевые слова: морфология реки, гидравлическая неровность, осадочные смеси, биогеоморфология, математическое моделирование
Key words: river morphology, hydraulic roughness , sediment mixtures, biogeomorphology, mathematical modeling
1. INTRODUCTION
WL | Delft Hydraulics was traditionally the centre of expertise in the Netherlands regarding river morphology and river engineering. At the end of 2007, WL | Delft Hydraulics, Ge-oDelft, the Subsurface and Groundwater unit of TNO and parts of Rijkswaterstaat joined forces in a new independent institute for delta technology, Deltares. Deltares combines knowledge and experience in the fields of water, soil and the subsurface. Its mission is to provide innovative solutions to make living in deltas, coastal areas and river basins safe, clean and sustainable. Synergy advantages of the new institute include the combination of surface water and groundwater in modelling and water management as well as the integration of dike stability, measures to reduce flood levels and measures to mitigate the consequences of flooding in studies to reduce flooding risks. Furthermore, detailed studies on erosion and scour benefit from a combination of hydrody-namic and geotechnical approaches (de Vriend & Barends, 2006). Inland Water Systems is one of the units of Deltares. Within this unit, the River Dynamics and Inland Water Transport department deals with river morphology and river engineering. It carries out research, develops models and provides specialist advice. Research and development are carried out in close co-operation
with universities, in particular in Delft with Delft University of Technology and UNESCO-IHE Institute for Water Education.
This paper provides a written account of the overview of river morphology and river engineering at Deltares that was presented at the round-table seminar at Saint-Petersburg State University for Waterways Communications on 26 March 2008. Chapter 2 describes the major demands from society that require knowledge on river morphology and river engineering. Chapter 3 highlights some of the corresponding co-operations and networks. Chapter 4 explains major lines of research and development.
2. KNOWLEDGE DEMANDS FROM SOCIETY
2.1. Safety against flooding: Room for the River
Protests against repeated cycles of dike reinforcement and environmental protests against transformation into heavily engineered rivers have led to policies of giving more space to rivers.
These policies aim at restoring or enhancing the functioning of natural ecosystems, but also at reducing flooding risks. They assume various forms. In smaller streams, the focus is on restoring straight, canalized water courses into freely meandering rivers by removing bank protections.
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Morphological analyses help in delimiting the erodible corridor in which banks are allowed to erode freely (Piegay et al, 2005). In larger rivers, a main purpose is the reduction of flooding risks. Here two philosophies of giving more space can be distinguished. Some countries, such as France and Italy, give more space to the river upstream by assigning retention areas to store part of the flood discharge temporarily. This results in a reduction of peak discharges downstream. In the Netherlands, the leading philosophy is to increase the conveyance. This reduces the water levels without altering the discharges.
The Dutch policy to give more space to rivers is currently being implemented in the Room for the River programme along the Rhine branches. An innovative feature of this programme is its high level of democracy. Stakeholders such as the central government, provinces, municipalities, waterboards, non-governmental organizations and individual citizens proposed 693 local measures to lower the water levels during a design flood. The effects of each measure, including flood level reduction, costs and secondary effects, were calculated. WL | Delft Hydraulics used the results to construct the 'Planning Kit', which is an easy interactive tool to compose integral strategies. The user can select combinations of measures on a map of the Rhine branches displayed on the screen of a computer, and visualize the resulting lowering of water levels, costs and secondary effects. By using this tool, he or
Fig. 1. The Rhine branches in the Netherlands
she discovers which measures are effective and which measures are not, taking the perspective of an engineer who tries to find the best solution. In this way, the most appropriate measures are not imposed by the authorities but discovered by the people themselves. The Planning Kit turned out to be a popular tool. It was really used by the population, including mayors etc., not only by those who have affinity with computers or technology. Moreover, this tool made the available information accessible to all parties and in the same manner for everybody.
