Sorting out river channel patterns

Article Progress in Physical Geography 34(3) 287–326 ª The Author(s) 2010 Reprints and permission: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0309133310365300 ppg.sagepub.com

Sorting out river channel patterns Maarten G. Kleinhans Utrecht University, The Netherlands

Abstract Rivers self-organize their pattern/planform through feedbacks between bars, channels, floodplain and vegetation, which emerge as a result of the basic spatial sorting process of wash load sediment and bed sediment. The balance between floodplain formation and destruction determines the width and pattern of channels. Floodplain structure affects the style and rate of channel avulsion once aggradation takes place. Downstream fining of bed sediment and the sediment balance of fines in the pores of the bed sediment provide the ‘template’ or sediment boundary conditions, from which sorting at smaller scales leads to the formation of distinct channel patterns. Bar patterns provide the template of bank erosion and formation as well as the dynamics of the channel network through bifurcation destabilization. However, so far we have been unable to obtain dynamic meandering in laboratory experiments and in physics-based models that can also produce braiding, which reflects our lack of understanding of what causes the different river patterns. Keywords bank erosion, bar pattern, floodplain sedimentation, riparian vegetation, river channel pattern

I Introduction This review is concerned with river morphology at a scale of, say, tens of bar or meander lengths, at which the channel pattern is the most striking characteristic. The storyline of this paper is the effect of various sorting processes on channel patterns that emerge as channels and floodplains form, erode and reform. Rivers sort the sediments that they receive from their hinterlands at sorting scales from particle size to river length. From mountains to the sea the bed sediment may fine from cobbles to mud. Across the river valley mud is found in floodplains and sand and gravel in the channels. Styles of sorting out sediments differ between channel patterns, but channel pattern strongly depends on the floodplain sediment and vegetation properties as well. This is partly linked to bed sediment sorting through the transition of fines from wash load

to bed material load. From bars to channels and through bends the bed sediment may vary horizontally from sand to gravel, which affects their morphodynamics and the steering of flow against banks by the bars. In short, morphological and sedimentological (sorting) patterns of rivers emerge as the result of interactions and feedbacks between the smallest turbulence and particle-scale processes up to reach-scale flow dynamics, sedimentation, erosion and vegetation patterns. The aim of this review is to unravel connections between various subjects related to river channel patterns, including bar theory, bank erosion,

Corresponding author: Faculty of Geosciences, Utrecht University, PO Box 80115, 3508 TC Utrecht, The Netherlands Email: [email protected]

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floodplain sedimentation, and interactions with vegetation. Interactions between morphology, sorting of fines and coarse sediment, as well as vegetation, are collected into a grand picture. Whether correct or not, I will argue that the time is ripe for making such connections quantitatively in integrative experiments and physics-based numerical models, which are already starting to accelerate our progress in the understanding of channel patterns. With the storyline of sorting I stress the intimate relation between channel and floodplain, and I will argue that this relation is the outstanding problem to be solved for better understanding of the morphogenesis of rivers. As a necessary prelude to the core of this paper, I will argue how and why combining field study, experiments and numerical modelling will lead to rapid progress in the coming decade. The next section is therefore devoted to basically different logical explanation structures of Quaternary geologists, sedimentologists, geomorphologists, engineers and so on. The main part of this paper, section III, is devoted to channels and channel patterns and how they emerge as the result of floodplain dynamics and channel dynamics and the basic size-sorting of sediment that may lead to this division. Furthermore, a strong link between the fractional bed sediment balance and the floodplain sediment balance and hence channel pattern is identified.

II Prelude: Three pillars of the Earth sciences Quaternary geologists, fluvial sedimentologists, fluvial geomorphologists, civil engineers and so on use three very different approaches that have their own aims, weaknesses and strengths. The basic logical explanation structure of Quaternary geology, for instance the reconstruction of the avulsion history of the Rhine-Meuse delta (Berendsen and Stouthamer, 2000), is completely different from the logical structure of physics-based modelling of river bifurcations 288

(Kleinhans et al., 2008), and from systematic experimentation on laboratory-scale deltas and fans (Bryant et al., 1995; Hoyal and Sheets, 2009), although they all refer to exactly the same natural phenomenon. Weaknesses of both logical structures and scope for progress will be discussed in the next section. The research methods are also very different: fieldwork, modelling or experiments. How and why these methods are fruitfully combined is explored in the following section. I hope this paper will clear up common blind spots and misconceptions of workers in different fields and promote mutual understanding and collaboration by providing a vocabulary. No straw man is slain here: misunderstanding between Quaternary geology and geomorphology has been reported (eg, Baker, 1996; Rhoads and Thorn, 1996) and nothing short of fear of being overwhelmed with facts or equations has been experienced on both sides. As further evidence, many papers have been written about the need for interdisciplinary approaches to prediction of Earth surface dynamics (eg, Paola et al., 2006), and the need for more quantitative physics-based work in the predominantly descriptive geographical sciences (Church, 2005). Finally, there remain many opportunities for Earth scientists to collaborate with geophysicists and civil engineers all over the world.

1 Earth scientists as detectives ... ‘Pure deduction, my dear Watson’? When asked what basic logical explanation structure is used in natural science, many scientists will answer ‘deduction and induction’. But this is incomplete. There are three logic components in our explanations: causes, effects and laws. Thus there are three ways to derive one component from the other two (Figure 1; Kleinhans et al., 2010a).

a Deduction. For deduction, the initial conditions (causes) are combined with laws of nature


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Figure 1. The three logical explanation structures based on causes, effects and laws, two of which are necessary to arrive at the third. Each has its own weakness (see text). Source: Kleinhans et al. (2010a)

to predict the effects. This is what happens in analytical solutions for linear stability analyses and physics- or chemistry-based modelling (except ‘reduced-complexity’ modelling; see later discussion). Although deduction is a solid form of logic, its Achilles’ heels are the choice of relevant laws included in the model, the inclusion of generalizations rather than laws, numerical issues, and the initial and boundary

conditions for the model which must be based on measurements that may be incomplete or contain errors (Oreskes et al., 1994). Deduction is also what happens in 1:1 and scale experiments. The limitation of experiments differs from that of models (illustrating their complementary benefit). The materials, and the implicit laws that come with it, are in a sense more real than in models (Morgan, 2003), but scale effects may


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limit the validity. This limit can be explored by proper non-dimensional analysis of relevant parameters as well as modelling on experimental and real scales (Peakall et al., 1996).

b Induction. Induction leads to (statistical) generalizations based on both causes and effects. Empirical hydraulic geometry relations between some measure of flow discharge and cross-sectional properties of natural river channels or stable canals are obvious examples (Ferguson, 1987). Also experimental relations between a measure of flow strength and sediment transport rate fall in this category, although relations of similar form have been derived from physics with some rigour. Furthermore, generalizations about avulsion based on field data of (sub)recent fluvial systems are inductive (Aslan et al., 2005; Stouthamer and Berendsen, 2007). The problems of induction are well known: the validity range of empirical relations is determined by the range and bias of the data included, and the amount of data is obviously never large enough to create a universally valid generalization – that is, law. Nevertheless empirical relations somehow contain knowledge about reality and have shown the way to underlying mechanisms in the past.

c Abduction. In abductive inference, final conditions, facts and so on are (often implicitly) combined with laws or generalizations of nature, to arrive at the best of a limited number of hypotheses that explain the observations. For example, crime scene investigations yield several clues that, combined with ‘laws’ of human behaviour and biology (blood and DNA evidence), all converge to the best explanation that Moriarty was the murderer. Pure deduction? No, Sherlock, this is an example of abduction, which Earth scientists commonly employ too (for background and historical references, see Baker, 1996; Kleinhans et al., 2005). 290

For instance, present-day landforms and conditions, in short, effects, are (often implicitly) combined with laws of nature and geoscientific generalizations – that is, major premises – to infer the best from a limited number of hypotheses that explain the observations, such as past conditions. Thus, the avulsion history of the Rhine delta was explained by a time-varying combination of sea-level rise, tectonics, climate change and autogenic processes (Stouthamer and Berendsen, 2000). Also the inference of formative conditions and processes from similar rivers as modern analogues is abductive. The major limitation of abduction is that one cannot be certain that all possible hypotheses, including the correct one, have been conceived. The right hypothesis might be one that no one thought of. Furthermore the geoscientific ‘laws’ of nature are often tacit so that it is far from proven that the inference to the best explanation does not contradict the laws of nature. For instance, a tacit law could be that climate change leads to a transition from braided river to meandering river, but as the precise reasons remain unclear this is far from a universal law.

d Mature explanation structures. Deduction, induction and abduction are used complementarily to great benefit to answer different questions. For instance, channel pattern discriminators provide empirical knowledge that points towards underlying physical causes. Abductive inferences to historical causes use generalizations or laws. The latter are tested with (scale) experiments or physics-based modelling to confirm that the inferred initial conditions indeed lead to the observed phenomena. Models produce diagnostic features that were hitherto not recognized or misinterpreted from images. In general, Earth scientific explanations appear to have an abductive structure with many inductive and deductive elements in it (Kleinhans et al., 2005). Note that various other explanation structures have been ignored, notably functional and teleological explanation. Functional explanation


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refers to the laws of past evolution of life. For instance, certain riparian reeds are flexible as they would otherwise break in the current. This trait of the species evolved in the past from mutations, was transmitted and lived on through genes, because it increased the chance of survival for the individuals that bear the trait. Teleological explanation refers to future-directed actions of man. For instance, the height of dikes along a river is explained by the fact that they were built with the intention to mitigate future flooding risk. Two things became clear in the discussion above. First, all three types of logical explanation commonly used in the Earth sciences have serious weaknesses; none is better than the other. Second, observations are somehow combined with experiments and modelling in natural science in general and by river researchers in specific. Why?

2 Complementarity of field data, experimentation and modelling Observations (including satellite imagery, measurements and so on) and experiments and models represent nature in different ways and have different weaknesses and strengths. The following discussion aims to provide a foundation for the truism that observation of nature, experimentation and modelling are complementary and should be combined where possible. By understanding why and how they are complementary, their application to the problem of river patterns will become more efficient and focused.

a The story of an Earth scientist’s life: Fieldwork and underdetermination. Earth scientific theories and hypotheses, ranging from mechanistical theories to explanatory reconstructions of past conditions, usually are underdetermined by the available evidence. That is, there is insufficient available evidence to choose one theory over its rivals. Typical examples of underdetermination problems are:

Measurement techniques may disturb the observed processes. The timescale of human observation is (much) shorter than that of the phenomenon under study. Many processes and phenomena cannot (yet) be observed directly or even indirectly. Erosional and sedimentary landforms of the past may have been obliterated by later erosion, and phenomena may not be accessible in practice. The relatively simple laws of physics can seldom be applied directly to the initial conditions to check whether they explain the observations, because they may be applied in models in many different ways that provide conflicting answers. Many processes are intrinsically random or chaotic, and may be very sensitive to initial conditions. This means that inference to the best explanation – that is, the most plausible combination of formative processes and initial conditions – may be impossible.

