What even is a meandering river? A philosophy-enhanced synthesis of multilevel causes and systemic interactions contributing to river meandering Open Access

Meandering rivers are an iconic geomorphic form, recognized since antiquity (Gürbüz and Kazancı 2019) and familiar to most students of geoscience. It is thus both surprising and significant that a comprehensive and unequivocal definition of river meandering is hard to provide. Ambiguity in the definition of what constitutes a meandering river has resulted in sometimes mutually exclusive explanations for the phenomenon, often arising because competing explanations have flexible demarcations of the term ‘river meandering’. The uncertainty concealed within the name ‘meandering’ stems from several angles. Firstly, many features that are part and parcel of the phenomenon (Fig. 1) are also defined as different patterns within other kinds of landscapes surrounding them. Secondly, there is a continuum of channel forms that are transitional between other planforms (e.g. Leopold and Wolman 1957; Parker 1976; Ferguson 1987; Kleinhans 2010), and the bracket in this continuum that may be considered true meandering rivers can be only inexactly delineated. Thirdly, the distinction from other kinds of channels with bends is often unclear, and the pattern bears similarity with other phenomena such as capillary flows (Davies and Tinker 1984), channels on glaciers (Marston 1983), channels on intertidal mud flats (Fagherazzi et al. 2004; Kleinhans et al. 2009), sinuous ridges on Mars (Williams et al. 2013; Cardenas et al. 2022), lunar rilles (Hurwitz et al. 2013) and channels on Titan (Gilliam and Lerman 2016), lava tubes (Orr et al. 2015), channels embedded in mud (Kleinhans et al. 2009; Candel et al. 2021), and both submarine (Corney et al. 2006; Deptuck et al. 2007) and bedrock channels (Seminara 2006). Fourthly, if we seek to answer the question ‘what are the causes of meandering?’, we clearly require that ‘meandering’ is understood to be a set of properties. Yet such properties themselves require clear definition: do they need to have a single common cause, or can they emerge from fundamentally different underlying physical mechanisms?

Five examples of approaches seeking causal explanations illustrate that meandering can be understood in terms of different properties, with causes studied by different methods and contrasting notions of causality:

1. Physics-based, analytical theories of periodic bars and bends predict wavelength from the morphodynamics in the channel.

2. Random-walk models predict sinuosity to depend on slope and floodplain resistance.

3. Data for meandering, straight and braided rivers show that transitions from one kind to another depend on bed sediment calibre, valley slope and mean annual flood discharge. Here, sinuosity above a certain threshold, braiding index and the presence of unvegetated bars are properties of meandering rivers.

4. Data for ancient braided and meandering river facies in sedimentary rocks show that meandering planforms were rare before the evolution of land plants and common thereafter.

5. Experiments with and without cohesive sediment result in meandering and braided rivers, respectively.

The interrogation of meandering as a concept is impeded by the multitude of approaches that have sought to explain this ill-defined phenomenon. The five simplified examples above illustrate this. A broad range of case studies suggest several causes and mechanisms for meandering, but the diverse scientific methods applied in these studies innately provide very different views of the phenomenon. Models and experiments have long appeared to produce meandering rivers despite their inherent simplifications (Friedkin 1945; Howard 1992). While they offers insights into meandering causes and mechanisms, these insights hinge on what is considered sufficient representation. For example, quantitative relations between flow discharge and meander length are similar for quasi-periodic sinuous and laterally dynamic channels across glaciers and tidal flats, but these are commonly not considered to be meandering rivers. Without clarity on causes, the notion of meandering can be imprecisely applied to a multitude of phenomena with distinct environmental and mechanistic properties. The significance of such properties can, in many instances, transcend simple form characteristics (e.g. length, amplitude and bend shape, migration dynamics and style of cutoff). The broadness of modelling and experimental approaches is a counterpoint to explanations for river meandering that are extracted from empirical geomorphological or geological observations of natural rivers and the alluvial strata that they leave behind. Such explanations suffer from their own lack of clarity, hampered by issues of underdetermination and a potentially misplaced deference to ‘most likely’ causes.

The philosophical vantage stressed in this contribution is frequently overlooked (Kleinhans 2021), but is essential to address because previous reviews suggest that what we think we know of meandering is strongly linked to partial observations that tend to be focused on particular aspects of the phenomenon, as well as to the specific applied methods that are used to extract this knowledge. In each instance, what is thought to be known has hinged on a researcher's particular perspective of concepts such as ‘fundamental cause’, ‘autogenic’ and ‘allogenic’. These loose concepts themselves warrant interrogation: if a cause is fundamental, does that mean that it is a necessary and sufficient condition for all forms and aspects of meandering? If only a fraction of total causes are fundamental, does this mean that there are also auxiliary causes, and if so what role do these play in meandering? Likewise, the notions of autogenic and allogenic imply that some forcings are an essential part of the meandering phenomenon whereas others are not but can be externally imposed. In all instances, there is an implicit assumption that ‘meandering’ is a clearly defined phenomenon and that it is possible for us to distinguish the essential characters that constitute meandering from those that are commonly associated but non-essential.

The contention of this paper is that a consensus understanding of river meandering requires appreciation of two issues, namely: (1) that a singular definition of river meandering is hard to pin down and almost always requires the inclusion of other properties or explication of implicit assumptions; and (2) that the scientific approaches used to interrogate the phenomenon have their own merits and limitations that must be understood before attempting to produce universal explanations or contest conclusions from alternative vantages. To aid future research into this field and uncover blind spots on the nature of meandering and the science of meandering, this contribution has two goals. Firstly, we aim to provide a review of sub-phenomena that may be considered components of river meandering, clarifying how interactions of different autogenic and allogenic processes affect meander kinematics and, by extension, channel morphology, sedimentology and internal stratigraphic architecture. Secondly, we aim to explore how scientists have interrogated the problem of meandering and to elucidate how multilevel fundamental causes of river meandering on Earth and Mars have been obtained from correlations and mechanisms. Here the conceptual apparatus of the philosophy of science will allow an independent perspective on the roles of theory and data, and on the multicausality of meandering.

The paper is organized into four sections. In the first one, we outline the inherent problems in searching for a causally effective description of river meandering. In the second and third sections, we offer two parallel reviews of meandering. The second section focuses on meandering from a channel perspective, where we emphasize how recent philosophical work on causality is useful for achieving a greater understanding of channel meandering (Illari and Russo 2014), clarify what is thought to be an inherent mechanism of meandering and what is external to it, and review how knowledge of meandering rivers was developed over the past decades amongst geomorphologists, physicists, statisticians, modellers, experimenters and geologists. In contrast, the third section discusses meandering rivers from a systems perspective, where floodplain causes of meandering are more important, which we demonstrate by unpacking notions such as ‘autogenic and allogenic forcing’, and ‘necessary and sufficient conditions’, emphasize the role of floodplains as inherent to the meandering phenomenon and discuss how meandering river systems manifest at several spatio-temporal scales. The fourth section summarizes the consensus so far on river meandering, as gleaned from a multitude of different scientific approaches and definitions of meandering.

A purely descriptive definition of meandering would need to avoid phenomena that are entwined in the cause–effect relations of meandering, such as bars and cutoffs, and instead must use terms from outside the geosciences such as sinuosity and shape (see Fig. 1). The problem with such a description is that it would inherently be less useful than a definition that collectively has the power to explain meandering, to distinguish it from other phenomena and to predict aspects of meandering in specific conditions and situations of interest. Such a causally effective description of meandering (and thus its demarcation from other river patterns) would provide understanding and possibilities to predict observable effects, and to manipulate rivers by engineering.

A causal description of meandering rivers (e.g. Fig. 1) could be as follows:

1. Meandering rivers have a sinuous, predominantly single-thread channel that is deeper at outer-bend banks. As a result, higher flow velocity occurs on the outside bank near the apex, driving outward and, often, downstream migration.