From all measures proposed, about 30 local measures have been selected for implementation. The increase of the conveyance is achieved by lowering floodplains, creating secondary channels or flood channels, removing obstacles, dredging the main channel, lowering groynes and setting back dikes. The associated sedimentation, however, decreases the navigability and limits the lifetime of the interventions. The adverse effects of main-channel sedimentation on navigation can be minimized by optimizing the design in such a way that the amount of sedimentation remains small or that the sedimentation takes place in deeper parts or in parts outside the fairway within the main channel. Sedimentation in the flood-plains requires periodic maintenance. In the fully trained Rhine branches in the Netherlands, where floodplains are no longer lowered by natural bank erosion, periodic excavation might have the additional advantage of providing habitats for pioneer species. This would help in maintaining a dynamic ecosystem with a continued presence of all stages of vegetation succession, instead of a static climax state only. Periodic floodplain maintenance could then be based on an ecological policy of cyclic rejuvenation.
The morphological effects of individual local measures are assessed by the engineers involved in the detailed design of the measures. Del-tares is responsible for auditing these assessments. A study on the combined morphological effect of all local measures
Fig. 2. Rhine bifurcation at Pannerden
still needs to be defined. A major issue here is the relation between morphological changes at bifurcations and the stability of the system of Rhine branches as a whole. It is for that reason that the morphology of river bifurcations has been a major research topic during the last years (Kleinhans et al, 2006, 2008).
2.2. Navigation: DVR project The river Waal is the main Rhine branch in the Netherlands and one of the busiest shipping routes in the world, connecting the sea port of Rotterdam with major urban and industrial areas in Germany. The total freight between Rotterdam and Duisburg amounted to 165 million metric tons in 1996. About every three minutes, a ship passes the border near Lobith, 24 hours a day and 7 days a week. Considering forecasts of further traffic growth, it was concluded in 1993 that safe, fast and efficient navigation in 2010 would require enlargement of the navigation channel from its present 150 m x 2.5 m profile to a 170 m x 2.8 m profile at OLR, i. e. at the low-water level that is exceeded during 95% of the time. The OLR criterion has been established internationally in 1947 and corresponds to a Rhine discharge of 1020 m3/s at Lobith. Alternative strategies to achieve this enlargement of the navigaton channel were elaborated in the Waal Programme. In 1996, the river management authority Rijkswaterstaat selected a preferred strategy composed of groyne extensions,
maintenance dredging and, above all, structural measures in river bends. The latter comprised bendway weirs, fixed layers (Sloff et al, 2006), longitudinal dams and bottom vanes. Inspired by the original work of Potapov (1950), WL | Delft Hydraulics had studied bottom vanes extensively in physical and numerical models (e. g. Flokstra, 2006). Bendway weirs were realized in the bend at Erlecom and fixed layers were constructed in the outer bend pools of the bends at Nijmegen and St Andries. However, bottom vanes in the bends at Hulhuizen and Haalderen were cancelled because doubts arose about their effectiveness and because the shipping sector was opposed to them out of fear that ships touching the steel bottom vanes might be cut open.
The original time horizon of the Waal programme was the year 2010. Recent extension of this horizon by 50 years till 2060 has created awareness that the ongoing overall bed degradation (incision, downcutting, entrenchment) of the river Waal causes problems. The fixed layers in the outer-bend pools at Nijmegen and St An-dries will not follow the degradation and hence become obstacles with limited depths at their downstream sides. The same applies to a natural non-erodible layer near Emmerich, just across the border in Germany. Sluices, river ports and canal entrances will not follow the degradation either and thus become obstacles for navigation too. Moreover, bed degradation may undermine the stability of banks, groynes and hydraulic structures, and may affect the morphological stability at bifurcations. The bed degradation results from a deficit in the sediment supply from upstream, from excessive dredging in the past and from a retarded adaptation to river training works and inland shift of the erosion base (river mouth), but the relative contribution of each of these causes is still unknown. The Waal is in this respect a particular case of a much wider problem, because most rivers in Europe and the USA exhibit overall bed degradation due to dams, torrent control works, aggregate mining, river training, lateral embankments, shortening of river courses, afforestation, vegetation encroachment and cessation of wood cutting and grazing.