Practising Earth scientists very often face situations in which theories are underdetermined by the available evidence, perhaps in contrast to physicists. In fact, it is hard to find papers that do not contain at least a paragraph on the way underdetermination was dealt with in practice (although it is usually not explicitly referred to as underdetermination). For example, consider a research project that aims to construct a spatiotemporal description and an explanation of the course of the river Rhine in the past 10,500 years (Berendsen and Stouthamer, 2000). The hypothesis that the Rhine has been present in the Netherlands is obviously practically unassailable. But what we really want is a description, and an explanation of the course of events (their order in time and their specific characteristics) that is generalizable to other, comparable river deltas in comparable circumstances. For such an explanation, much more detailed evidence is needed


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to distinguish between competing theories on, for instance, causes (initial conditions), relevant circumstances, and principles of evolution of bifurcations and avulsion. However, empirical data often leaves room for a wide range of different, often incompatible, hypotheses. In sum, the inferences are hampered by problems of underdetermination (Kleinhans et al., 2005).

b Physics-based modelling. A physics-based model may be used to test whether a hypothesis does not conflict with the laws of physics. But a model contains various sets of laws, of which some at best are derived from physics, but even then are simplified to allow numerical solutions. Also, for many problems it is not obvious which laws apply, and to what extent simplification is possible. Thus one cannot be certain that a mismatch between model results and observations is not due to the simplifications and numerical techniques or the initial and boundary conditions used in the model (Oreskes et al., 1994). For instance, conservation of momentum and conservation of mass are simple physical laws that apply to fluid flow. These simple laws are the basic components of the Navier-Stokes equations that describe fluid flow, but cannot be solved analytically. These equations are therefore implemented in so-called mathematical or physical computer models. There is a trend in geomorphology at present to develop ‘reducedcomplexity models’ (discussed later), but clearly every model reduces the complexity of reality. In this paper the focus is on physics-based modelling rather than rule-based modelling. For physics-based numerical models to work in practice, the equations have to be simplified, and discrete timesteps and grid cells must be used to model the flow in space and time. The discretization brings a host of necessary numerical techniques to ensure conservation laws and to minimize numerical (computer-intrinsic) error propagation. When initial or boundary conditions are specified for this model, certain laboratory or field conditions can be simulated. 292

In this way, a model is used to test whether a hypothesis does not conflict with the laws of physics, and the model results can then be compared to the observations. As such, models are not very useful for simulating the details of a concrete existing case, because then the initial and boundary conditions must be specified in great detail so that it is no longer clear whether the results are due to these empirical parts or due to autogenic modelled behaviour. Nevertheless this is common practice in river and coastal engineering, which is clearly justified because of the societal need, and is further justified by using established models, and is commonly (but unfortunately not universally) complemented by healthy mistrust and careful specification of prediction uncertainties. Furthermore, models can be tested thoroughly if not verified on highly detailed experimental data of simple setups, as is increasingly done for morphodynamics and CFD codes (Hardy et al., 2003; Lesser et al., 2004). This removes some of the need for detailed initial conditions, although boundary conditions still present problems, because they were relatively simple in the experiments compared to a field verification case. However, models are very useful to present results of complicated sets of equations that the unaided human mind cannot comprehend, sensitivity to certain parameters and to explore scenarios (Oreskes et al., 1994; Kleinhans et al., 2005). As such, models are used to mediate between theory, based on physics, biology and perhaps chemistry, and nature (Morrison and Morgan, 1999).

c Experiments. Many different types of experiments represent reality in one way or another (Peakall et al., 1996), and all have limitations similar to those listed above for models, in particular the simplified initial and boundary conditions. One important additional problem is that of scale effects. In Froude-scaled models it is attempted to obtain a Froude number and a


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Shields mobility number as close as possible to the prototype, but, as one depends on velocity and the other on velocity squared, it is impossible to satisfy both requirements. The Reynolds number may be lower but this is considered to have small effects on the mobile bed as long as the flow remains turbulent. Moreover, the sediment can usually not be scaled down with the same factor as the length of the model because of cohesion problems with finer sediments. The sediment mobility scale problem is commonly solved by applying a higher bed slope to the model, called distortion, so that the relatively coarser is equally mobile (Peakall et al., 1996). But then scale problems arise, such as in the reproduction of alternating bars (Struiksma et al., 1985). It has been attempted to forego the necessity of distorting the model by using lightweight sediments such as bakelite, but this appeared to give other scale effects, for instance in the transverse bed slope (de Vries et al., 1990). For the scale modelling of suspension, however, lightweight sediments may still be necessary, but this is in its infancy. It matters which aspects are under scrutiny whether scale effects are really problematic. Against experiments with vegetation on braided rivers it has been argued that the stems of the plants scale like Sequoia gigantea. In terms of size this may be correct, but size does not matter here. The relevant properties of the plants are their hydraulic resistance as well as the strength their roots provide to the sediment (Tal and Paola, 2007). This added strength can be quantified with geotechnical experiments. In analogue experimental models the scale factor may be much larger. Consequently the aforementioned scale effects prohibit a straightforward quantitative comparison to the natural system. Seen from the perspective of a quantitative replication of the detailed morphology of a prototype, analogue experiments are therefore poor tools. In particular, the particle sizes are ridiculously large compared to water depth. Highly subcritical flows and meandering are

nearly impossible to obtain because the flow is commonly near-critical or supercritical (Peakall et al., 1996). It could then be argued that such experiments represent braided gravel systems well. However, in the lab the flow is commonly hydraulically smooth (Postma et al., 2008) which leads to other bedforms and bar patterns as well as unrealistically deep scour holes. Thus, the mean grain size of the sediment should be larger than about 0.5 mm, or the conditions should be made hydraulically rough by the presence of larger sizes in a mixture. However, for somewhat different aims analogue experiments are excellent tools. The sediment can be seen as the material that builds up stratification. Given the large scale factor, time is very much compressed so that long periods can be studied. The detail at the particle scale is irrelevant for such sequence stratigraphy, which is more referring to geometry of sediment package stacking and sediment budget than to the detailed sediment transport and morphodynamics of how the sediment got there (Paola et al., 2001; van Heijst et al., 2001). Moreover, analogue experiments can be studied as interesting cases of natural systems in themselves. They are, after all, composed of the same material as in nature: water and sediment. The simple fact that many patterns in nature are similar over a wide range of scales suggests that the basic factors controlling their nature are similar (Paola et al., 2001). For a number of relevant systems and aspects we know this to be true: alluvial fans and fan deltas occur in nature at the scale of hundreds of kilometres as well as at the scale of 1 m on the bank of a river or on the beach. Their autogenic processes of incision, sheet flow and avulsion as well as response to allogenic forcing (changing boundary conditions or simulated tectonics) can well be studied in experiments (Bryant et al., 1995; Ashworth et al., 2004; Hoyal and Sheets, 2009; van Dijk et al., 2009). With carefully chosen materials of different density, large-scale sorting along sedimentary basins can also fruitfully be studied


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reality

conceptual model, description

numerical model

experiment

Figure 2. ‘To twist the lion’s tail’ and observe what would happen, Lord Bacon’s view on doing empirical science, is not commonly possible with large rivers because it is dangerous. Instead, we twist tails of representatives of lions. These all have shortcomings and limitations, calling for combination of the three pillars of the Earth sciences: observations, experiments and modelling. (Note: whatever the cartoonist wants the reader to believe, the author does not have a tattoo.)

(Paola et al., 2001). This is not to say there are no scale problems at all, but they are not fatally problematic.

d Twisting a lion’s tail. Lord Bacon once stated that scientists want to twist the lion’s tail. That is, we want to manipulate reality to see what will happen, so that we find out how the beast works. This is unwanted, dangerous or impossible in many Earth scientific cases. Large rivers were avulsed by man for warfare purposes, which cost many lives (Parker, 1999; Slingerland and Smith, 2004). In lowlands such as the Netherlands great effort has been put into preventing such disasters for centuries, with some success. 294

But we can twist the tails of representatives of lions (Figure 2). We can twist our qualitative images of lions, albeit myopic, bald and threelegged ones, as we reconstruct the workings and the past of rivers. But then we run the risk of twisting our own imagined tail rather than the real tail. We can twist tails of downscaled relatives of lions (kittens) in a variety of ways to experiment systematically. There are scale effects: when the experimental lion size reduces to that of a beetle, their mode of locomotion dramatically changes. We can play exhaustively with model lions, albeit simplified. But then there are numerical problems, particularly with rectangular model lions representing rounded real lions, and their behaviour is limited by the


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laws we apply a priori. Henceforth any feeling of superiority of modellers or fieldworkers or experimentalists can therefore safely be abandoned. Now to be serious. This is my answer to the fundamental question of why and how we fruitfully combine fieldwork, experimentation and modelling. Field data is as close as possible to reality, but may be seriously hampered by incompleteness, inaccessibility and other problems of underdetermination. Experimentation allows a much larger control and accessibility while maintaining materiality, but may suffer from scale problems. Modelling allows full control over boundary conditions and laws, but the representativeness of reality is considerably decreased. The only hope is to base such models on the laws of physics, chemistry and perhaps biology so that they can be used to test whether hypotheses derived from field data and experiments are not in disagreement with established laws. There are many more roles for models, but this is the most important for basic research. Most importantly, when the results from all three epistemic approaches converge, we foster hope that we possess good explanations for natural phenomena.

III Riverland: Patterns of selfformed channels and floodplains Natural rivers on plains and deltas can have distinctive planforms such as meandering and braided. Why these patterns emerge is only qualitatively understood. Their self-organization involves feedbacks between channel morphodynamics and channel migration, and the evolution and subsequent erosion during floods of floodplains including the vegetated levees that flank the river as natural dikes. This dynamic self-organizing landscape and its sedimentation patterns will simply be called ‘riverland’ in this paper. In this section, problems of classical classifications and explanations for channel patterns

will be reviewed as well as the potential for bar theories, bank erosion and floodplain formation to contribute to a comprehensive explanation. Recent work on experimental channel patterns and on modelling will be reviewed to identify where progress can be made in the near future. For clarity, some matters were simplified; for example, channel patterns other than meandering and braiding (Figure 3) are not thoroughly considered. It is to be expected, however, that a more complete understanding will also cover anabranching, anastomosing, wandering rivers and so on.

1 Classifications Many qualitative classifications have been forwarded in the past. Leopold and Wolman (1957) emphasized that a braided river is defined by the number of active channels being larger than one, while each individual channel may meander as in single-thread rivers. More importantly, Leopold and Wolman emphasized that a continuum of channel patterns with many intermediate classes represents natural rivers better than a hard discrimination between meandering and braiding such as they induced. Schumm (1985) distinguished straight, meandering and braided classes from suspended load-dominated to bedload-dominated and from low to high flow strength at the same time, from small to large width/depth ratios and high to low pattern stability. Since then, anastomosing, anabranching, wandering and many other patterns have been identified. Ferguson (1987) reinterpreted the qualitative channel pattern classification of Schumm (1985) in terms of streampower on the one hand, and amount and size of bedload as well as width/depth ratio and channel instability on the other (Figure 4). Interesting groups of patterns then emerge. For sand and increasing streampower, straight, meandering and anabranching patterns emerge. For gravel, straight, slightly sinuous with alternating bars, and braided rivers emerge.