2. The channel has shallow bars at the inner-bend banks that are likely to become emergent in low-flow discharge conditions, meaning that they primarily develop at higher discharge.

3. The entire channel position shifts over time but generally maintains a constant width and a constant meander wavelength (periodicity) with limited variation.

4. The geomorphic expression of this movement is revealed because many meandering rivers have their inner bends flanked by sets of curved ridges, called scroll bars, that are former inner-bend bars.

5. The sedimentary-stratigraphic expression of this movement is revealed by the internal anatomy of former inner-bend bars, which comprise laterally inclined deposits that record previous iterations of the slope of the inner bend bars.

Empirical geomorphological evidence is also suggestive of recurring processes and phenomena that emerge in regions adjacent to a meandering channel. The presence of curved, unconnected lakes formed by cut-off and abandonment of bends is evidence for the continued growth and migration of bends and the dynamic nature of meandering, even as channel dimensions remain approximately constant. Such abandoned bends and scroll bar bundles transition into floodplains covered by mud and, often, riparian vegetation, which is visible where flood waters spill from the extant channel but themselves contain no clear channel (yet, or anymore). The floodplain is destroyed where outer banks retreat and where meander bends are cut off. When the width and general pattern of river meandering are maintained, it is evident that floodplain destruction and formation in a reach must be balanced (e.g. Lauer and Parker 2008). In instances where floodplain formation processes are outweighed by destruction through channel migration, braided rivers may form with relatively larger widths and lower depths and with multiple channels separated, during low flow, by bars (Ferguson 1987; Kleinhans 2010). From this perspective, the meandering pattern can be thought in some way to be caused by the set of processes that reforms the floodplain following destruction, and characteristics associated with those putative causes are included in some definitions of meandering.

Shape description, theory and observation from nature converge on a consensus that it is the combination of three groups of properties – sinuosity, pattern-preserving dynamism and the association with a floodplain – that matter most in isolating the concept of a meandering river. Yet this set is not without problems. For example, sinuosity without dynamics may not be considered meandering, but what characteristic timescale is considered dynamic? Sinuous rivers in bedrock show migration, cutoffs, etc. but at a much longer scale, but does this exclude them from the class of meandering rivers? Here, the characteristic timescale can be defined as the duration over which a bend migrates by a distance equal to the channel width, or as the average duration between bend cutoffs. For rivers with the same sediment mobility (Shields number), which is nearly independent of river size when expressed as width, depth or discharge (Candel et al. 2021), this timescale would have an inverse relation with river size. Such a reference to the dynamics is problematic because the ‘characteristic timescale’ of meander displacement varies greatly in time and space, and even between rivers of different sizes, all of which are poorly understood. The presence of a floodplain with cohesive material and/or vegetation may also be necessary as discussed later, or otherwise the channels are expected to migrate, cut off and shift so frequently that a braided river with multiple parallel channels and cohesionless bars would emerge.

Clearly, a meandering river is a perpetually reforming pattern, and as such it is more a process than a thing or entity (i.e. ‘river meandering’ rather than a ‘meandering river’) because it ceases to be meandering and is merely sinuous when the channel is, or has become, static in position. In other words, ‘meandering’ has geometric and kinematic aspects. Moreover, a large set of causes is involved, which leaves open the question of what the fundamental cause of river meandering is and what are auxiliary or even irrelevant aspects. This shows the need to separate the effects, namely aspects of the meandering patterns such as bars, from fundamental or auxiliary causes, and it shows the need to distinguish relevant aspects from independent effects of meandering, such as bedforms, that have the same causes. The relevant causes make a difference in the sense that the river pattern would be different, possibly not meandering, without them. The irrelevant aspects can be put in the background (such as constraints by surrounding landscapes that may change while the pattern is still discernable). Additionally, factors that can be believed to be constant can also be put in the background, such as the nature of the fluid, the presence of granular material and the gravitational acceleration at the planetary surface, unless other meandering landforms such as those of turbidites are studied, which operate under a wholly different range of force balances. From this perspective, ‘meandering’ also has dynamic aspects in the sense of sediment mobility (Shields number), relative velocity (Froude number) and other dimensionless ratios of relevant forces and strengths.

From the above it becomes clear that the description of the meandering pattern, and any published definitions of meandering rivers and river meandering that go beyond simplistic pattern descriptions such as ‘curved’, are all laden with theory, whether explicit or implicit. The theory-ladenness and the hidden assumptions of backgrounded causes are problematic when meandering is both argued to have a single cause, such as ‘fundamental instability’ or ‘vegetation’, and argued to be generic. Perhaps there is one cause that makes the difference between meandering rivers and all other river patterns, with other auxiliary causes explaining the variations within meandering patterns.

Problems arise when a meandering theory developed from modern Earth observations is applied in ancient or alien conditions where vegetation was different or absent, where gravity is different, as on planet Mars (Howard 2009), or where the fluid and sediment are composed of different materials, as on Titan (Burr et al. 2013). Finally, there is a worry that inferences for meandering patterns based on terrestrial, fluvial phenomena are potentially biased.

To be able to differentiate the various causes and mechanisms underlying river meandering, more precision is needed as to what causes and mechanisms actually are. Here, use is made of recent advances on causality in the philosophy of science (Hall 2004, see Illari and Russo 2014; Johnson et al. 2019 for an overview). These advances have focused on the practice of how scientists gain knowledge of causes and mechanisms (epistemology) rather than on theorizing about what the nature of causation is (ontology). For reasons of readability, theories of causality are presented here in simplified form, which can roughly be divided into two groups:

1. Theories in the first group, called ‘production’, are about how causes produce an effect and how mechanisms can be traced to produce an effect (Machamer et al. 2000). For example, flowing water produces instability forming bar patterns by inducing nonlinear sediment transport (a mechanism), orbital variations of the Earth produce climate change by causing regular insolation variations, and vegetation produces strengthened river banks by emplacing roots within the sediment pile.

2. Theories in the second group, called ‘difference making’ or ‘probabilistic’, are about causes that do make a difference, or raise the probability of an effect. Probabilistic causation can detect that certain causes raise the probability for the effect of interest to occur. For example, just like obesity raises the probability of mortality by Covid-19 (Popkin et al. 2020), a lower slope raises the probability of meandering and lowers that of braiding. Intervention and manipulation are often effective methods to find out whether putative causes and putatively necessary conditions are indeed causes rather than effects.

The isolation of causal mechanisms thus requires many cases. In fields such as medicine, this can be achieved through random controlled tests, but in planetary science such approaches are usually impossible (without experimental theology put into practice). Without strong control, there is a risk that putative causes may be confounded in statistical relations. For example, the relation between meander dimensions and channel width is confounded by the flow discharge as this causally influences both meander dimensions and channel width (as discussed later).

In sum, the two kinds of causation have different issues: production suffers from masking of effects of mechanisms by other mechanisms operating in the opposite direction, and probabilistic causation suffers from confounding causes by spurious correlations. Russo and Williamson (2007) argued that both production and probabilistic causation are therefore needed to establish a claim, because the combination can be used to avoid masking and confounding. For example, to establish a claim that river meandering requires floodplain vegetation, the mechanisms are needed to know why that is, and statistics are needed to quantify how important the causally relevant variables are and how strong their relations are. As such, statistics are needed to gain confidence that the mechanisms indeed make a difference to the pattern and are not merely an effect of the pattern, or have a cause in common with the pattern. This is how the combination of multiple viewpoints for causation provides stronger evidence of causation (Johnson et al. 2019).