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Fig. 3. Navigation bottleneck at non-erodible bed due to downstream bed degradation
Maintenance and improvement of the navigability of the river Waal thus requires dredging of local shoals but sediment nourishment to arrest the overall bed degradation. This calls for smart strategies of sediment management, to be developed under the DVR project (DVR = Duurzame Vaardiepte Rijndelta = sustainable navigation depth in the Rhine delta). It is within this framework that Rijkswaterstaat has commissioned first WL | Delft Hydraulics and now Deltares to develop a two-dimensional morphological model of the river Waal and adjacent Rhine branches. This model, based on Delft3D, will be the major instrument for the optimization of sediment management strategies and the assessment of the effects of these strategies on navigation. The model is basically a descendent of the first two-dimensional morphological model by Van Bendegom (1947). It contains a parameterization of helical flow in river bends that can be traced back to Ro-zovskii (1957). Struiksma et al (1985) present an earlier major milestone in the development of the model. Havinga et al (2005, 2006) provide more background information about the projects to improve navigability and the corresponding knowledge requirements.
2.3. Nature rehabilitation The current Water Framework Directive of the European Union ignores sediment and river morphology. The awareness is now growing, however, that hydromorphological changes are the main cause why European rivers are not attaining a good ecological state. Current policies of giving more space to rivers do aim at restoring or enhancing the functioning of fluvial ecosystems, but they often lack insight in how this
functioning depends on hy-drodynamic and morphological processes. In turn, the morphological development of rivers depends significantly on vegetation in ways that are still difficult to model quantitatively. Interactions between abiotic processes of hydrodynamics and morphology on the one hand and biotic life cycles of species and communities on the other hand are studied in the interdisciplines of ecohydraulics and biogeomorphology. They pose new questions to river morphologists. For instance, river engineering focused traditionally on outer-bend scour and bank erosion, whereas ecological processes depend more on inner-bend deposition, colonization by vegetation and bank accretion. Another example is that temporary deposition of mud on a gravel bed, frequently washed away during floods, has little importance for traditional engineering assessments of the morphological development of a river. However, this deposition can harm the reproduction of fish species if it occurs in certain parts of the year. More generally, temporal and spatial variations are more important for ecology than they are for engineering river morphology. Finally, the functioning of ecosystems depends also on the chemical composition of polluted sediments and the decrease of light penetration by turbidity due to suspended sediments.
2.4. River training As all major rivers in the Netherlands have been trained, further work here is merely a matter of subtle modifications, such as the lowering of groynes in Room for the River and the construction of fixed layers or bendway weirs in the Waal programme. Training of wild large rivers abroad, however, still poses great challenges. WL | Delft Hydraulics was involved in such river training projects in Bangladesh, a country where bank erosion is considered to be the number one natural hazard, well before drought as number two and flooding as number three. An example is the twelve-year FAP21/22 project of bank protection and river training along the braided Brahmapu-
tra-Jamuna river in Bangladesh (Mosselman, 2006). This is one of the world's largest and geo-morphologically most active braided rivers. Its enormous bank erosion leads in places to hundreds of metres of annual bank retreat. Mosques, schools and hospitals fall into the river and the losses of land often leave no other choice to the riparian population than to move to the slums of the capital Dhaka.
by the river within the time frame of the project, which originally was assumed to span five years. At the same time, the test sites should not be eroded during the first years of land acquisition and construction. This led to the need to predict bank erosion and planform changes two to three years ahead. For the rapidly changing Brahmaputra-Ja-muna with its sensitive dependence on small perturbations, such a prediction span is too long to be
Fig. 4. Forty kilometre length of the Brahmaputra-Jamuna River, showing low-water planforms on 25th of February 1985 (left) and 20th of February 1986 (right), with corresponding change-detection image (centre). Water levels at Bahadurabad on days of image recording: 13.49 m + PWD in 1985 and 13.38 m + PWD in 1986. Courtesy National Aerospace Laboratory NLR, the Netherlands
Test structures were built that were lighter than traditional structures for the conditions along the Brahmaputra-Jamuna River. They were assumed to be more cost-effective and thereby affordable for a developing country such as Bangladesh. Limited damage could be allowed as long as they maintained their function of stopping local bank retreat during the critical summer monsoon period. The damages could be repaired during the low-water season. The testing of the structures required that those structures had to be attacked
covered by either simple extrapolation or detailed two- or three-dimensional mathematical modelling. Therefore, Klaassen et al (1993) developed a new probabilistic prediction method based on empirical laws derived from a large set of satellite images (Klaassen and Masselink, 1992; Klaassen et al, 1993; Mosselman et al, 1995). Jagers (2001, 2003) implemented the prediction method in a computer model and tested it against observations. He also constructed and tested an artificial neural network for the prediction of low-water planform
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changes in the Brahmaputra-Jamuna. Despite work are limited. The original prediction method some fair results, it appeared that the predictive has nonetheless proven to result in a good selec-powers of the computer model and the neural net- tion of sites for bank protection structures.