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Figure 3. Examples of rivers of various patterns and flow strength versus bank strength ratios. From top to bottom: Brahmaputra River, India (10 km wide braid plain), Rakaia River, New Zealand (1.7 km wide braid plain), Allier River, France (0.8 km wide meander belt), Koyukuk River, Alaska (10 km wide meander belt), Columbia river, Canada (2.1 km wide fluvial valley), Escalante River, Utah (60 m wide channel) and Nanedi Valles, Mars (2 km wide channel). Source: Google Earth and Mars (accessed May 2009)

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important new insight. Two classical explanations were given for channel pattern that are still debated: (1) pattern changes with flow strength; (2) pattern changes with sediment feed rate.

a Flow strength and sediment feed. The first

Figure 4. Typification of channel patterns Source: After Ferguson (1987)

Alabyan and Chalov (1998) showed how every known pattern can be described as a combination of three main configurations (straight, meandering and branched) on three main relief or structural scale levels (low water channel, flood channel and valley bottom). Thus, a single discharge is no longer used; rather, the pattern is determined at a range of discharges involving larger scales. Moreover, the effectiveness of a discharge, expressed by the product of sediment transport rate and frequency of occurrence, was shown to exhibit more than one peak for Russian plain rivers, only one of which is near the mean annual flood and one other determines where the rivers appear to be branching. Two classes of factors emerge that determine the channel pattern: flow strength and sediment characteristics. A similar distinction between pattern at bankfull and flood conditions was found useful in the Brahmaputra River (Thorne et al., 1993), where weak meandering of the entire braid belt in its valley was found.

2 Classical explanations for channel patterns Ferguson (1987) provides an insightful review of the state of the art c.1987 and presents an

explanation is that the pattern changes from meandering to braiding with increasing flow strength. Flow strength is determined by channel-forming discharge and energy gradient, which can be combined in a (unit) streampower or a bed shear stress. Many discriminators between meandering and braiding rivers have been proposed based on flow strength (Leopold and Wolman, 1957; Ferguson, 1987; van den Berg, 1995). Note, however, that all these workers stress that there are no hard thresholds; rather, gradual transitions exist between channel patterns, indicated by a discrimination line. Various modifications were proposed on how exactly the necessary parameters should be determined in practice. This is not a trivial problem. For instance, energy gradient depends on channel pattern (the more sinuous, the lower) and can therefore not be used as an independent predictive variable. Valley gradient, on the other hand, is independent of the timescale over which channel pattern may change (Ferguson, 1987; van den Berg, 1995). For instance, the field determination of reach-averaged channel-forming discharge, width and depth is difficult and somewhat subjective in natural rivers (see Soar and Thorne, 2001, for an impressively complete review). The second classical explanation for channel pattern is that it depends on supply and type of sediment (Ferguson, 1987). The type of sediment is relevant for bank stability (discussed later): sinuosity has been found to decrease with increasing proportions of silt and clay or vegetation in the channel banks and bed. Changes in the supply of bed sediment feed to a channel had been observed to provoke changes in the channel pattern. Overloading a river with more sediment than transport capacity may result in braiding


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Figure 5. Experimental river with pattern between braided and meandering (at St Anthony Falls Laboratory, 10 m long, 2 m wide). Bars are less mobile due to vegetation and the number of active channels is reduced. Source: Tal and Paola (2007; 2010)

whereas reduced load may result in meandering (Church, 2006). But again practical problems proved difficult to overcome: for instance, the definition of a representative particle size is not straightforward as it varies rapidly and strongly over short distances; sediment transport is notoriously difficult to measure, and, moreover, the overloading or underloading suggests disequilibrium of the longitudinal profile. A disequilibrium between sediment feed and sediment transport capacity would lead to deposition near the sediment source rather than overloading along a considerable length of the river. Yet equilibrium meandering and braiding rivers are found; in particular the latter have been reproduced in laboratory experiments. An important problem applying to both classical explanations is that there appear many different channel pattern classifications (Leopold and Wolman, 1957; Schumm, 1985), pointing at a deeper source of uncertainty of what exactly the deterministic characteristics of channel patterns are – for example, whether braid bars are vegetated or not, how many braids there may be, what the minimum sinuosity is for meandering, at which flow stage bars should (not) be emergent, and whether there is channel sediment or floodplain sediment in between the channels (eg, Leopold and Wolman, 1957; Knighton and Nanson, 1993). The obvious answer is that 298

channel patterns should be considered along a morphological continuum rather than as separate classes, and are moreover likely to be continuously but hysteretically adapting to changing boundary conditions rather than in equilibrium (Leopold and Wolman, 1957; Ferguson, 1987; Vandenberghe, 1995), but this still leaves unclear how the patterns should best be characterized.

b Experimental channel patterns. Lack of experimental evidence was another major problem for both explanations, as it proved very difficult to produce self-formed dynamic meandering in the laboratory, while braided channels are relatively easily reproduced in laboratory experiments (Ashmore, 1991). This either reflects a lack of understanding of the basic conditions in which a self-formed dynamic meandering river emerges, or reflects serious scale problems that, once understood, could lead to better understanding of meandering. Only three sets of experiments have reproduced aspects of dynamic meandering rivers with floodplains. Friedkin (1945), Schumm and Khan (1972), Jin and Schumm (1987) and Smith et al. (1998) produced meandering channels in cohesive sediment. The major advance of these experiments was that meandering was produced at all and the strength of the banks was found to be a crucial factor. In Friedkin’s experiments the channels usually developed into braided channels


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Figure 6. (a) Relation between channel pattern, flow strength (unit streampower) and bank strength. (b) Relation between channel pattern, streampower based on valley slope and bankfull discharge, and channel bed sediment size. Source: (a) After Figure 6.4c in Ferguson (1987); (b) van den Berg (1995)

Peakall et al. (2007) produced one dynamical low-sinuosity meandering channel in sediment ranging from fine gravel to silt (silica flour), whereby the silt appeared to add strength to the banks. Many features of meandering rivers were observed in the experiment, such as point bar formation and chute cutoff. Discharge was kept constant. Gran and Paola (2001) and Tal and Paola (2007) seeded alfalfa to an initially braided experimental river in non-cohesive uniform sediment during low flow. A dense vegetation resulted in a static stream or slowly wandering rivers whereas less dense vegetation resulted in single-thread sinuous channels with some characteristics of meandering, such as point bar formation and chute cutoff (Figure 5; Tal and Paola, 2010). Meandering has also been obtained from an initially straight channel using alfalfa and a lightweight sediment that filled in lower areas of the floodplain so that recapture was prevented (Braudrick et al., 2009). In both cases three combined factors leading to meandering probably were the reduction of floodplain flow strength by the hydraulic resistance of vegetation and the concurrent increase of focus and strength of channel flow, the increased strength of eroding banks, and the filling of abandoned channels and lows by vegetation and lightweight floodplain sediment, so that multiple channels and reoccupation were prevented.

c Flow strength and bank strength. Ferguson through the process of chute cutoff; in Schumm’s experiments the channels remained straight with low-sinuous thalwegs or eventually removed the emplaced cohesive floodplain that was not selfformed; and in Smith’s experiments channels ultimately became (nearly) static as they had cohesive sediment in the bed as well as in the floodplain. However, natural meandering rivers are dynamically meandering by continuous creation, expansion and translation and cutoff of meander bends (eg, Camporeale et al., 2005).

(1987) qualitatively clarified that the ratio of flow strength and bank strength determines channel pattern and how this would explain many observations (Figure 6). Nanson and Croke (1992) based their classification of channel pattern on the cohesion of floodplains in relation to flow strength. Indeed spectacular results were obtained in the experiments by adjusting the supply of silica flour or plants (Peakall et al., 2007; Tal and Paola, 2007). Thus, the key to producing self-formed dynamic meandering is a proper scaling of bank strength


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relative to flow strength. Conversely, the key to producing self-formed dynamic braiding is by having very weak banks. This explains the following observations qualitatively (Ferguson, 1987): (1)

(2)

(3)

(4)

sand-bed rivers braid at lower slopes than gravel-bed rivers of similar discharge because sand is more easily entrained; braiding requires a larger gradient than meandering, given a discharge, because braiding involves a greater amount and rate of channel modification and bank erosion; changing bank vegetation correlates to pattern changes along rivers and in time, because vegetation often increases bank strength; inactively meandering channels have such strong banks that they cannot be eroded; this would include a number of the earlier experiments (Smith et al., 1998) as well as many small rivers in glacial tills, meandering channels on intertidal mudflats and so on (Ferguson, 1987; Fagherazzi et al., 2004; Kleinhans et al., 2010a).

Quantitatively the two classical explanations had not been unified at the time of Ferguson’s review. This is still true because it has remained difficult to quantify bank strength and because a physics-based explanation for channel and bar pattern is lacking. In addition diagrams such as those in Figure 6 are empirical and have dimensional parameters on their axes. Further progress requires that such parameters are more rigorously based on physics and perhaps appropriately non-dimensionalized. But this is the key difficulty we face; for instance, it remains unclear how exactly something complicated as vertically varying bank strength could be expressed in a meaningful physical parameter. We will turn to a physics-based theory for bar pattern in the next section and discuss bank erosion later. 300

3 Role of channel bars in channel pattern Flow over non-cohesive sediment almost never creates a plane, a smooth surface or a straight plain-floored channel. On the contrary, patterns emerge at many scales. The reason is that sediment transport rate depends on flow shear stress to a power higher than unity. A slight local irregularity on the bed surface causes flow deceleration and local curvature, which then leads to a relatively large local gradient in sediment transport that may grow into bedforms, bars, channels, sand waves and so on. This tendency is predicted even when flow and sediment transport equations are dramatically simplified and linearized. Linear stability analysis explores how this fundamental instability mechanism causes infinitesimal perturbations to grow to regular patterns (eg, Federici and Seminara, 2003).

a Bar theory. For clarity, we start with cases in a straight channel. The length and celerity of bars can be predicted from stability analyses to depend most of all on the width/depth ratio of the channel and to a lesser extent on friction and on sediment mobility (Parker, 1976; Struiksma et al., 1985). For very narrow and deep channels, a perturbation would result in alternating bars but their amplitude decreases, so that a plane bed develops. For somewhat wider and shallower channels, alternating bars grow. At some point, higher mode bars appear – that is, mid-channel bars emerge in between the alternating bars (Crosato and Mosselman, 2009). This can be called braiding, but it is important to realize that these bar patterns already appear in a constant flow discharge so that the bars would be submerged at all times, however far they grow up to the water surface, so that the river appears straight rather than braiding. More importantly, the width/depth ratio is used as independent parameter in these analyses and cannot be predicted.