For an effect to occur, causes are often said to be necessary and sufficient conditions. Consequently, removal of such a cause should then inhibit the effect. In the geosciences, however, three prominent issues arise, showing that the phrase ‘necessary and sufficient condition’ oversimplifies the causation:

1. Equifinality. A common issue is equifinality, meaning that some phenomena could also be caused by a different cause leading to the same effect (von Bertalanffy 1950a, b, also see Schumm 1991). For example, sinuous channels can form by river flow, turbidites and lava flow (see the introduction).

2. Multiplicity. Another common issue is that a set of causes are involved to produce an effect, and these causes cannot in isolation produce the effect. This relates to the complexity of phenomena and is called multiplicity in Schumm (1991). For example, Quaternary glaciation cycles could not only be caused by orbital variations, because orbital variations also occurred during the Neogene but major glaciations did not. Rather, multiple causes and background conditions were at play, such as the thermic isolation of Antarctica. Not all of these causes are necessary but some combinations of these causes are needed to obtain the effect. In other words, a cause is often part of a set of multiple causes that produce the effect together, while each individual cause is insufficient.

3. INUS conditions. A third common issue (although uncommonly recognized in the geosciences) is a combination of the previous two. For some phenomena, there are alternative sets of causes that produce the same effect, where each individual cause is unnecessary. Thus, while the said single cause is neither necessary nor sufficient, it is nevertheless a cause that can bring about the effect. In the example of orbital forcing, the set of causes of orbital variations is unnecessary because it is not the only possible cause of glaciation (consider the Paleozoic icehouse), which is a form of equifinality. Neither is this single cause sufficient because other conditions are required as part of a set that produces the glaciations (multiplicity, consider the thermic isolation of Antarctica and the composition of the atmosphere). As such, the orbital variation is an INUS condition: an Insufficient but Non-redundant part of Unnecessary but Sufficient condition (Mackie 1974, see Illari and Russo 2014). In view of the recent debate on vegetation as a ‘necessary and sufficient condition’ for meandering, the INUS concept will be useful, but note that many more models of causality exist (Illari and Russo 2014).

Until the 1980s, the cause of river meandering was sought mainly in physical mechanisms within the channel. Observations led to the idea that bars cause meander initiation. For example, Leopold and Wolman (1957, p. 59) observed similar bars and pools in straight channels as in meanders and suggested that ‘the processes which may lead to meanders are operative in straight channels’. In several scale experiments (Friedkin 1945; Einstein and Shen 1964; Schumm and Khan 1972), a train of bars formed on the sides of the straight channel in an alternating pattern, such that only the deepest part of the channel was mildly sinuous. This, in turn, directed the main force of the flow onto parts of the bank such that bends expanded. Bank erosion was observed mainly during floods as occurring in discrete steps of bank collapse and subsequent removal by the flow of collapsed material at the bank toe (e.g. Parker et al. 2011). However, over the long timescale associated with the sustained meandering pattern, it was considered appropriate to focus on the average erosion process at the outer bank and assume that its displacement is mirrored by the inner bank, while the stochasticity of bank erosion was considered of less interest at this timescale (Callander 1978; Seminara 2006). Here, the stochasticity, or natural variability of meander dimensions, was ascribed to unrelated irregularities, such as in the soil, valley walls and vegetation.

Running concurrent to these observations, statistics were used to identify which independent causes made a difference in meandering. Meander dimensions correlated well with such variables over many orders of magnitude. In addition, plotting rivers at typical flood conditions for a few variables divided them reasonably well into two groups, with mostly braiding rivers above, and meandering rivers below, a certain ‘discriminating’ relation between gradient and flow discharge (summarized in Friedkin 1945; Leopold and Wolman 1957; Callander 1978; Ferguson 1987). However, the probabilistic causation could neither refute nor confirm putative physical mechanisms behind the meander dimensions: ‘no physical or mechanical principle has been identified which explains qualitatively the size and geometry of meander curves’ (Leopold and Wolman 1960, p. 788). Leopold and Wolman also attempted to address possible confounding issues. They pointed out that the scatter in the relation between bar and meander wavelength and flow discharge was larger than in the relation between wavelength and channel width. They suggested therefore that bar and meander wavelength is more directly dependent on width than on discharge, while width is related to discharge and channel slope, and later data collection has confirmed that local channel width is statistically the better predictor for wavelength (e.g. Leuven et al. 2018).

Physical mechanisms were needed to show why width was important. The initiation of quasi-regular bars and meanders from a perfectly straight, simple channel with water and sand was ideal for analysis by first principles deduced from physics. Bar theories were developed from linearized equations of coupled hydraulics and sediment motion. The resulting systems of equations were subjected to minor periodic, along-channel perturbations in the flow, to calculate which wavelengths are ‘unstable’ in the sense that they grow. Such growth arises, simply put, because sediment flux is a highly nonlinear function of flow strength such that the slightly deeper zones rapidly lose sediment as the slightly shallower zones supply a much lower flux (see Crosato and Mosselman 2020 for review). The pattern is mainly formed by the wavelength that has the fastest growth of incipient channels and bars. This theory allowed for analytical solutions of the wavelength of the fastest growing incipient bars, which is considered the predicted pattern by the theory (e.g. Parker 1976; Seminara and Tubino 1989). Parker also produced a quantitative relation for the wavelength and number of parallel bars and channels in braided rivers based mostly on first principles, and a quantitative discrimination criterion between meandering and braided rivers with supposedly independent variables.

The most important variable in the physical theories turned out to be the channel aspect ratio, or width-to-depth ratio (e.g. Parker 1976). This ratio also determines how steep the transverse bed slopes can become, which, in turn, determines the gravitational effect on particles in the downslope direction (Baar et al. 2018), with a negative feedback on bar growth. In channels narrower than c. 20 times the depth, bars hardly form. Above that, alternate bars develop that may initiate meandering. Within relatively wide channels more parallel channels and bars emerge to form the braiding pattern. The statistical relation between meander dimensions and channel width was now causally explained and quantitatively acceptably produced from the physical mechanisms, while the relation with flow discharge was shown to be indirect. The unified treatment of meandering and braiding made the theory stronger as it showed the limits of the conditions in which meandering occurs (Parker 1976).

Hitherto, meandering was strongly associated with the three-dimensional flow pattern that occurs in bends (Thomson 1876). This pattern is caused by a combination of the conservation of momentum and bed friction: a positive transverse water surface slope sets up against the outer bank owing to the momentum vector of the flow. Given that flow velocity is higher near the water surface than near the bed owing to friction, helical flow results from the water surface slope, with the rotation towards the inner bend near the bed surface (see for review Blanckaert 2018). On average, this transverse bed shear stress causes sediment transport towards the inner bend, where it can form a bar (Struiksma et al. 1985). However, other bar-forming mechanisms act at the same time, so this is not the only possible cause of inner-bend bars and outer-bend scour. For one thing, the bend displaces the core of high flow momentum towards the outer bank, where it can lead to bed scour and bank undercutting in the outer bend and bar formation in the inner bend, which is one important cause of meandering as assumed in meandering simulation models (Johannesson and Parker 1989, discussed later). Mixing of the turbulent flow, which depends on friction, redistributes momentum in the curved channel with some spatial delay, of which helical flow is an effect. The velocity magnitude of helical flow is at best a fraction of the total flow velocity magnitude and is itself not a cause of outer bank erosion. Yet the association of meander bends and helical flow has long been thought to be of causal significance and this myth is still perpetuated in papers and handbooks. Many theories allowed refutation of such competing theories and confounding observations (Parker 1976; Struiksma et al. 1985; Seminara and Tubino 1989). Thus, secondary flow became relegated from consideration as a cause of meandering to a concomitant process or even an effect of meandering.