Fig. 5. Construction of FAP 21/22 bank protection along the Brahmaputra-Jamuna River in Bangladesh
Fig. 6. Bottom vanes as low-cost river training structures for smaller rivers in Bangladesh
More knowledge remains required for the design of river training structures, on both large and small rivers. Predictors of local scour are still empirical and might become more reliable in the future if they will be based on the physics of detailed hydrodynamics and corresponding appropriate descriptions of sediment transport. More knowledge is also required on innovative river training structures that produce less scour or are more cost-effective. Bottom vanes appear to be a successful measure for the smaller rivers of Bangladesh. They have been investigated extensively in the laboratory and in the field (Hossain & Mosselman, 2006; Hossain et al, 2006 a, b).
3. CO-OPERATION AND NETWORKS
Groundwater unit of TNO, have merged into Del-tares. Representatives from the different institutes convene three times a year to formulate, to co-ordinate and to evaluate the research in river morphology on a national level. The Morphological Triangle has defined three main research themes. The first and most important theme is directed towards better understanding and better predictions of the morphology at Rhine bifurcations, because the stability of the Dutch Rhine branches depends strongly on the sediment distribution at these bifurcations. The second theme regards morphological phenomena during floods. The third theme regards biogeomorphology of floodplains. Within this theme, Utrecht University co-operates with Moscow State University.
3.1. Saint-Petersburg State University of Waterways Communications, Russia Co-operation with the Saint-Petersburg State University of Waterways Communications (SUWC) is the rationale for this article. In March 2008, Deltares representatives Sanjay Giri and Erik Mosselman visited SUWC to explore the possibilities of co-operation in the fields of river morphology and river engineering. The cooperation will start with exchanges of researchers. Problems of mutual interest were identified to include the distribution of sediment transport at bifurcations and the deterioration of navigation conditions due to bed degradation. River morphology and river engineering are important for Russia's entrance into the World Trade Organization in 2010. The entrance will require studies and investments to bring the navigation routes up to standard.
3.2. Morphological Triangle, the Netherlands Different institutes in the Netherlands have joined their forces by founding a 'Morphological Triangle'. This triangle is a thematic subdivision of the Netherlands Centre for River Studies (NCR) in which universities, technological institutes and Rijkswaterstaat-Waterdienst (formerly Rijkswa-terstaat-RIZA) co-operate. The participating universities are those of Delft, Utrecht, Twente and Wageningen. Also the UNESCO-IHE Institute for Water Education is counted among this group. The two participating technological institutes, WL | Delft Hydraulics and the Subsurface and
3.3. Bundesanstalt für Gewässerkunde, Germany Germany nourishes the river Rhine with sediments in order to arrest undesired river bed degradation. Volume, composition and location of the sediment nourishments can be optimized by using a two-dimensional model for river morphology. First computational results showed that Delft3D could realistically reproduce the development of the river bed after nourishment (Sloff & Sieben, 2007). The Bundesanstalt für Gewässerkunde and Deltares are currently verifying the model against field data from sediment nourishments at Iffezheim near Karlsruhe.
3.4. SedNet, European Union SedNet is a 15 -year European Union project regarding sediment management on river basin scale. It has a strong international network and was originally initiated and co-ordinated by the Subsurface and Groundwater unit of TNO. Since this unit became a part of Deltares, SedNet and the River Dynamics and Inland Water Transport department have joined forces.