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Conceptually, bars can be divided into free and forced bars. Free bars grow from perturbations. If they migrate, then their migration rate depends on their size. Forced bars grow from a local channel constriction or a change in curvature of the channel. Free and forced bars are not entirely mutually exclusive; mixed forms exist in low-sinuosity channels (Seminara and Tubino, 1989). But the more prominent the forced bars, the less likely free bars are to exist. Forced bars occur in most real rivers because they have curved banklines (Crosato and Mosselman, 2009). Very large free bars may move so slowly that they effectively form the forcing for smaller-scale bars.

b Alternating bars and incipient meandering. In meander bends, bars are forced to their positions by the bends. It is therefore tempting to infer that alternating bars lead to alternating bank erosion so that meandering rivers emerge. Many meander simulation models were set up from bar theories (Camporeale et al., 2007) with the condition that bank erosion was matched on the other bank by an equal amount of sedimentation so that the channel maintains a constant width. As such, meander simulation models reproduce many properties of meandering rivers (Camporeale et al., 2005; Crosato, 2007). But a physics-based explanation that independently predicts channel pattern may not be that straightforward: alternating bars in an initially straight channel migrate so fast that over time the banks will be eroded everywhere (Seminara and Tubino, 1989). The channel then widens so that the pattern evolves towards braiding. An obvious solution for the high bar celerity is that vegetation or cohesive sediment on the bars and banks would retard bar migration rate and at the same time reduce bank erosion so that meandering rivers might form. Alternatively, alternating bars in meandering gravel or cobble bed rivers may be covered by very coarse sediment at their surface – that is, armoured – to such

extent that their celerity is reduced because only extreme floods would mobilize such armour layers (van den Berg, personal communication, 2009). A third possibility is that large alternating bars are no longer forced to migrate, because the upstream sediment feed is no longer fluctuating strongly over the width (as is the case in braided rivers) but rather follows the alternating pattern (Crosato and Mosselman, personal communication, 2009). Either way, bar theory provides one element that was missing from the classical explanations of channel patterns: it predicts whether alternating bars focus bank erosion or whether braid bars develop that shave the banks more uniformly. With weak banks natural and experimental channels commonly evolve into wide and shallow rivers with irregular bars that form the braiding pattern (Parker, 1979; Xu, 2002). Rivers with strong banks and bar surfaces become narrow and deep (Hey and Thorne, 1986; Soar and Thorne, 2001; Xu, 2002; Parker et al., 2007), and as a result have alternating bars (Struiksma et al., 1985; Camporeale et al., 2007). Stronger flow in pools between alternating bars cause only localized bank undercutting (Johannesson and Parker, 1989; Camporeale et al., 2005; Crosato, 2007) and mass wasting during floods (Osman and Thorne, 1988; Thorne and Osman, 1988; Darby et al., 2000; 2007) which may lead to meander bend growth, migration and cutoff. In short, the balance between floodplain formation and bank erosion determines channel width, and channel width determines bar pattern, which in turn determines where the banks are eroded, while floodplain formation or armouring on the bars as well as resistive floodplain material in the banks reduces bar migration. Depending on these processes and feedbacks, different channel patterns emerge. Before we turn to floodplain formation and destruction, first the effect of bed sediment sorting on the bar dynamics must be discussed.


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4 Effect of bed sediment sorting on morphology Sorting of bed sediment affects the morphology of rivers and channel patterns in a variety of ways (see Powell, 1998, for a review) at different scales, in particular at the length scale of bars and beyond the length scale of channel patterns. The sorting itself is a result of mobility differences related to flow-particle interaction (discussed below). Thanks to the usually high efficiency of sediment sorting processes, much of past fluvial morphology and engineering could progress by considering only uniform channel or floodplain sediment, as attested by a large body of literature on sediment transport predictors and morphodynamic modelling for uniform sediment. Nevertheless the importance of sorting has long been recognized and has been unravelled in fluviosedimentological geological studies. Recent progress by the powerful combination of geological and process field studies, experiments and numerical modelling has made clear that further progress requires consideration of the entire particle size distribution and the sorting processes (for reviews, see, for example, Powell, 1998; Bridge, 2003; Parker, 2004; Blom, 2008; Frings, 2008). Sorting has direct relevance for the morphology at the bar scale. Two categories of sorting mechanisms can be distinguished: particleparticle interactions and flow-particle interactions. These two categories will be reviewed and their effect on bar dynamics will be discussed.

a ‘Shaken and stirred’: Particle-scale sorting and large-scale effects. Particle-particle interactions comprise granular effects that would also occur in still water or vacuum. When a sediment mixture is in motion due to stirring, flow shear, or gravity-driven sediment flow, the sediment expands and the pore space increases because of the extra space taken by the individual and colliding particles. The fine sediment is able to move into the pores between the large 302

particles under the influence of gravity, meanwhile working up the coarse sediment because that cannot move into the pores. This promotes a coarsening upward sorting. A related phenomenon is percolation, which occurs when gravel particles rest on each other and have empty pore spaces, so the fine sediment may fall through the pores even if the gravel is immobile (Kleinhans, 2004; Gibson et al., 2009). In any sediment mixture, a range of larger sizes contribute to the bed structure, whereas a range of smaller sizes partially fill the pores but could be removed without a drop in bed level. For a bimodal sediment with much gravel, the gravel is bed structure sediment whereas the sand is pore-filling sediment. When the relative amount of sand increases, the pores become filled entirely so that the sand contributes to the bed structure. Adapting a model from chemical engineering, Frings et al. (2008) were able to predict the porosity of an arbitrary noncohesive sediment mixture from the particle size distribution. Moreover, a cutoff size is predicted at which the abundance of a certain size is such that this size no longer contributes to the structure of the sediment. That is, sediment finer than the cutoff size only partially fills the pores of the coarser sediment. When more is added up to the point where this fine size starts to participate in the structure, the cutoff size becomes smaller. This behaviour may very well explain the difference in incipient motion dynamics between sand and gravel in mixtures. Wilcock and Crowe (2003), for instance, arbitrarily attribute a cutoff size of 2 mm to the two behaviours, but application of the Frings et al. (2008) model by Vollmer and Kleinhans (2008) showed that a cutoff size of 1.4 was more likely. Also rapid changes from gravel-bed to sand-bed rivers can be explained by the filling of gravel pores by sand (Frings et al., 2008; Frings, 2008). In the past the pore-filling sand in gravel-bed rivers was ignored as ‘wash load’, and silt and clay in sand-bed rivers was likewise ignored as ‘wash load’. Recently the sand received much


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more attention because of its relevance for salmon spawning and invertebrates in view of fine sediment release from dam lakes and in case of dam removal. In highly bimodal sediments the sand percolates a certain depth into the gravel bed depending on the coarse tail of the sand distribution and the fine tail of the gravel distribution (Cui et al., 2008; Frings et al., 2008; Wooster et al., 2008). There are large-scale effects of this sorting behaviour on incipient motion, bar dynamics and, most importantly, downstream fining.

b Incipient motion. Incipient motion and the transport rates of particles of different sizes have been studied for half a century (see Egiazaroff, 1965; Parker et al., 1982; Wilcock, 1998; Kleinhans and van Rijn, 2002; Wilcock and Crowe, 2003; Vollmer and Kleinhans, 2008). The differences in mobility of the particle sizes in a mixture have been described empirically in hiding functions, where ‘hiding’ refers to smaller particles that are hidden in the lee of larger particles. There are, unfortunately, about as many hiding functions as there are experiments or field data sets. The entrainment of the size fractions depends on the drag and lift forces on the particles. Calculation of these forces must include effects of flow turbulence, pressure fluctuations into the bed, bed slope and the exposure of particles above the average bed. Models based on various combinations of these factors have been presented (eg, Bridge, 1981; Wiberg and Smith, 1987; Zanke, 2003; Vollmer and Kleinhans, 2007; 2008), but a large number of issues remain to be resolved. In particular, the structure of turbulent flow over rough beds requires a more sophisticated formulation than the law of the wall (Nikora et al., 2001). Furthermore, the configuration of particles on the bed (Buffington et al., 1992; Aberle and Nikora, 2006) require attention. The incipient motion models may now contain more physics of the flow, but merely displace the empiricism to the particle-particle interactions

and water-working history that lead to the particle configuration. The mobilization of sand from within a gravel layer and the relative mobility of sand and gravel in any mixture has been studied in isolation from the infiltration problem. The connection between the two issues lies in the exposure of particles to the flow, where ‘flow’ includes turbulent pressure fluctuations in the bed that may mobilize sediment that is not exposed to direct flow. Infiltrated sand is negatively exposed but may nevertheless be entrained in some conditions (Vollmer and Kleinhans, 2008). Thus, incipient motion models based on force balances and near-bed and in-bed flow parameterization promise to provide physics-based explanations and models for the hitherto empirical hiding functions. However, for further progress the particle configuration of water-worked beds must be described first – perhaps partly based on the porosity model of Frings et al. (2008) combined with the percolation model of Cui et al. (2008).

c Sorting versus morphodynamics of bars. Local sorting on the bed affects the dynamics of bars because the sorting changes the mobility of the sediment. In particular, the bed surface may react in two ways to a gradient in flow shear stress and sediment transport: by changing bed elevation and by changing bed surface particle size distribution (Hirano, 1971; Parker and Klingeman, 1982; Dietrich and Whiting, 1989; Mosselman et al., 1999). In morphodynamic models, sorting has been modelled with the active layer concept (Hirano, 1971; Parker, 2004; Blom, 2008), but the balance between changing bed elevation and changing bed surface composition is extremely sensitive to the thickness of the active layer. Hence, the effects of armour layers (Parker and Klingeman, 1982) and dunes (Kleinhans, 2001; Blom, 2008) on large-scale morphology is significant but as yet not well enough understood. A promising model concept is to replace the active layer with a continuous (but numerically discretized) distribution of bed elevation


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and a vertical sorting function (Blom and Kleinhans, 2008; Blom, 2008). Vertical growth of bars is not only limited by water depth but also by the increase of the effect of gravity on particles on increasing transverse slopes. The first effect is that particles on a transverse or longitudinal slope are more easily mobilized (Parker et al., 2003; Vollmer and Kleinhans, 2007). The second effect is that gravity pulls the moving particles down the slope while the flow drags the particles along a slope. In bends, near-bed inward directed flow due to the helicoidal motion of flow in bends counteracts this effect. The transverse slope effect can be used for the prediction of channel bed morphology (Odgaard, 1981; Struiksma et al., 1985; Talmon et al., 1995; Parker et al., 2003), but the different formulations appear to have a considerable effect on the morphology, including the bar properties and channel depths in braided rivers and estuaries. It is by no means resolved how transverse slope models can be extended to sediment mixtures. The helicoidal flow in bends drags particles upslope towards the inner bend while gravity pulls particles downslope towards the outer bend. As drag depends on the surface area of particles while gravity depends on the volume, the balance turns out to give the classical bend sorting of coarser sediment in deeper outer bends and finer sediment towards the shallower inner bends (Parker and Andrews, 1985; Bridge, 2003). But transverse slope models are, in particular, poorly developed for cases where dunes are present (Talmon et al., 1995), although for plane bed in bedload there is progress (Parker et al., 2003). The actual bed slopes on the dunes differ much from the spatial average slope through the bend, and dunes sort sediment vertically as well and modify near-bed flow. Ideally, the Blom (2008) model must be incorporated somehow in a 2D or 3D model in combination with a submodel for fractional sediment transport on transverse slopes. Furthermore, these models must be integrated with the incipient motion and particle 304

configuration models discussed above. This is necessary to predict bar pattern, dimensions and dynamics for the entire range of sand-bed to gravel-bed rivers and single-thread to multimode bar rivers.