Parker's (1976) theory is an example of the idea that the bar formation mechanism in the channel initiates meandering by changing the distribution of flow in the channel, specifically by eroding banks alongside deep pools at a certain wavelength. However, there was an alternative theory where the link between the flow and the forming of meanders was more direct. As Seminara (2006) remarks: ‘Bend instability theory predicts that any small random perturbation of channel alignment eventually grows, leading to a meandering pattern’ (Seminara 2006, p. 285). Simply put, the momentum of flow going through an initial bend, or other planimetric (horizontal) perturbation in the channel centreline, leads to faster flow near the outer-bend bank, which locally increases the bank erosion. Owing to the deviation of the local flow direction from the general river gradient, the flow then crosses the channel and the opposite, downstream bank is under attack. Following bank erosion and local, temporary channel widening, a new bar forms in the inner bend while the outer bend continues to be scoured. Here, the bar is a consequence of meandering rather than a cause. Bars need not even be present, as narrow meandering rivers were not predicted to have bars. Moreover, problems emerged for the bar theory, which was now sufficiently developed to be able to predict bar migration speed. This speed was found to be so high that there was not enough time for focused bank erosion such that meanders built out, meaning that meandering could not arise from it (review in Seminara 2006). On the other hand, the bar theory explained bar formation in both braided and meandering rivers in agreement with observations, so it could not be rejected either. Further theoretical development suggested that resonance between the two mechanisms in meandering rivers (Blondeaux and Seminara 1985) reduces the migration of alternate bars with overdeepening under some conditions (Struiksma et al. 1985).

In the 1980s a group of theoreticians, experimenters and fieldwork-oriented researchers produced a monograph about the ‘bend or bar stability’ problem, entitled River Meandering (edited by Ikeda and Parker 1989). One chapter expanded on the resonance of bend and bar instability, such that the bars migrate no faster than the bends, which effectively reconciles both mechanisms in one theory (Seminara and Tubino 1989). In agreement with experimental observations, meander initiation was now thought to proceed as follows: (1) starting with an initially straight and narrow channel without cohesive banks, the channel widens; (2) at sufficient width, alternate and bars and pools arise that ‘push’ the flow towards the bank on one side such that it erodes; (3) this initiates perturbation in channel alignment, triggering a planimetric instability so that the bend mechanism takes over. Alternatively, the planimetric instability can also be triggered by other, unrelated factors in the landscape, such as engineered constructions, as well as poorly erodible substrates and banks (Struiksma et al. 1985; Smith et al. 2011; Leuven et al. 2017; Candel et al. 2020), which can in fact make the planimetric instability mechanism prior to the bar instability mechanism. However, the unified theory was not out of the woods yet. As Seminara states:

Does this picture exhaustively answer the question of why meanders form and is it generally accepted? […] it is fair to say that the question of meander formation is somewhat academic because it is hard to substantiate any answer by field observations. On the other hand, rather surprisingly, laboratory observations have so far been unable to provide conclusive answers. In fact, cohesionless sediments are typically employed in these experiments. (Seminara 2006, p. 284)

In other words, the choice between alternative theories was underdetermined by the available field data and experiments, a situation typical in the geosciences (Kleinhans et al. 2005; Turner 2005; Currie 2018). Moreover, a closer look is needed at the role of floodplain destruction and formation, and at the possibilities and limitations of scale experiments.

There are other theories of meandering that could be discussed, such as a set of theories built on minimization or optimization principles of friction connected to thermodynamics and starting from the random walk principle (Langbein and Leopold 1966; Nanson and Huang 2008). This was based not only on theory but also on some observations that meandering rivers were perhaps more efficient in flow conveyance and sediment transport than straight rivers (Langbein and Leopold 1966; Callander 1978). Rivers can adjust to changes in boundary conditions not only by changing width and depth, but also by changing the degree of meandering, which changes the slope as far as bend expansion is not offset by cutoff. However, these theories have been treated with suspicion, for example, Ferguson (1987) states that ‘minimization of channel slope is a metaphysical assumption that does not explain how meanders or braids develop’ (Ferguson 1987, p. 147). In more than one conference, heated debates were sparked by the teleological smell of such theories, which many engineers and geomorphologists rejected the idea that rivers are not agents and that such minimization principles are either invalid or should be derivable from the usual physics of flow and sediment transport (Mosselman 2004) and are, in that case, not an addition to the knowledge. However, it is interesting to note that estuarine morphology employs thermodynamic principles without question (Langbein 1963; Savenije 2015; see Zhou et al. 2017 for discussion). This suggests that a study is needed where the unified physical theory and a theory built on entropy for meanders are compared at a theoretical and empirical level.

The erosion of outer-bend banks received much attention in civil engineering for obvious reasons, but fieldwork-oriented geomorphologists and sedimentologists had long noted that cohesive outer banks in many cases formed as part of the river floodplain (e.g. Schumm 1963). In other words, the eroding outer banks formed as part of the set of mechanisms that form the entire dynamic fluvial landscape, including the floodplains. Another important aspect of meandering dynamics also depends on the floodplain properties: bend cutoffs, which proceed from two meeting meander loops with different bank erosion rates or from channel carving into the inner-bend during overbank flood conditions (Osman and Thorne 1988; Howard 1992). This meant that the physical channel mechanisms of bar instability and planimetric instability were at best INUS conditions for meandering, and a more complex mechanism was needed for the explanation of meandering.

Over the past decades, floodplain destruction and formation mechanisms have been integrated into models of meandering. Most meander simulation models reduce the rivers to single lines that move through two-dimensional space (Johannesson and Parker 1989; Howard 1992; Camporeale et al. 2007). These models usually, but not always, uniformly fill space around the river and assume that bank erosion is mirrored by inner-bend accretion. Many of these models represent the planimetric instability theory. To represent the effects of rooting from vegetation, the erosion threshold of the banks is increased. This means that the entire set of physical and biological mechanisms in floodplain formation and destruction are represented by a single parameter. Consequently, such models only reasonably reproduced observed meander dimensions and displacement after considerable calibration of the bank erosion coefficients (Camporeale et al. 2007). However, these models were designed to meander, and cannot produce contrasting (braided) river patterns, weakening their claim of demonstrating the causes of meandering considerably (Kleinhans 2010).

Another limitation of meander theories and models was that the precise driver of the meandering dynamics remained unresolved. The net result of outer-bank erodibility and inner-bend bar growth rates is the width of the channel, but since these involve very different mechanisms, either one or the other is likely to limit the bar and meander migration rate. The eroding bank may ‘pull’ the bend, or the growing bar and vegetation encroachment may ‘push’ the bend. Too fast a pull or too slow a push would lead to unimpeded widening until the threshold channel state is reached with a braided pattern. To resolve the bar push or bank pull problem, meander models were created where the two are uncoupled (Parker et al. 2011; Eke et al. 2014), showing that combinations exist of the many variables where outer-bank erosion and inner-bend sedimentation occur at equal rates. Several of the variables represent floodplain properties and mechanisms of floodplain formation and destruction, such as vegetation encroachment, and protection of the outer bank toe against unimpeded sand erosion by an armoured layer of decaying slump blocks from the collapsing bank top. As such, the bank pull–bar push balance was demonstrated for the kind of meandering river where these mechanisms are important. However, such models fail to reproduce the remarkable lateral periodicity of scroll bars (Strick et al. 2018), which are important morphological features and form recognizable deposits.

Like numerical models, physical laboratory experiments allow intervention and manipulation. In experiments known to produce dynamic meandering, interventions have been made on scroll bars and banks (van de Lageweg et al. 2014). For example, to test bar push mechanisms, scroll bars were manipulated by feeding in more sediment, but this had no effect. Conversely, sediment removal on the outer bank led to scroll bar formation and bend migration. This finding was further supported by control experiments without interventions and with fixated outer banks. The bank pull dominance was a surprising result, because the outer banks in these experiments are neither cohesive nor vegetated. Additionally, the scale experiments have the same low mobility as gravel-bed rivers, rendering the flow on the scroll bars near and below the threshold for motion. This effectively protects the bars, with stronger flows regularly causing bank toe erosion on outer banks. These physical experiments therefore probably represent different styles of meandering than those considered in the numerical model.