3.5. US Geological Survey, United States of America The US Geological Survey carried out a controlled flood experiment in the Grand Canyon in March 2008. The purpose of this experiment was to study how sand bars and beaches can be restored. These bars and beaches are currently disappearing due to erosion, but the Colorado River once managed to maintain them when it still
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had a natural discharge regime. The study has a modelling component, using Delft3D as the most advanced model available, with features such as large-eddy simulation and the erosion, transport and deposition of graded sediments. That is why Deltares participated in the experiment. Another area of co-operation is the American Community Surface Dynamics Modeling System, where the US Geological Survey is one of the partners and where a Deltares specialist, Bert Jagers, is a member of the steering committee. The Community Surface Dynamics Modeling System develops, supports and disseminates integrated software modules that predict the erosion, transport and deposition of sediment and solutes in landscapes and their sedimentary basins. It involves the Earth surface as the dynamic interface between lithosphere, atmosphere, cryosphere and hydrosphere.
3.6. Nanjing Hydraulic Research Institute, China The Nanjing Hydraulic Research Institute (NHRI) and Deltares have identified a number of projects of mutual interest that are suitable for cooperation. The projects include river morphology, sedimentation in navigation channels, impact of dredging and dumping, and sand-mud transport. The co-operation will start with a period in which Chinese engineers from the NHRI River and Harbour Engineering Department and the NHRI Hydraulic Engineering Department will work at Deltares for some time. They will study Chinese problems in close consultation with Deltares colleagues and they will participate in research projects of Deltares.
4. SOME LINES OF RESEARCH AND DEVELOPMENT
4.1. Bed forms and hydraulic roughness Dunes on the river bed play several important roles. First, they produce form rough-« ness as a major component of the hydraulic re-
| sistance. Second, dunes may represent shoals
hindering navigation at lower discharges. Third, __ their growth and migration are intimately related to sediment transport, which implies that accurate modelling of sediment transport requires a good representation of dune dynamics (and vice versa). Fourth, spatial variations in dune shapes and dimensions and, hence, spatial variations in
hydraulic roughness affect the pattern of bars and pools (Mosselman et al, 1999). The hydraulic resistance is particularly important during floods as it determines the relation between discharge and water levels. The development of dunes during a flood is complex. Initially dunes grow higher and make the river bed rougher, but in later stages the dunes grow longer with the opposite effect of making the river bed smoother. Subsequently, in a way still not well understood, new bed forms develop on top of the elongated dunes that make the river bed rougher again. Rijkswaterstaat and the University of Utrecht measured dunes in the Rhine branches (Wilbers & Ten Brinke, 2003; Wilbers, 2004). Toyama et al (2007) and Giri et al (2007) present physics-based models of dune development. Knowledge from this type of detailed models needs to be parameterized, because dunes are subgrid features in operational morphological models for larger spatial scales.
4.2. Sediment mixtures The sediment in a river is usually a mixture of particles with differing physical characteristics. Segregation of these particles during processes of erosion, transport and deposition gives rise to spatial grain sorting phenomena such as coarse bed surface layers, downstream fining and finer-grained point bars (Kleinhans, 2002; Blom, 2003; Frings, 2007). These grain sorting phenomena affect the patterns and dimensions of pools and bars (Mosselman et al, 1999). The sorting may be due to differences in mass density (mineralogical composition), grain size and grain shape. For the sorting of coarse sediments, differences in grain size are the dominant cause and it is for these differences that graded-sediment functionality has been implemented in numerical models such as SOBEK and Delft3D. This functionality is based on (i) division of the sediment mixture into separate fractions, (ii) transport formulae and mass conservation equations for each of the separate fractions, (iii) hiding-and-exposure corrections to the critical shear stress of each of the fractions, and (iv) an active layer or transport layer affected by erosion and sedimentation. A current line of development is to replace the simple active-layer concept by a more detailed description of vertical sorting in bed forms along the lines of Blom (2003) and Blom et al (2003).
The models for graded sediment are complex and contain many parameters. This makes it difficult to evaluate model results on the basis of numerical outputs alone. Mosselman & Sloff (2007) and Mosselman et al (2008) therefore ana-
lyze the models theoretically to obtain a better understanding of their fundamental behaviour. The analyses are based on Ribberink's (1987) original theoretical analysis of morphological models with graded sediment.