d Downstream fining as a template for channel pattern. Many rivers show a downstream fining pattern from gravel to sand. Coarse gravel-bed rivers and fine and medium sand-bed rivers are abundant, but rivers with intermediate sizes such as pea gravel are rare (Parker, 1991; van den Berg, 1995). Downstream fining in gravelbed rivers is caused by selective transport processes as well as abrasion of sediment, as is well known (eg, Parker, 1991; Hoey and Ferguson, 1994; Paola and Seal, 1995). Downstream fining in sand-bed rivers is much less well understood (Frings, 2008), but may be related to a reduction in suspended bed sediment transport capacity caused by concavity of the long profile of rivers (Wright and Parker, 2005). Mixtures of sand and gravel as well as intermediate sizes might be abundant locally where the river transforms from gravel-bed to sand-bed. The gravel-sand transition may be rapid or gradual depending on lithology, sediment mobility and particle size distributions (Sambrook Smith and Ferguson, 1996; Cui and Parker, 1998; Frings et al., 2008; Frings, 2008), but it is by no means clear how and why exactly. Furthermore, the downstream fining trend can be broken at points where tributaries contribute different sediments (Rice, 1999). One ramification of particle-scale sorting (and the resulting division between bed structure particles and pore-filling particles) for downstream fining is that the transition from gravelbed river to sand-bed river depends strongly on the sediment composition: bimodal sediment may have a sudden transition whereas unimodal sediment may have a more gradual transition (Frings et al., 2008; Frings, 2008). Indeed Iseya and Ikeda (1987) found experimentally that the transition is sudden for bimodal sediment, which


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also causes a sudden change in water surface slope, mobility of the sediment and sediment transport. More generally, although the porefilling sediment does not contribute to bed level change, it does contribute to the formation of floodplains and the downstream fining pattern that affects the morphology along the river. This means that all particle size fractions of the upstream sediment feed should be considered in the mass balance of one-dimensional morphological models with sediment mixtures even if they do not contribute to morphological change locally. One way of adapting the current models is by allowing bed sediment porosity to vary (Frings et al., 2008).

Having elaborated on the roles of bars and sorting, the key element that is still missing from a comprehensive theory for channel pattern is how floodplains form and how they are destroyed when the channel banks erode. Clearly, there is a strong link with another classical problem of fluvial geomorphology: that of hydraulic geometry, where it has been found that coefficients determining channel dimensions were correlated to the nature and strength of the channel banks. I first turn to hydraulic geometry to see what can be learned from it before the review continues on bank erosion.

coefficients. Furthermore, some relations include a particle size parameter and some include the mud fraction of the banks (Ferguson, 1987; Soar and Thorne, 2001) which is empirically better for some data sets but has no more universal validity. Thus, it has empirically been shown that hydraulic geometry depends partly on bank strength, caused by cohesive sediment or vegetation (Leopold and Maddock, 1953; Parker, 1979; Hey and Thorne, 1986; Soar and Thorne, 2001; Xu, 2002; Church, 2006; Eaton and Church, 2007; Parker et al., 2007). Many workers attempted to provide physical justification for hydraulic geometry relations by extremal hypotheses, assuming that channel flow friction is minimized, sediment transport in the channel is maximized, etc (for an overview, see Knighton, 1998). The key argument was based on the Least Action Principle and has been extended to channel pattern explanation (Huang and Nanson, 2007; Nanson and Huang, 2008). Even if the least action principle is valid for water and sediment, which is heavily debated, then it remains to be seen whether vegetation adheres to the principle. Just as a frivolous note, there is a Harvard law of animal experimentation which says that under highly controlled laboratory conditions laboratory animals do just as they please. This may well be true for laboratory plants as well.

a Channel-forming discharge. Most hydraulic

b Channel-forming Shields mobility number. As

geometry relations predict channel width and depth from discharge, from which a third relation obviously follows for the flow velocity (eg, Lane, 1935; Leopold and Maddock, 1953; Hey and Thorne, 1986; Knighton, 1998). The number of empirical hydraulic geometry relations is about as large as the number of empirical sediment transport predictors and is likely even more uncertain in its predictions. Coefficients were derived by semi-analytical derivations and by fitting to data sets. Implicitly a friction relation is then also present in the

an alternative to the hydraulic geometry relations based on a formative discharge, Parker et al. derive an elegant non-dimensional set of hydraulic geometry relations for bankfull single-thread gravel-bed rivers without bank vegetation (Parker et al., 2007) and for sandbed rivers with floodplains with cohesive sediment and/or vegetation (Parker et al., 2008). The basic idea evolved from the concept of a threshold channel: when the banks of a river are composed of the same non-cohesive sediment as the bed, the banks will be eroded by flow until

5 Bank erosion and channel geometry


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the river is so wide and shallow that both banks and bed are barely mobile (Parker, 1978a; 1978b; 1979). The parameters discharge, width and depth are made dimensionless with median particle size of the bed surface and gravitational acceleration, perhaps inspired by the Shields and Einstein non-dimensionalizations in sediment transport relations. The coefficients were derived from the Manning-Strickler flow resistance equation, a nearly constant critical Shields parameter, a sediment transport relation and a channel-forming relation expressed in terms of bankfull Shields number to critical Shields number. As these relations are not exactly universal, the resultant coefficients in their hydraulic geometry relations are also approximations. Hence Parker et al. (2007) present their relations as ‘broad brush’, although it is fair to say that they are much less broad brush than the purely empirical hydraulic geometry relations of the past. The use of a channel-forming Shields number (Paola et al., 1992; Parker et al., 1998) differs fundamentally from the use of a channel-forming discharge as in most other relations. Empirically, sand-bed rivers have a bankfull Shields number well above the threshold for suspension, and gravel-bed rivers have a bankfull Shields number slightly above the critical Shields number. The simple fact that gravel and sand are both abundant but intermediate particle sizes are not (Smith, 1996; Frings, 2008) permits the use of a constant channel-forming Shields number for sand-bed rivers and a constant for gravel-bed rivers. These are then not valid for pea-gravel rivers and rivers with sand-gravel mixtures. To generalize the relations to both sand- and gravel-bed rivers, the channel-forming Shields number can be expressed as a bank strength surrogate parameter times the critical Shields number. This means that the bank strength of all sand-bed rivers is lumped into one single value, and another value for gravel-bed rivers. Paola et al. (1992) used this for the derivation of a diffusive sediment transport law applicable to gravel- or sand-bed rivers for the purpose of 306

long-term and large-scale sedimentary basin modelling. For such purposes small-scale details such as channel pattern are less relevant. Parker et al. (1998) explicitly mention that factors like cohesive banks and vegetation cause uncertainty and variation in the empirical coefficients of their relations. For our purpose, unfortunately, bar pattern is very sensitive to width/depth ratio so that these relations will not be very useful for explaining channel patterns. We must therefore dive deeper into the detailed processes of bank erosion and the floodplain formation that precedes it.

6 Floodplain formation and destruction The term ‘floodplain formation’ covers a suite of very different processes that lead to the sortingout of fine and coarse sediment in different locations. Hence various combinations and intensities of these processes lead to very different floodplains. The detailed structure of a cut-bank in a floodplain, in turn, determines the erodibility of the bank.

a Floodplain formation by overbank sedimentation and vegetation. Fine sediment is deposited on growing inner-bend bars, which transform into levees (Brierley et al., 1997). Natural levees grow during floods by sediment diffusion from channel onto banks (Pizzuto, 1995), sediment advection in focused overbank flow (Middelkoop and Asselman, 1998; Nicholas and Walling, 1998) and crevasse splays growing from levee breaches (Walling and He, 1997; Cazanacli and Smith, 1998). Pioneer vegetation settles on the higher and better drained grounds (or perhaps on the wetter near-channel grounds in semi-arid regions) and successes into riparian forest (Johnson, 1994; Tal and Paola, 2007; 2010; Perucca et al., 2007), causing hydraulic resistance, sediment trapping and added strength. The vegetation on inner-bend bars promotes the transition from bar to levee by reducing flow velocity, adding strength and trapping sediment so that the


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channel maintains its width (Johnson, 1994; Darby, 1999; Baptist et al., 2006; Geerling et al., 2006). Furthermore it transforms bars into islands in a multi-thread channel (Gurnell and Petts, 2002; Tal and Paola, 2007). Levees in turn control natural flooding frequency and overbank sedimentation (Wolman and Leopold, 1957; Brierley et al., 1997). Further away from the channel fines settle in the floodplain to form cohesive sediment (Middelkoop and Asselman, 1998; Nicholas and Walling, 1998). Sediment compaction and vegetation succession (Geerling et al., 2006) strengthen floodplain deposits over time, providing bank strength when eroded in a meander bend. Thus, discharge regime and flooding frequency of a river determine the deposition of levees, of cohesive fines and the settling of vegetation, which in turn determine width and depth of self-formed channels, while the flooding frequency is determined by the dimensions of the self-formed channels (Wolman and Leopold, 1957).

b Floodplain destruction by bank erosion. Floodplain is simultaneously formed and destroyed. Floodplain destruction occurs not only by bank erosion but also by channel cutting. Both are strongly affected by the composition of the sediment, which may have layers of different composition and strength, and by vegetation including peat. Channel cutting will be discussed later in the context of avulsion. Bank erosion occurs in two steps: bank undercutting by fluvial erosion and bank failure by mass wasting. First, banks are undercut by fluvial erosion at the base and lower portion of the banks (Thorne and Tovey, 1981; Osman and Thorne, 1988; Thorne and Osman, 1988; Darby et al., 2000; Simon et al., 2000; Simon and Collinson, 2002; Darby et al., 2007). This depends on the flow shear stress and the strength of the sediment at the base of the banks. If this is cohesionless sediment similar to the bed sediment of the river,

then the angle of repose and lateral sediment diffusion are important (Parker, 1978a; 1978b). Although shear stress can be argued to increase with depth, the flow pattern in bends, particularly sharp bends, has not been clarified to such extent that it is clear how exactly banks are eroded. Channel bends lead to secondary flow patterns as has been known for a long time (van Bendegom, 1947; Allen, 1978). The basic reason is that flow is faster near the water surface, so that conserved momentum of a flow through a bend leads to a developing helicoidal motion of which the magnitude depends on water depth, bend radius and friction (de Vriend, 1977; Blanckaert and de Vriend, 2003). In sharper river bends, however, a smaller counterrotating cell develops near the outer bank water surface, which leads to a reduction of flow strength and turbulence at the bank and possibly to a reduction in bank erosion (Thorne et al., 1985; Blanckaert and Graf, 2001). In very sharp bends the flow separates from the inner-bend channel boundary and impinges directly on the bank on the opposite side of the channel (Leeder and Bridges, 1975; Ferguson et al., 2003), which leads to a very different bank erosion and bar formation pattern. This style is likely associated with channels with relatively very strong banks and limited to no dynamical meandering (Ferguson, 1987; Kleinhans et al., 2009). For simplicity, we will further assume that a good model would provide sufficient detail and realism in the flow for modelling the bank erosion but this clearly deserves more work. This detail would be needed for the coupling of bank erosion to the flow in realistic cases (Mosselman, 1998; Duan and Julien, 2005; Darby et al., 2007). The second step in bank erosion is the mass failure of the bank (Thorne and Tovey, 1981; Osman and Thorne, 1988; Thorne and Osman, 1988; Darby et al., 2000; 2007; Simon et al., 2000; Simon and Collinson, 2002). This process is more akin to mass wasting in mountainous areas than to other fluvial processes. There are various failure mechanisms that occur