The two processes discussed so far indicate that the aforementioned ‘stochastic variability’ should be considered. Bend migration leads to changing directions and magnitudes of the momentum vector and the sediment transport vector, which in turn affect downstream bend migration (Lanzoni and Seminara 2006; van Dijk et al. 2012). This is a punctuated process as the formation of scroll bars, an important aspect of the pattern of many meandering rivers, is periodic (Strick et al. 2018). Furthermore, bend growth is limited by cutoffs across the bars (chute cutoff) and between bends (neck cutoff, Camporeale et al. 2007), which also leads to strong local gradients in flow and sediment transport (Zinger et al. 2011). While these perturbations may dampen out in the downstream direction, continued triggering dynamicizes the meandering pattern irregularly and is in some cases found to trigger a series of cutoffs (e.g. van Dijk et al. 2014). In the bend and bar theories, the perturbations are not causes but triggers, because they make no difference to the effect, which is initialization of periodic bends or bars. Mathematically, two kinds of instability can be distinguished: absolute instability and convective instability. Absolute instability develops upon a single trigger, whereas convective instability requires continued perturbation, usually on the upstream boundary. Lanzoni and Seminara (2006) first showed that perturbations in channel planform develop as convective bend instabilities, often migrating in the downstream direction. This was independently discovered in experiments with some cohesive sediment, where meandering only developed upon periodic upstream perturbation of the inflow position (van Dijk et al. 2012). Mathematically it was proven that all finite reaches in real rivers, numerical models and experiments require continuous perturbation on the inflow position or direction (Weiss and Higdon 2022). In nature, such perturbations can occur owing to many processes unrelated to meandering, but in numerical models and experiments of limited length, continuous perturbations must be imposed, usually on the upstream boundary. While the perturbation does not simply drive the meandering like arm-waving causes waves in a rope to propagate, for sustained, irregular meandering as the explanatory target, the amplitude and timescale of perturbations make a difference to the meandering dynamics (Lanzoni and Seminara 2006; van Dijk et al. 2012; Weiss and Higdon 2022). This complex dependence on the boundary condition implies that the perturbations can also have effects on morphological properties such as scroll bar bundle characteristics and depositional architecture, as was evidenced experimentally (e.g. van de Lageweg et al. 2014).

River meandering clearly arises not only from mechanisms in the channel but also from interactions with the floodplain. Such entanglement of physical and biological mechanisms is typical for the geosciences (Kleinhans et al. 2005). Studying these interactions requires more than a recognition of multiple INUS causes: it requires a systems approach.

'System’ derives from the Greek systema, which means an organized ‘whole’ that is compounded of parts. However, the modern notion of system is more than systematically ordered knowledge or a system of mathematical equations: a system has behaviour and it produces phenomena. The notions of a complex mechanism and a complex system are related and overlap. A mechanism for a phenomenon consists of entities and activities organized in such a way that they are responsible for the phenomenon (Illari and Williamson 2012). Mechanisms can be complex in the sense that they have feedbacks and inter-level interactions (Machamer et al. 2000), where level refers to various scales of organization. Feedbacks and inter-level interactions are also properties attributed to systems. Indeed, Glennan's (2002) preferred definition for a mechanism could be equivalent to that for a complex system:

A mechanism for a behaviour is a complex system that produces that behaviour by the interaction of a number of parts, where the interactions between parts can be characterized by direct, invariant, change-relating generalizations. (Glennan 2002 p. S344)

The system-level behaviour is the explanatory target, the interactions between parts are what can explain the system behaviour, and manipulation of parts can indicate which factors offer control and make a difference to the behaviour.

The one-way causal relations discussed earlier are all change-relating generalizations of the interactions between parts, but the positive feedback that leads to bar growth and meandering is in fact a systemic property of the water–sediment system in a channel. Tracing the causal pathways through a system may, however, show a sequence of events following a trigger, rather than a web of simultaneous interactions. Here the notion of a causal pathway is helpful, which refers to a group of causal factors that are ordered in a sequence that leads to the effect of interest (Ross 2018a, b). The sequence arises as some signal flows through the system, such as the downstream propagating pulses of flow and sediment and the upstream propagating backwater effect caused by bend cutoffs (van Dijk et al. 2012), or the changes caused by the spreading and seed-banking of propagules of an eco-engineering plant species (Corenblit et al. 2007). Furthermore, the causal pathway selects causal sequences that regulate system behaviour and background causes that are relevant in an exhaustive description of a mechanism, such as gravity, but do not immediately make a difference to the system behaviour (Ross 2018a, b).

In von Bertalanffy's (1950a, b) General Systems Theory, a system is defined as a complex of interacting elements, meaning that they stand in a certain relation so that their behaviour in that relation is different from their behaviour in another relation. The interacting elements are herein quantified by a system of differential equations in which the change of a quantity of an element is a function of other quantities (von Bertalanffy 1950a, p. 143). Wimsatt (1994) generalized the complex systems approach. In the case of river systems, these are the equations for flow, sediment transport and morphology with prescribed initial conditions and usually with constant boundary conditions (Fig. 2, Seminara 2006). Systems are often open in the sense that matter, energy and information (in the case of rivers, water, momentum and sediment) are exchanged with their environment. Despite the openness, the persistent activity of a system creates a robust, or resilient, order out of the disorder of the environment, while disorder is the result of the second law of thermodynamics. The order is what makes it possible to distinguish such a system easily from the jumble of the world. An important consequence of being an open rather than a closed system is that a simple and single equilibrium state does not exist, but there are multiple alternative states. This was quickly realized to apply to geomorphological systems (Chorley 1962) and has been echoed in many earth-scientific publications since (e.g. Schumm 1991; Thorn and Welford 1994; Phillips 1999; French et al. 2016; Steffen et al. 2020), although the idea of a single equilibrium is still prominently used in the practice of simulation of earth-scientific phenomena. According to bar and bend theories, the plane bed is an unstable equilibrium, which, after perturbation, develops into a periodic pattern of bars or bends that is a stable equilibrium. The homeostasis found in living organisms is not accomplished by earth systems, which are more clearly open to influences of the surroundings (also see Zuchowski 2018).

This leaves us to expand on what is meant by complexity and how it differs from complicatedness. Phenomena and models thereof in the geosciences are often called ‘complex’, but a recent questionnaire revealed that geoscientists have no consensus on what they mean precisely by complexity (Baartman et al. 2020). The number of explicitly included processes and their interactions, or feedbacks, are deemed the most important characteristics of complex models, followed by processes that ‘act over multiple scales’ (Fig. 2), ‘number of input variables’ and ‘nonlinearity of processes’, but what ‘multiple scales’ and ‘nonlinearity’ mean is not explained. The majority of the questionnaire does not see mere complication in the spatiotemporal resolution or the number of equations and output variables as characterizing complexity. For conceptual clarification, an outside view is needed. Ladyman et al. (2013) reviewed measures for complexity in the scientific literature and characteristics and concepts associated with complex systems in the philosophical literature. The emerging definition for a complex system is an ensemble of many elements which are interacting in a disordered way, resulting in robust organization and memory. Without the order emerging from disorder, a system may be merely complicated, such as a gas at equilibrium, which can be studied at the level of its components by statistical mechanics and thermodynamics. As also suggested by von Bertalanffy (1950b, p. 140) and Wimsatt (1994), the robust, but not static organization of an open complex system allows the investigation of the cause–effect relationships and dynamics at a higher level of organization.