2000 2500
3000 3500
distance (m) ->
4000 4500
10.005
10.0045
10.004
0.0035
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Fig. 7. Computed bed topography (top, in m above NAP) and bed sediment composition in terms of mean sediment grain sizes (bottom, in m) in the area of the Rhine bifurcation at Pannerden (using Delft3D with a constant
discharge of 6000 m3/s, Mosselman & Sloff, 2007)
4.3. Biogeomorphology Biogeomorphology studies the interactions between organisms and the development of land-forms. Vegetation represents the most important organisms for interactions in fluvial biogeomor-phology, although other organisms affect river morphology as well (e.g. perturbation of the river bed by fish and trampling of river banks by cattle). The hydraulic roughness of vegetation can be included straightforwardly in morphological modelling (Baptist & Mosselman, 2002; Baptist, 2005), but still much research is needed to develop quantitative models for colonization of newly accreted land, vegetation succession, vegetation effects on erodibility and vegetation effects on sediment transport. In this field Deltares co-operates with UNESCO-IHE Institute for Water Education and Utrecht University. The latter institute co-operates with Moscow State University in biogeomorphological studies of floodplains of the Volga.
of two to three years is too long for deterministic two- or three-dimensional mathematical modelling. A framework has been developed in which different aspects of the development of this river, operating at different time scales, have different predictability horizons and require different types of modelling for different quantities, in analogy to the differences between weather forecasts and predictions of climate change. The development of an individual channel during a single flood may be modelled using a two-dimensional morphological model, changes in braided river planform over a period of two or three years may be assessed using the approaches of Klaassen et al (1993) and Jagers (2001, 2003), and overall migration of the braid belt on time scales of decades may be estimated using mere statistical methods.
The morphological development of the Rhine branches, albeit more predictable than the development of the Brahmaputra-Jamuna, has also uncertainties due to model incompleteness,
Fig. 8. Tree uprooted by bank erosion promotes bar formation
4.4. Stochastic morphology, uncertainty analysis and predictability The rapidly changing Brahmaputra-Jamuna with its sensitive dependence on small perturbations has been an important inspiration for considerations on uncertainty and predictability in river morphology. For this river, even a prediction span
insufficient data and unknown future discharges. This is addressed in the stochastic river morphology developed by Van der Klis (2003) and Van Vuren (2005).
4.5. Other lines of research Other lines of research include the effects of gravity and helical flow on transverse bed slopes
(Talmon et al, 1995), the stability of river bifurcations (Kleinhans et al, 2006, 2008), reservoir sedimentation (Sloff, 1997; Sloff et al, 2005, 2007 a, b), the morphology around groynes (Yossef, 2005, 2006; Yossef & Rupprecht, 2006; Yossef et al, 2007) and bank erosion (Mosselman, 1992; Mengoni & Mosselman, 2005; Akkerman et al, 2006; Rinaldi et al, 2008). Andrej Alabyan extended Mosselman's (1992) model RIPA for the morphology of rivers with erodible banks with more detailed bank failure algorithms (Darby et al, 2002). The research on bank erosion has close links with the modelling of river meandering at Delft University of Technology (Crosato, 2007 a, b).
5. MATHEMATICAL MODELLING
Deltares has been implementing morphological functionality in the two- and three-dimensional modelling system Delft3D and in the one-dimensional modelling system SOBEK for decades. These modelling systems have thus become major carriers of river morphological knowledge. Numerical models such as SOBEK and Delft3D are not only a final product of research, to be used for predictions in practical applications. They play a central role in all phases of research and development. They help
in sharpening research questions, in specifying requirements for field measurements, in integrating knowledge, in testing hypotheses and in communicating ideas.
Theoretical analyses of mathematical equations are complementary to numerical solutions, as they offer additional insights into the fundamental behaviour of the corresponding physical system. Designing numerical models requires theoretical analyses to determine the appropriate numerical scheme and the type and location of the boundary conditions to be imposed. Theoretical analyses also help the optimization of calibration strategies for numerical models, as they reveal which parameters are responsible for different aspects of the solution. They help the interpretation of results from numerical models as well, because numerical solutions may exhibit spurious wiggles, phase lags or attenuation that in this way can be distinguished from real physical phenomena. Furthermore, analytical solutions provide exact solutions for certain idealized cases that may serve as validation cases for numerical models. Finally, analytical solutions can be used as rapid assessment models or rules of thumb. Examples at Deltares are the theoretical analyses of river morphology by Struiksma et al (1985), Mosselman et al (2006), Mosselman & Sloff (2007) and Mosselman et al (2008).
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