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depending on bank height, bank oversteepening due to the fluvial undercutting, composition and layering of the bank material, presence, nature and age of vegetation (Pollen, 2007), and history of water levels and groundwater level. Once a failed block deposits on the river bed, it modifies the morphology and the flow locally. Eventually, it will be eroded by the flow. The composition and possible vegetation in it determine how long the block remains in place and to what extent it protects the bed (Fagherazzi et al., 2004). The influx of material different from the bed sediment means that various particle size fractions and their properties must be included in the prediction of morphology. Channel pattern stability is intuitively associated to a balance between sediment import and export from a reach. However, the volume of floodplain sediment eroded from banks is likely larger than the deposited volume of this sediment on point bars, because the flow energy on the point bars is higher than on the floodplain and because their proportion of length along the channel is smaller than that of the eroded outer banks (Wolman and Leopold, 1957). This means that, for a system to be really balanced, the excess eroded floodplain sediment must be deposited in oxbow lakes and on the overbanks (Lauer and Parker, 2008). The layering of the bank material is basically a ‘memory’ of floodplain processes that affects the channel dynamics. Furthermore the flood history determines groundwater level and outflow, which is well known to be very important for bank stability. The fact that advanced morphodynamics models have barely been coupled to advanced bank erosion models may therefore well be explained by full awareness of the apparent arbitrariness and spatial variation in bank layering as well as the intricacies of coupling of groundwater flow and open channel flow. But to progress with riverlands we must understand and model them, so we now turn to future modelling of this complicated set of processes. 308

7 Physics-based modelling of self-formed channels and channel patterns Rivers self-organize their pattern/planform through feedbacks between channels, floodplain and vegetation, which emerge as a result of the basic spatial sorting process of wash load sediment and bed sediment that leads to channels and floodplains. The balance between floodplain formation and destruction determines the width and pattern of channels. The role of the composition of banks was already acknowledged in classical explanations for channel geometry and channel patterns. Past work was based on bankfull flow in channels, but floodplains are not formed during bankfull flow; floods are essential. This shift in emphasis from channel to floodplain dynamics is significant. It leads to a system of processes and feedbacks that is so complicated that it can no longer be comprehended by the unaided human mind. The practical consequence is that a comprehensive computer model and an experimental scaling strategy are urgently wanted. The time is ripe: the variables governing floodplain dynamics that were once hidden in empirical coefficients can now be addressed in advanced models for three-dimensional flow and bar dynamics, bank erosion and floodplain sedimentation and vegetation. These are all relatively well understood on their own, and comprehensive models could be used to explore the feedbacks between components of the selforganizing riverland. Then it remains to be explained how this suite of processes and their interactions lead to the emergent empirical property that channel dimensions and pattern are well correlated to bankfull flow.

a Meander models or braided models. Since the advent of the computer, simulation models have been constructed for meandering rivers and for braided rivers. Meander simulation models based on advanced bar theories strongly simplify the bank erosion and levee and


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Figure 7. Braided river emerging in a morphodynamic model calculation (using Delft3D; Lesser et al., 2004; Kleinhans et al., 2008) after 30, 50, 70, 90 and 110 years, loosely based on the river Rhine. The initial quasi-regular mode 4 bar pattern (four bars across the width) is also predicted by linear stability analysis (Crosato and Mosselman, 2009). The large alternating bars nearly stabilize after several decades despite the lack of vegetation or floodplain sediment. Discharge 2500 m3s 1, gradient 110 4, particle size 2 mm. Flow is from bottom to top. Channel width is 2000 m and length is 40 km. White-to-black scale indicates sedimentationerosion relative to original bed (scale –5 to 1.5 m for left image and –10 to 3 m for later steps). Discharge was based on time series of the Rhine with 5% random noise along the upstream boundary, and initial bed level was seeded with a few centimetres of random noise. More noise or larger fluctuations on the discharge would lead to more bar migration. Sediment transport was calculated with the Engelund-Hansen formulation.

floodplain formation in stark contrast to the modelled complexity for channel morphology (Ikeda et al., 1981; Johannesson and Parker, 1989; Camporeale et al., 2005; Crosato, 2007). Moreover, the width is specified as a constant which implies that bank and floodplain formation exactly mirror bank erosion. This ignores the feedbacks with antecedent deposits and vegetation that determine channel width, pattern, floodplain dynamics and, in short, the entire riverland self-organization. Braided river simulation models based on cellular automata (Murray and Paola, 1994; 2003) strongly simplify the channelization process. In many such models the channels are one grid cell wide. The models can easily be adapted to include vegetation and other aspects, but the behaviour remains the result of rules that are only distantly related to the laws of physics that govern flow and sediment transport. Although these models reproduced many characteristics of these rivers, braided river simulation models cannot produce meandering, and meandering simulation models cannot produce braiding. Hence transitions from one to another river pattern cannot be modelled, which reflects our lack of understanding of either pattern and illustrates that essential riverland elements are still missing from these relatively simple models.

b Future development of channel pattern models. Future models should be able to allow channels to form autogenically and form their own geometry as well as pattern. At present, two- or three-dimensional codes model channel processes well but only within fixed banks (Figure 7) and at local reach scales for short periods (eg, Lesser et al., 2004; Kleinhans et al., 2008). Nevertheless, with increasing computer power, such models can soon be applied to large braided rivers over periods of millennia. Furthermore, the predicted bathymetry, when stored in small increments, allows the creation


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Figure 8. Portion of a geological profile of the braided river modelled in Figure 7 (near top), automatically created by comparing bed levels in four timesteps. The dotted line indicates initial bed level and the bar tops are only just submerged. Darker colours are old, while lighter colours are young. The four lower panels show channel planform in four timesteps (covering 110 years; black is deep and light grey is shallow), while the black horizontal line in the panels indicates the position of the profile.

of geological profiles (Figure 8) that provide another way to combine models and field data. Flow has been modelled in this context at many levels of details, from gradually varied flow to Large Eddy Simulation. Much has been said about computational fluid dynamics 310

modelling elsewhere (Bates et al., 1996; Hardy et al., 2003; Keylock et al., 2005). The issue here is how much detail in the turbulence is necessary to obtain the flow structures that cause bank erosion and floodplain deposition at the scale of channel patterns. Secondary circulation is


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relatively easily resolved in quasi-3D codes (Lesser et al., 2004) and even parameterized in 2D codes (Struiksma et al., 1985; Blanckaert and de Vriend, 2003) but the extra near-bank cell as well as near-bank flow pattern is not (Blanckaert and Graf, 2001). Two-dimensional or quasi-three-dimensional flow models with k-e-type turbulence closures have the advantage of computational efficiency and have proven in the past capable of prediction bar and sorting patterns as well as compound channel flow with vegetated floodplains (Darby, 1999; Baptist et al., 2006). Large Eddy Simulation, on the other hand, can now resolve flow structures that simpler models cannot resolve (Hardy et al., 2003; Keylock et al., 2005), but implementing sediment transport and morphology in such models has barely begun (Ferguson et al., 2003; Keylock et al., 2005). It remains to be seen how much detail in the flow models is needed for floodplain dynamics. Bank erosion has been modelled well in a single cross-section (Simon and Thomas, 2002; Darby et al., 2007) but the antecedent river deposits that strongly determine bank erosion had to be specified exhaustively as initial and boundary conditions. Relatively simple bank erosion has been implemented in a 2D morphodynamic model surprisingly long ago (Mosselman, 1998; Darby et al., 2002) but demonstrated significant discrepancies between data and model. Similarly, Duan and Julien (2005) modelled incipient meandering in spectacular agreement with the Friedkin (1945) experiments, but evolution beyond incipient meandering is not yet possible because the floodplain formation was not included. The same was true of Friedkin’s experiments. Furthermore, a serious numerical problem arises (Mosselman, 1998; Darby et al., 2002; Duan and Julien, 2005): that of the representation of a curved channel with a cut bank on a grid. A regular rectangular grid produces many cut cells. A curvilinear grid may follow the bank lines much better and can be deformed as the banks

erode, but cannot become too curved as orthogonality is required. Moreover, it is hardly possible to model chute or neck cutoffs. An irregular unstructured grid may follow bank lines well and allow bend cutoffs, but still incremental bank erosion will result either in small cells near the bank or in numerical diffusion due to frequent regridding. A porosity treatment of partially filled grid cells may also allow the use of regular grids with irregular banklines without the need to remesh (Keylock et al., 2005; R. Hardy, personal communication). Levee and overbank formation were so far ignored but should be the result of sediment sorting over bars as well as in floodplains. This will require adaptation of the active layer concept so that it includes the routing of pore-filling particle size fractions including sand, silt and clay as discussed earlier. It also requires more work on sand-mud interaction. Vegetation development in relation to river morphodynamics is in its infancy, but a number of things are clear from the bank erosion studies reviewed above. Considering that a comprehensive model would run long enough for vegetation to develop, existing models of vegetation succession could be integrated (eg, Temmerman et al., 2003; Geerling et al., 2006) in combination with rules of the tolerance of vegetation to duration of flooding, as well as rules for peat growth, seed banks and perhaps even delivery of large woody debris (Gurnell and Gregory, 1995). Perhaps the hydraulic resistance of vegetation is sufficient to describe the interaction between vegetation and sedimentation, but this obviously needs to be explored.

c Increasing computational capacity: The sky is not the limit. Regardless of the immediate challenges for numerical models, application of a riverland model including bank erosion and floodplain formation over periods of millennia will require much more computational power than is typically available in the geomorphology community. Perhaps as a result of this