The vegetated river system conceptualized in Figure 2 and numerically modelled in Figure 3 is an example of an open, complex system. The composite of a dynamic river meander belt with sustained meandering, bend cutoffs and floodplain formation is the higher level of organization emerging from the (lower-level) dynamics of water, sediments and vegetation. This system may be in a dynamic steady state of meander expansion and cutoff and of floodplain destruction and formation such that on average the characteristic spatial scales of meander dimensions stay constant. This system also has characteristic timescales of meander displacement and cutoff. Scale experiments of rivers are, in fact, open systems with controlled, steady boundary conditions, such as constant flow discharge and sediment input and fixed downstream water surface elevation (Peakall et al. 1996). Numerical models are likewise open system representations with the same control on boundary conditions and a choice of mechanisms and interactions (e.g. Johannesson and Parker 1989).

The notions of allogenic and autogenic forcings can be elucidated by the system representation (Fig. 2, see the introduction). Autogenic refers to mechanisms and feedbacks within a system, whereas allogenic factors are enforced from outside the system as initial conditions and boundary conditions, such as those imposed in physical experiments and numerical models (Hajek and Straub 2017; Kleinhans et al. 2018). These are assumed to be independent causal factors in nature, meaning that feedbacks from within the system on the boundary conditions are ignored, as are feedbacks between the processes that cause the boundary conditions to the system of interest. Of course, whether a variable is constant depends on the timescale of interest. This is not merely to say that ‘there is no such thing as “equilibrium” in geology’, but a key point for the recognition of, and representation as, an ensemble of interacting elements as a system. For the case of river meandering in Figure 1, the floodplains were formed in interaction with, and at a similar timescale to, the meander bends, but the surrounding glacial landscape formed much longer ago by processes not interacting with the river on the characteristic timescale of the bends. Likewise, the valley slope and width are variables on a timescale longer than that of meander development. On smaller timescales, on the other hand, a meandering river perpetually perturbed by cutoffs has a statistically definable equilibrium state of length and sinuosity to which it tends (Howard 1992). Geologists, with their focus on the inference of specific past causes rather than generic mechanistic explanations, had long realized that:

any account of winding rivers in terms of the slope of a valley floor or other surface leaves out a factor that would be very important to a geologist: the origin of that valley floor or surface. Evolving, mobile rivers, as opposed to quasistatic channelized flows, create the valley floors and surfaces upon which they flow at any moment in time. From this geological viewpoint, an explanation for the pattern of any evolving river must include something about the historical development of that river. (Mackin 1963, in Baker 2013, p. 8322; also see Macklin et al. 2006)

This is meant to say that the sediments building up the river channel and the floodplain together determine the development of the valley, the gradient of which has previously been identified as a cause of meandering rather than an effect of the mechanisms forming the river. In other words, the valley slope is a causal factor that is often taken as a fixed initial condition but is on longer timescales an interdependent variable (Fig. 2, also see Schumm and Lichty 1965; Phillips 2006).

One way to identify INUS causes that make a difference is to study cases where the river pattern changed owing to an identifiable change in conditions. Many cases exist since recent human interference drastically changed the hydrodynamics and the morphodynamics of rivers. For example, the braided Platte River in Nebraska narrowed rapidly while riparian vegetation expanded and aged following flow regulation by an upstream hydropower dam in the river (Schumm 1985; Johnson 1997). The effect was a transformation from a braided river to something closer to meandering as the vegetation resisted flow over the newly constructed floodplains and focused flow into fewer channels. In Yellowstone National Park, on the other hand, the eradication of wolves over a century ago caused the elk population to expand in number. In turn, the elk expanded their browsing areas towards river banks, resulting in greater bank erosion and river widening, and ultimately the cessation of meandering (Beschta and Ripple 2006). These are examples of causation by difference-making, namely by tracing the effects of a large change in conditions as it cascades through the system (Millstein 2019). However, the specific causes remained confounded: riparian ecosystems are clearly strongly correlated to meandering, but it remained unclear whether vegetation affected river patterns specifically through resistance to flow and/or through bank stabilization by rooting.

One way to gain more control over the variables is to combine data from many meandering rivers and rivers with different patterns. The classic channel pattern stability diagrams show fields for meandering and braided rivers separated by a discriminator, but none of the causal variables are directly related to floodplain characteristics. In the empirical diagram of Leopold and Wolman (1957) the separation is determined by the causal variables channel slope and bankfull discharge, whereas in the theoretically derived diagram of Parker (1976) the causal variables separating braided, meandering and straight rivers are the channel slope, the Froude number (flow velocity normalized by shallow wave celerity) and the width-to-depth ratio of the channel (Fig. 4). Note that relations exist between the bankfull discharge on the first diagram and the Froude number and channel dimensions on the second, but neither diagram is empirically superior for an independent dataset (Candel et al. 2021). In both cases, the authors stress that transitions are gradual, with river patterns forming a continuum rather than distinct classes (also see Ferguson 1987). However, none of these channel-related variables are independent of the river pattern: the channel slope is determined by sinuosity (van den Berg 1995) and the width-to-depth ratio is determined by the outer-bank bank resistance against erosion and the rate of floodplain formation on the inner bank (Eaton and Giles 2009; Lapôtre et al. 2019).

Various channel pattern diagrams have since been developed that incorporate variables related to the floodplain, some empirical and some incorporating bank stability mechanisms (see Candel et al. 2021 and Fig. 4 for comparisons), and some showing that chute cutoff-dominated meandering, in contrast to neck cutoff-dominated meandering, occurs on the transition between meandering and braiding. Candel's diagram extends the van den Berg diagram with data on the mud fraction in the river bank and empirical relations for critical shear stress for mud erosion. This diagram includes the causes of a new class of tortuous rivers with fairly stable banks but locally occurring meandering with scroll bars. It is as yet poorly quantified to what degree river pattern is modified by barely erodible substrates, but recent findings indicate that purely alluvial rivers are rarer than hitherto assumed, even in net accreting settings such as deltas (see Pierik et al. 2022 for a first data compilation).

The extensions of the channel pattern discrimination diagrams show that floodplain character underlies the chosen causes of pattern and makes a difference to the generated pattern. However, it is challenging to populate these with data as floodplain properties require much more data collection, and it is unclear which characteristics of the complex of vegetation, sediment and floodplain stratigraphy precisely to measure. Both the history and the boundary conditions come into play. Vegetation cover is a result of historical settling of vegetation. Fine sediment properties and concentration are determined by hinterland characteristics. Floodplain stratigraphy has formed in relation to channel shifting, long-term sedimentation and vegetation dynamics (see for reviews Ferguson 1987; Lewin and Macklin 2003; Kleinhans 2010). Furthermore, the meandering river is demarcated from braided rivers on one end and laterally (nearly) immobile rivers on the other. When including the floodplain in a system, richly complex interactions between many processes become relevant, and the history, or memory, of the system becomes important (Fig. 3, Phillips 2006; Kleinhans et al. 2018).

The historicity of complex floodplain systems has been demonstrated for small tortuous rivers with self-formed floodplains that increasingly confine channels to produce very sharp bends in terms of bend radius relative to channel width (Candel et al. 2020). In such bends, flow separation, or even recirculation, can occur (Bagnold 1960; Ferguson et al. 2003; Blanckaert 2018), and may form a variety of different deposits including concave bank benches, lee bars and counter-point bars (Smith et al. 2011).

In view of the desire to reconstruct past conditions from the rock record, one question is whether subtly different meandering styles can be recognized in preserved stratification. The commonly used ‘lateral accretion surface’ offers an incomplete vantage as scroll bar migration has marked upstream and downstream components. It does so at an average transverse bed slope determined by the force balance on the sediment of helical flow and gravity, allowing reconstruction of flow conditions and bended channel dimensions (van de Lageweg et al. 2014; Baar et al. 2018). The depositional products of very sharp bends differ in recognizable aspects (Smith et al. 2011). In low-energy conditions, fines may deposit on the lateral slope surfaces that are preserved in between the channel sediment deposits, and are collectively known as inclined heterolithic strata (IHS) (Figs 5 & 6, Thomas et al. 1987; Choi et al. 2004; Olariu et al. 2015; D'Alpaos et al. 2017; McMahon et al. 2022). While IHS are found in rivers, they are more frequently found in tidal environments, where the propensity for flocculation and mud settling is greater as clay suspensions encounter saline waters (e.g. La Croix and Dashtgard 2014; Seiphoori et al. 2021).