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limitation, there is an ongoing debate on the feasibility of the micro-reductionistic stance. The question is whether we need to incorporate as much physics and detail into models as possible to reproduce as many relevant aspects of reality as possible. Or perhaps it is feasible to simulate aspects of reality with rule-based, so-called ‘reduced complexity’ models, as long as addition of rules is not arbitrary (Murray and Paola, 2003). For example, can we simulate river channel patterns with such reduced complexity models or do we need to address the full set of processes and conservation laws deemed relevant? Reduced complexity models have the obvious advantage over physics-based flow and morphodynamics models that they are computationally much more efficient. The obvious disadvantage is that ad hoc rules and empirical constants replace well-established (conservation) laws of physics and well-established semi-empirical relations, so it can by no means be ascertained that reduced complexity models correctly reproduce the dynamics of natural systems for the correct reasons. It is therefore essential to ground models on the truth by evaluating model output quantitatively against real data, while allowing for errors in the latter. While the debate rages on, it is useful to consider one argument: computational efficiency. In the Earth sciences, models are run on single personal computers or perhaps clusters of tens of PCs. Now imagine that we could use the supercomputers with hundreds or thousands of parallel processors that are presently used and developed further for Global Circulation Models and models for galaxy formation and astronomical data processing. These disciplines have already paved the way for full complexity models in our discipline. The reduced complexity models would remain useful, if only for organizing thoughts, teaching and inspiration, but increased complexity models could – within years – provide us with increasingly powerful tools to test whether our hypotheses agree with the laws of physics, and to explore the parameter 312

space of riverlands that is only sparsely filled with real rivers on Earth, and perhaps on Mars and Titan (Irwin et al., 2008; Kleinhans, 2010).

d Grounding models on the truth. To verify model results, data on channel patterns is required. The recent explosion of data types such as lidar DTMs and remote sensing imagery will allow not only for qualitative comparison of river patterns and quantitative comparison of morphometrics, but also for new techniques such as pattern recognition. After all, patterns are emergent characteristics that are easily but subjectively recognized by the human mind, while physics-based numerical models that produce patterns, or pixel-based images of rivers, do not involve recognition of any pattern. Data are increasingly available at many spatial and temporal scales. To date, about 40 years’ worth of satellite imagery have been collected. Aerial photographs have been available for much longer, and in some cases historical maps have been available since about AD 1600. Also digital terrain models are often available from stereo photography, surveying and lidar. In principle this creates new opportunities for excellent specification of initial conditions and for testing models. But a point-by-point comparison of model results and real rivers is not sensible for several reasons. First, the details in the real system cannot have been covered by the model if the initial and boundary conditions have not been supplied in endless detail, perhaps much more than in the new techniques, as argued at the beginning of this paper. Second, important processes that slightly modify the morphology may not be included, such as size-sorting of the sediment. Third, the physics for, eg, bars may be about correct but the empirical constants in the model may not be exactly correct (in the transverse slope components, for instance). If we were to follow the point-by-point approach, a morphodynamics model would modify the initial topography of a braided river until it fitted the wavelengths and


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other properties emerging only from the specified physics and constants, and would then dramatically fail the test of comparison to a terrain model collected at a later date. But this is neither fair nor fruitful: the model may have done very well in terms of bar dimensions and dynamics but a slightly different wavelength would render the point-by-point comparison useless. Instead, general characteristics of rivers must be compared, such as dimensions and derivatives of bars, bends, channels, networks, etc. This has been moderately successful in the study of the largest rivers on Earth (Thorne et al., 1993; Klaassen et al., 1993; Latrubesse, 2008) and for experiments (Egozi and Ashmore, 2008), but also suggests that general statistics are not critical enough. Parameters such as braiding indices are usually binary (channel/no channel) and strongly depend on stage. We need quantifiers for subtle patterns to reveal structure objectively, which has been a major challenge in many natural sciences for decades. Two promising approaches with spectacular recent progress are pattern recognition in remote sensing images and network analysis. Objects can be recognized by spectral signature and heterogeneity of the spectral signature (Addink et al., 2007). Such objects can then be classified with shape parameters, such as compactness, roundness and convexity. This has already successfully been used to distinguish river channels, thaw lakes and oxbow lakes (van der Werff and van der Meer, 2008). Combinations of pixel-based and object-based classification result not only in present water-filled channels but also in former channels, oxbow lakes and fills with bare sediment as well as vegetation cover (Addink and Kleinhans, 2008). It is conceivable that these techniques yield semiautomated maps of various channel and floodplain elements, in which patterns and structure can quantitatively be explored. Furthermore, evolution over time can be explored through changes in subsequent images, particularly for

the largest rivers of the world as the image resolution of the earlier imagery is limited. The evolving channel network structure could be studied with techniques developed for the world wide web, the internet, road networks, neurological networks, food webs and social networks including that of scientists, expressed in their citing behaviour (for a review, see Strogatz, 2001). River networks may be similar to other networks, or completely different, which would tell us something as well. Complex networks may be scale-free, random, regular or something in between random and regular called small-world, which come with varying degrees of clustering and connectivity. From this perspective, a river channel network is nearly regular, nearly chain-like because a channel splits into two or at best three channels at a node (or bifurcation), and directional because the water obviously flows only in one direction (although it would be interesting to apply this to deltas with tides). Fluvial plains are characterized by an equal number of bifurcations and confluences, whereas deltas are characterized by more bifurcations than deltas. Temporally, channel networks evolve, so that the network may change, and links between nodes may have time-varying proportions of the total flow and sediment flux. Ideally, the usefulness of network characterization will be tested on rivers that changed their pattern over time. Given the limited number of good data sets available, there is a clear role for experiments.

8 Dynamics of riverlands Channel patterns may change. A certain channel pattern may be stable in some characteristics, but the pattern itself is usually dynamic, even if no net aggradation takes place. Channels migrate, meander or form new braid bars and channels, crevasse splays form, vegetation grows, and channels avulse over the fluvial plain so that it starts all over again while the old channel fills up and is buried. Avulsion is an autogenic


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process – that is, an intrinsic result of the dynamics of riverlands while the allogenic controls, or boundary conditions to use a nongeological term, are constant and tectonics is insignificant. To understand channel patterns and changes in patterns, the style of autogenic avulsion must be understood as well, so this will be the focus of the next section. Channel patterns may also change more dramatically. Braided rivers are known to have transitioned to meandering at timescales of centuries to decades. In geology the reasoning that braided rivers change to meandering at the transition from glacial to interglacial is still occasionally heard. Such changes may be allogenically forced, or, in non-geological terms, be forced by a change in boundary conditions that has nothing to do with the internal dynamics of riverlands. Two fundamental problems will be discussed in the latter part of this section. First, the reaction of a river to changes in forcing is strongly affected by the history of a river as stored in the sediments and surface morphology. Second, perhaps there exist thresholds in the river systems that, when crossed, lead to a sudden dramatic change in pattern, and, when crossed in the reverse direction, lead to hysteresis.

a Splitting rivers at their seams: Avulsion and bifurcations. Avulsion means the shift of the course of a river over a considerable length – say, several meander bend wavelengths. At bifurcations, water and sediment are divided over two downstream branches. An avulsion site is at least temporarily a bifurcation because the new channel develops while the old one is still active. The terms avulsion and bifurcation are derived from medicine, where they are used for blood vessel topology and change thereof. Avulsion and bifurcations are found on alluvial fans and coarse-grained fan deltas, fluvial plains and lowland deltas. Braided rivers are characterized by many bifurcations, bars and confluences within a single channel. Anastomosing rivers may have bifurcations that are 314

stable in a nearly symmetrical division of flow and sediment for a relatively longer time. Occasionally trifurcations, with three downstream branches, are found in nature, particularly in deltaic environments. Avulsion wreaked havoc for society (Parker, 1999; Slingerland and Smith, 2004; Kleinhans et al., 2010b) in the recent past. Furthermore the rate and style of avulsion determine fluvial architecture, in particular connectivity between sandy bodies, which is relevant for hydrocarbon exploration. It is therefore surprising that avulsion and bifurcations barely received attention until two decades ago. Avulsions can be divided into four classes (Slingerland and Smith, 2004): avulsion by annexation of a small active or abandoned channel (eg, Stouthamer and Berendsen, 2007); avulsion by incision and formation of a new channel (eg, Stouthamer, 2005); avulsion by progradation of splays or lacustrine deltas through which a dominant channel may eventually develop (eg, Smith et al., 1989); and avulsion following the formation of a mouth bar with an unstable bifurcation (eg, Edmonds and Slingerland, 2007). These four classes have distinct initial conditions, distinct processes and different relative control of upstream and downstream conditions. Several conditions are known to be necessary for avulsion (for reviews, see Slingerland and Smith, 2004; Stouthamer and Berendsen, 2007), in particular a gradient advantage of the new course over the old course, and slight underfeeding of the new channel with sediment. The gradient advantage should here refer to energy gradient, which is often not understood. Even if the bed gradient along a crevasse (across a channel belt) into a flood basin is much larger than the bed gradient along the channel, then the flow into the flood basin may still pond so that sediment deposition clogs up the crevasse channel and the avulsion fails (Aslan et al., 2005). Hence, long flood basins and breached flood basins would promote avulsion. Ponding is


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basically a backwater effect and therefore a downstream control, even though it may seem as if the sediment feed conditions at the crevasse entrance matter more (Slingerland and Smith, 1998). A slight underfeeding of the new course is necessary so that the new channel enlarges. This is an upstream control. Key factors are the size of the crevasse, the absence of resistive layers in the old channel bank and under the deepening new channel, the suspended sediment concentration of the flow entering the channel (Slingerland and Smith, 1998) and the presence of an upstream bend in which helicoidal flow modifies the direction of the sediment just upstream of the bifurcation (Kleinhans et al., 2008). The formation of a mouth bar and bifurcation is akin to that of splay formation. A body of sediment is formed where the flow expands (Edmonds and Slingerland, 2007), after which the body is in the way of the flow so that the flow decelerates through the backwater effect. If the flow decelerates on the upstream side of the mouth bar, then it grows in the upstream direction (Kriele et al., 1998; Hoyal and Sheets, 2009; van Dijk et al., 2009; Kleinhans et al., 2010b). Furthermore, the channel with the two bifurcates in it is lengthened compared to other main channels, so that a gradient disadvantage occurs. Continued long enough, the entire channel upstream of the mouth bar aggrades, reducing its flow capacity compared to other channels, and avulsion occurs on the delta. In all avulsion types, there is, averaged over a period longer than a few avulsions, net sedimentation. There is hardly opportunity for avulsion in incised rivers (Stouthamer and Berendsen, 2000). A gradual rise of flood levels above the surrounding floodplain is another necessary condition for avulsion, fulfilled by a slight overloading of the upstream feeder channel with sediment (upstream allogenic control; Bryant et al., 1995; Ashworth et al., 2004), or with general progradation of a delta or fluvial plain (downstream autogenic control; van Dijk et al.,

2009), or with base level rise (downstream allogenic control; Stouthamer and Berendsen, 2000), or with tectonics (allogenic control; Stouthamer and Berendsen, 2000), or with compaction of floodplain sediment and floodplainfilling peat (autogenic control; Aslan et al., 2005; Stouthamer and Berendsen, 2007). Morphodynamics models have reproduced various emergent phenomena, including upstream-driven avulsion on fans, mouth bar formation and upstream migrating sedimentation causing upstream avulsion (Edmonds and Slingerland, 2007; Kleinhans et al., 2010b). Potentially, a comprehensive morphodynamic model for channels and floodplains, as described above, could autogenically create avulsions. To include tectonics and compaction is relatively straightforward, although differential vertical movement of the bookkeeping grid for sediment composition may provide another numerical challenge.

b Avulsion, bifurcations and channel pattern. Bifurcations are rarely stable, so that the number of active channels does not increase ad infinitum because they silt up. The period over which a bifurcation becomes unbalanced – in other words, avulsion duration – depends on the length of the downstream bifurcates, the difference in gradient, and presence of upstream bars or bends (for reviews, see Slingerland and Smith, 2004; Kleinhans et al., 2008). It can be postulated that single-channel systems exist in principle when the average interavulsion period is larger than the avulsion duration, and alternatively multi-channel systems such as anabranching and anastomosing rivers exist. If activity of a channel is defined as having significant bed sediment transport (say, more than a fifth of the transport integrated over all channels), then many seemingly anastomosing rivers are in fact single-channel systems wherein the abandoned branches convey a limited portion of the flow discharge, in particular during floods (Makaske et al., 2002; Kleinhans et al., 2008).


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The pattern of wandering gravel-bed rivers, also confusingly called anabranching rivers, is thus related to bifurcation stability. New channels are sometimes cut into the floodplains at flood or into abandoned channels, when the flow has been diverted by very large floods, log jams, sediment waves or ice dams (Burge, 2006). The relation between avulsion parameters and pattern raises the question of whether large anabranching rivers, such as the Russian plain rivers (Alabyan and Chalov, 1998) and the GangesBrahmaputra system are simply single-channel systems with recently abandoned floodconveying branches or multi-channel systems in which the interavulsion period is larger than the avulsion duration. One argument against this is the existence of relatively stable, vegetated and inhabited islands (chars) in the Brahmaputra (Thorne et al., 1993). No megariver with a discharge larger than 17000 m3 s 1, except the Mississippi, satisfies the usual discriminators between braiding and meandering, possibly because there exists a stable state of anabranching (Latrubesse, 2008). Interestingly, some of these megarivers have straight, sinuous or braided channels within the larger-scale anabranching, which would argue for a separate explanation as suggested by Alabyan and Chalov (1998), rather than one at the same level as meandering and braiding. Latrubesse (2008) suggests that the Least Action Principle (Huang and Nanson, 2007) explains the emergence of islands and hence anabranching because narrowing channels would be the only way to transport all the sediment at such small energy gradients. As megarivers cannot well be scaled in experiments, anabranching must urgently be explored with physics-based models for an explanation.

c Changes in channel patterns: Thresholds and transitions. Channel patterns are known to have changed, sometimes called metamorphosed (Schumm, 1985). An increase of vegetation density on bars transformed a braided river into a transitional river (Schumm, 1985; Johnson, 316

1994; Tal and Paola, 2010). Changes in the hinterland as well as human interference such as gravel dredging led to incision of a braided river which then transformed to meandering (van den Berg, 1995). Climatic change, both in the hinterland and the fluvial plain, transformed a large braided river into a meandering and straight river in the transition from interglacial to glacial (Vandenberghe, 1995; Berendsen and Stouthamer, 2000). A simple correlation between cold climate and braided rivers or warm climate and meandering rivers has been used and debated in the past. The key explanation is discharge regime and coupled sediment feed to the river. In a cold climate there may be one large spring melt flood event causing a large discharge peak, whereas in a temperate climate there may be several relatively smaller rainfall-generated flood events, and in a tropical climate the monsoon provides a different flood regime again. The annual discharge may still be the same but the distribution over the year changes, with shorter and higher peaks in the cold climate case. Thus the effective channel-forming discharge in a cold climate may be larger. The sediment feed rate and vegetation respond to climate change as well, so that a suite of changes may cause a transition of river pattern (Vandenberghe, 1995; Busschers et al., 2007). It would be extremely interesting to study the effect of each individual factor in a physicsbased model capable of producing different river patterns. Careful geological reconstructions have led general hypotheses for the relation between channel pattern and climate to move away from the cold-braided and warm-meandering simplicity. In particular, the tardy reaction of vegetation, destruction of strong interglacial soils and permafrost to climate change causes out-ofphase changes in supply of water and sediment (Vandenberghe, 1995; Busschers et al., 2007). Furthermore, it takes time to convey coarse sediment, generated by strong physical weathering, to downstream fluvial plains so that its delivery


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may be delayed to the end of a glacial (Busschers et al., 2007). Before this arrival of different sediment the local sediment is therefore reworked. Thus, both irregularity of discharge and sediment supply are thought to cause channel pattern change. The period of instability could be short, so that the pattern is braided only some time after the onset of a glacial period (Bogaart and van Balen, 2000; Vandenberghe, 2003). In some climates, however, the adaptation period may be much larger (Church and Slaymaker, 1989). The transition from meandering to braiding is poorly recorded, whereas the braidingmeandering transition is well preserved. The preservation potential of the braided river deposits is much larger than that of the meandering deposits because braided rivers tend to erode widely, removing the former meandering river floodplain, while meandering rivers erode deeply (Vandenberghe, 2008), although sometimes a large-scale avulsion allows for some preservation of meandering channels (Busschers et al., 2007). As an example, the following interpretations were given for the transition from braiding to meandering of the river Rhine (Busschers et al., 2007). In the Last Glacial maximum (before 21,000 BP) the river was braiding and incising. This strongly supported observation may conflict with the idea that sediment overloading causes braiding, but part of the incision is due to isostatic effects so that the overloading with sediment basically occurs from below the entire river. While the warming initiates 18,000 BP, the transition from braided to meandering is first observed 14,500 BP. Both far downstream in the Rhine basin (Netherlands) and upstream (Upper Rhine Graben in Germany) the meandering initiates in the larger channel or channels of the braided river. Two or three channels were active over several millennia, although it is not known whether one or all of the channels had most of the discharge and sediment transport. So, initially the braided river temporarily forces characteristics onto the meandering river

(Busschers et al., 2007; Erkens et al., 2009), which is important to take into consideration when quantifying and explaining measures such as meander length over the transition. By 8000 BP there is only one active meandering channel. But what exactly caused these channel pattern changes? To answer this question, the channel patterns first require explanation in their own right as argued in most of this paper. Once such understanding has been gained, abductive inference could lead to answers that are better constrained. At present, a number of causes have been forwarded, including climatic change in the hinterland, deforestation and sea-level change. But climatic change and deforestation may mean a change in annual flow discharge, flow discharge regime, upstream sediment feed and the composition of the upstream sediment supply, all timed differently because of their different timelags, and all with delayed effects because the inherited channel pattern belonging to past conditions is only slowly modified. Once a comprehensive model and an experimental scaling strategy have been developed, these questions can be addressed systematically.

9 Towards understanding river channel and pattern formation The following questions emerge from this review.

What is the role of flow strength and variation in flooding magnitude and frequency on the pattern? What is the effect of nature, magnitude and variation of upstream sediment feed on patterns? Is added bank strength due to sorting of fines and bed sediment a sufficient condition for meandering, and is vegetation also a sufficient condition? What extra interactions and feedbacks on channel pattern would arise if both occur?


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What is the role of bar pattern and sorting at the bar scale? Is the emergence of fixed alternate (complex) gravel bars a sufficient condition for meandering in gravel-bed rivers? To what extent do small-scale sorting processes cause large-scale sorting patterns relevant for channel pattern? In what parameter space of vegetation, fine sediment deposition (density, size, thickness) and flooding (magnitude, frequency) do transitions between river patterns take place? In what manner, to what extent and why do channel patterns and their causes in sandbed and in gravel-bed rivers differ? In what manner, to what extent and why do channel patterns and their causes in small rivers (1 m width) differ from those in large rivers (10 km width)?

Fieldwork successfully helped raise these questions but will not alone answer them. A quantitative experimental scaling methodology must be developed and a comprehensive physics-based model must be developed in which the various river channel patterns emerge as a result of different initial and boundary conditions of water and sediment. The challenge for experimenters is clear: a systematic geotechnical study of scaling relations of bank strength must be done. The precise process that causes bank strength in silica flour or vegetation must also be studied in order to be able to develop new scaling rules. With such scaling rules, various combinations of vegetation (like vegetation in reality; Tal and Paola, 2007), silica flour (like suspended sediment; Peakall et al., 1996) and polymer (like cohesive floodplain sediment; Hoyal and Sheets, 2009) could be pursued to combine the best of all and create the richest experimental riverlands. The challenge for modellers is also clear: to combine physics-based models that produce accurate three-dimensional flow, bar patterns with (preferably) physics-based models for 318

floodplain sedimentation, including the effects of cohesive sediment and of vegetation, and a submodel for the erosion and failure of channel banks. In short, hydrodynamics, sediment transport of particle sizes ranging from cohesive fines to the coarsest bed sediment, effects of vegetation and morphodynamics must be fully coupled. The obvious way to do this is to implement a bank erosion model into a hydrodynamics and morphodynamics code as has already been done, but perhaps with more advanced 3D flow, with a solution for the deforming grid problem and capability to deal with fine sediment transport and settling and vegetation. The time is past that such a reductionistic modelling approach is unrealistic. In short, a greater engagement with fundamental mathematical and physical principles is both required (Church, 2005) and feasible. Once experimental and model tools have been developed, modelled evolution of fluvial landscapes at longer timescales can be compared to geological studies that reconstruct environmental and climate conditions and change. Then questions such as the following can fruitfully be asked.

What are necessary conditions and timescales for transitions between river patterns in response to changes in forcings? How do channel patterns develop when the initial condition is another channel pattern rather than a hypothetical plane valley? Are there hard thresholds (rather than transitions) that the system can cross during changing forcings and due to extreme events? Can such threshold crossings lead to hysteretic or irreversible change due to vegetation or inherited floodplain structure?

Significant progress can be made in the next decade by modelling and experimentation. Existing models can be combined for bed and floodplain sediment dynamics and morphodynamics, rules for vegetation and bank erosion into a comprehensive physics-based model.


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Experimental floodplain and channel processes can be enriched with essential characteristic by adding vegetation, fines and polymer to the sediment mixture. While following such a physical reductionistic approach, the empirical concepts, relations, diagrams and geological reconstructions developed in the past will provide necessary verification data, constraints, exceptional cases and realistic boundary conditions. Acknowledgements As for most reviews, the author is indebted to many colleagues in the Geoscience faculty and all over the world for exchanges and discussions. In particular, I thank Janrik van den Berg and Gary Parker for their inspiring guidance in the past that, to my initial surprise, echoed on while I move from sorting into channel patterns. I thank Christopher Keylock and Wietse van de Lageweg for comments on the draft; Piet Hoekstra for his ongoing support and comments on an earlier version of this review; and Gilles Erkens, Kim Cohen, George Postma and Wim Hoek for their attempts to hammer some geological sense into my thinking. I also thank the MSc students who did so much work to chase the fundamental questions raised here. Fred Trappenburg of Geomedia, Utrecht University, painted and virtually modelled two representative lions and seduced a third to have its tail twisted (gently) on camera; Kimberlee Kessler Design granted permission for the use of their real lion photograph. MGK is supported by the Netherlands Organisation for Scientific Research (NWO) (grant ALW-Vidi- 864.08.007).

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