The characteristic spatiotemporal scale of a river determines which temporal scale is important. This point generalizes to non-alluviated rivers that are confined by antecedent landscapes with erosion-resistant materials, which inhibits the formation of river patterns, and to rivers that are confined by rapidly growing and dense vegetation, which likewise inhibits meandering river formation (Fig. 3, Ghinassi et al. 2016; van Oorschot et al. 2016). In both cases, floodplain formation remains possible but floodplain destruction is inhibited, removing a condition for dynamic meandering. This is more probable in smaller rivers with lower flow strength, of which there are many on the planet (e.g. Candel et al. 2021). The range between the limiting conditions where either physical or biological dynamics dominates the landscape is called the ‘biogeomorphic feedback window’ (Eichel et al. 2016).

Historicity is also evident in the patterns of the largest rivers on Earth that are affected by the complex Quaternary history of their floodplains and the antecedent slope and variations in lithology spanned by a river (Latrubesse 2008; Ashworth and Lewin 2012). In both small, tortuous rivers and large, anabranching rivers, the system is in fact determined more by the antecedent landscape and the development of the boundary conditions, because under these conditions the characteristic timescale of forming a river pattern is much longer than that of the change in conditions during the Holocene for the tortuous rivers and during the Quaternary for the large rivers, as far as their inherited conditions are not entirely dominated by bedrock. Historicity is also evident where memory, delayed response or hysteresis effects render a pattern unsteady (or in disequilibrium), such as in the Platte River that became vegetated, or the river in Yellowstone Park that changed dynamics as the effect of the introduction of wolves cascaded through the levels of the system from the grazers to the vegetation and the river banks (also see Millstein 2019).

Channel pattern discrimination diagrams fail in the above cases where rivers are not close to some dynamic steady state, but are rendered morphologically less active. In terms of system science, there is a mismatch in these cases between the selection of the river system and what is outside the system of interest, in terms of both the spatial extent and the timescale of relevant history. Much of the river pattern research has focused on smaller, dynamic rivers for all kinds of theoretical and practical reasons. The physical dimensions of most geological outcrops, for example, are inferior to the dimensions of many geomorphic components of contemporary large river systems (Fig. 2, McMahon and Davies 2018a). This includes the question as to what determines steady-state river patterns and the fact that smaller rivers are more numerous than continent-spanning rivers, both active at present and recognized in geological outcrops.

Some two-dimensional models are cellular automata that, without vegetation, simulate aspects of braided rivers and with vegetation produce single-thread channels (Murray and Paola 2003). Unlike many meander simulation models, these cellular automata have no representation of the fluid but only directly move the sediment by slope-driven diffusion. Vegetation reduces the slope-driven diffusion. Compared with the control runs, vegetation changes the river into a single-thread channel that wanders over the fluvial plain, similar to meandering rivers but without quasi-regular bends, which would have required more physical channel processes. However, Murray and Paola (2003) were careful not to claim that their single-thread river simulations were meandering.

Various attempts to produce meandering rivers at a very small scale in laboratory flumes with sand had shown that bank erosion outbalanced floodplain formation, so that the continued widening ultimately resulted in a braided river (e.g. Friedkin 1945). These experiments showed that the alternate, periodic bar stage can be the onset of meandering, but when sustained for a longer time leads to chute cutoff and braiding. This in turn resulted in a stronger realization that floodplain development was part and parcel of river meandering, while the periodicity of the meandering pattern is related to the bar formation mechanism. Three specific, often-cited experiments contributed to the idea that fluvial floodplains play an important causal role in meandering: Schumm and Khan (1972), Smith (1998) and Gran and Paola (2001). The experiments of Schumm and Kahn were supplied with clay over an initially cohesionless sand bed, which led to floodplain formation that, in turn, changed the river from braided into weakly meandering with alternate bars. The experimental control was that only the variable that made a difference in floodplain formation was changed, and these experiments, with those of Friedkin (1945), have frequently been cited as evidence for the theory that meanders grow out of alternate bars if the banks are cohesive enough to avoid widening to the point that braiding emerges. In a similar vein, Smith conducted experiments with an initially fully cohesive sediment bed, which produced sinuous, displacing channels without bars. With their smaller scale, the dynamics are considerably faster in experiments. Yet they still take weeks to months, while the properties of the used materials change over time owing to physical and chemical changes and the growth of fungi and algae. This makes it a methodological challenge to convincingly demonstrate sustained meandering, which would require a reworking of the entire floodplain such that it becomes self-formed by fluvial processes rather than emplaced by the experimenter. This arguably failed in the experiments of Friedkin, Schumm and Smith, but later succeeded in van Dijk et al. (2012, 2013a, fig. 5).

Gran and Paola (2001) used live sprouts as vegetation in river experiments, which was a methodological breakthrough. Repeated manual reseeding led to a sustained single-channel river with differences in pattern and dynamics from the initial braided river, on which seeds were spread regularly. Gran and Paola (2001) frequently referred to the bank stabilization effect of roots in their observations, which is valid because the rooting of the sprouts was sufficiently deep in comparison with channel depth. However, they also observed that flow over the vegetated floodplain reduced proportionally with vegetation density, while the main channel(s) deepened to accommodate the unchanged discharge. The co-occurrence of both mechanisms meant that they could have caused the transition in river pattern either alone or in concert. In other words, one may have masked the other. Later experiments confirmed this (Tal and Paola 2007), but also showed that the distribution of seeds, either uniformly by wind or on the channel margin and bare floodplain by water, made a difference in single-threadedness and meander sinuosity (Braudrick et al. 2009; van Dijk et al. 2013b).

The quantitative similarities in form and dynamics between the laboratory rivers and their natural target population, even though the laboratory rivers are approximately 100 times smaller, are strong reasons for assuming that the analogy is strong enough to infer causes and mechanisms of meandering in nature from the experiments (Peakall et al. 1996; Paola et al. 2009; Kleinhans et al. 2014). However, the quantified causal mechanistic relations have not been applied to natural rivers because there are certain unavoidable effects of downscaling on the physical mechanisms, such as the reduction or disappearance of turbulence. Hence the translation between experimental and natural ‘materials’ is not straightforward enough to allow unambiguous quantification of causal pathways and mechanistic relations between particular causes and effects (Reynolds 1887; Kleinhans et al. 2014). As Schickore (2019) argues, control experiments were important to find out how scale experiments work, and what the relevant conditions are that need to be imposed and controlled to avoid confounding and masking effects. For example, experimental river patterns are sensitive to the boundary condition of sediment supply, which is intended to keep the system in a dynamic equilibrium. Supplying insufficient sand at the inflow boundary leads to incision and the abandonment of smaller, parallel branches, so that a single-thread meandering river might temporarily emerge. This masks the effects of the (absence of) floodplain formers (Kleinhans et al. 2014). Furthermore, in relatively short experiments with a few bends, a lateral dynamic imposed on the upstream boundary is necessary to obtain dynamic meandering, for which the naturally occurring random perturbations in the channel subsystem are insufficiently large and merely lead to a static alternate bar pattern of low amplitude (van Dijk et al. 2012; see earlier discussion on convective instability).

While vegetation or cohesive floodplain sedimentation were found to have similar effects on river pattern, scale experiments demonstrated that the underlying mechanisms are quite different from what is regularly assumed. The vegetated experiments emphasize the increased strength of erosive outer banks owing to rooting in the floodplain, as well as increased flow resistance on the floodplain (Tal and Paola 2007; Braudrick et al. 2009). The experiments with weakly cohesive suspended floodplain sediment had neither, but instead had an increased threshold for erosion on the inner-bank floodplain and a decreased flow strength owing to accretion of fines in zones where sand was no longer in motion (van Dijk et al. 2012, 2013a). On the erosive outer banks, on the other hand, the effects of unvegetated floodplains were negligible, as undercutting of the cohesive floodplain veneer went unimpeded. Yet in both vegetated and unvegetated cases a meandering river morphology emerged. The meandering dynamics between different laboratories are difficult to compare because of the different conditions. A comparison of van Dijk et al.'s experiments (2013a, 2013b) reveals that the channels with vegetation had similar bend wavelengths and a larger channel depth, but lower sinuosity and higher bend migration rates when compared with the experiment with suspended floodplain sediment. The experiment with water-distributed seeds is most affected by vegetation because the settling during floods mostly occurred on channel margins. Likewise, Braudrick et al.'s (2009) experiments showed bend expansion in response to reseeding on the newly formed inner-bend bars. In sum, scale experimental meandering rivers formed mainly because of floodplain formation on inner bends, either by riparian vegetation or by floodplain sedimentation, while erodibility of the outer banks appeared hardly relevant. Note, however, that these experiments all produced low-sinuosity, chute cutoff-dominated meandering, where inner-bend erosion leads to the cutoffs, whereas it is outer-bank erosion that leads to the cutoffs in high-sinuosity neck cutoff-dominated rivers (Constantine et al. 2014).

Notwithstanding the insights yielded by one-dimensional modelling, two-dimensional or three-dimensional models are needed to model floodplain development. Without some form of reduction of floodplain erodibility compared with the channel sediment, numerical model rivers and rivers in scale experiments inevitably become braided. Over the past 10 years, various codes for two-dimensional models have been developed that simulate many aspects of both meandering and braiding, depending on conditions and the intensity of floodplain formation by mud sedimentation and vegetation settling (Nicholas 2013; Iwasaki et al. 2016; Kleinhans et al. 2018). However, there are many differences between these models in choices of implemented mechanisms, numerical and semi-empirical aspects and the imposed initial conditions and boundary conditions. This may be the reason why the appearance and dynamics of the meandering produced by the different models differ, for example in the pronounced formation of scroll bars and occurrence of cutoffs. Nevertheless, these models were used to isolate mechanisms that cannot be isolated in nature. For example, Iwasaki et al. (2016) and Kleinhans et al. (2018) included the flow resistance mechanism of vegetation that hinders bend cutoff, but not the rooting. Even though this left the outer banks entirely unprotected, a dynamic meandering pattern emerged in their models with vegetation (Fig. 3) while the control without vegetation produced braiding. These comprehensive models are used in the production of causality in ways that are impossible with experiments and through observation. They lend themselves a greater degree of control over initial and boundary conditions, in addition to isolation of those mechanisms of plant effects in the water and soil that cannot be separated into live organisms, such as those imposed by below- and above-ground biomass. In the aforementioned models, the inner-bank surface cover by cohesive sediment, vegetation or both was sufficient to change the river pattern to low-sinuosity meandering with chute cutoffs. However, denser vegetation and higher mud concentrations did not lead to high-sinuosity meandering. This remains a challenge for two-dimensional models and experiments alike.

A closer look at one of the models shows how well the notion of open, complex systems describes the modelling approach (Figs 2 & 3, van Oorschot et al. 2016; Kleinhans et al. 2018). The model assumes an initial setting where a river flows over a gently sloping surface of sand. The system is open in that water and sand are supplied as temporally varying boundary conditions on the upslope boundary and are allowed to flow out of the spatial model domain. An existing hydromorphodynamics code (Delft3D, created by Deltares) was used to solve the equations of shallow flow for the supplied water on a grid, to calculate the flux of sand caused by the flow and to change the elevation of the river-bed surface caused by the difference between in-flux and out-flux of sand in each grid cell. Mechanisms at spatiotemporal scales much smaller than meandering, such as turbulence, friction and particle-scale sediment motion, are causally backgrounded by parameterization (Fig. 2). Other conditions implicitly assumed to be necessary for a river to form are specified as constants, such as gravitational acceleration. In a separately coded subsystem, vegetation is modelled. Settling, growth and mortality are determined by physical processes and occur on a similar timescale to the development of a river bend. Willow and poplar are settled on the grid depending on flow conditions and duration of inundation, which causes stress for the plants. Plant mortality also depends on the flow and on the excavation or burial of roots, depending on plant size. Plant growth is calculated seasonally (van Oorschot et al. 2017), which is considerably longer than the hydrodynamic time-step. The additional flow friction caused by the plants depends on this cover fraction and size and is added up as parallel resistors for each grid cell. The resistance operates through turbulence, but this occurs on sub-grid scales and is here parameterized. The effective resistance is fed back to the flow model, which is how the vegetation affects the flow, the motion of sand and the resulting development of the underlying bed surface. In turn, the bed surface elevation and the flow over it determine the fate of the vegetation, and thus the system is propagated in time. The choice of the specific biological and physical phenomena, which are connected by feedbacks, has backgrounded all other physical and biological phenomena as well as phenomena at chemical, societal and other possible levels. Likewise, the much faster and much slower interactions are excluded: turbulence, moving particles and individual plants are parameterized, the river valley is initially specified to ignore the tectonics that create the relief, and climate and fluvial erosion processes are kept under control as boundary conditions. In contradiction with the meandering experiments described before, this model did not require a dynamic upstream boundary for meandering to be sustained. It is at present unclear why not, but one hypothesis is that the small-scale structure and dynamics of vegetation produce sufficient perturbations for the convective instability to be maintained.

Vegetation interacts with the physical mechanisms that comprise both the physical and the ecological aspects of landscape change (Eichel et al. 2016; see Viles 2020 for review). Notably, riparian vegetation (vegetation along rivers) has been found to have specific assemblages of species that have adapted to survive the dynamics close to rivers (Geerling et al. 2006; Corenblit et al. 2011). Sustained rejuvenation of the land surface by frequent floods and the erosion and sedimentation by meandering and braiding lead to an interrupted succession of vegetation, such that unique species find a niche along rivers. In fact, certain species have been argued to modify their physical environment to their own benefit. This idea was first formulated by Jones et al. (1994):

Ecosystem engineers are organisms that directly or indirectly modulate the availability of resources (other than themselves) to other species, by causing physical state changes in biotic or abiotic materials. In so doing they modify, maintain and/or create habitats. (Jones et al. 1994, p. 373)

In other words, certain species have direct effects on the physical mechanisms in their environment that increase their own probability of survival. This eco-engineering concept provides bridge principles between the biological level of organisms and the physical level of fluvial mechanisms (Kleinhans et al. 2005) that are needed in the model system of meandering (Fig. 2). Willow trees and beavers are classic examples of ecosystem engineering species. The effects of beaver dams on rivers are spatially much more extensive because a backwater zone develops upstream over distances orders of magnitude larger than the dam itself (Larsen et al. 2021). This has long-lasting effects on the landscape, for example through the formation of peat on the river floodplain and the displacement of entire river reaches, which is another example of the ‘memory’ of the complex river system. The distinction between eco-engineering species and other species is not black and white, though, but a subset of pioneer species makes a much greater difference to river floodplain formation than others.

A concept closely related to eco-engineering is that of niche construction. Many species alter their environment in physical, chemical, ecological, and at least for humans, cultural ways. In turn, the niche construction modifies the selection processes, and thus codirects evolution (Odling-Smee 2009; Corenblit et al. 2011). Plant species actively change their environment, which is part of the model, but evolved over much longer timescales to adapt to the physical (and chemical) environment (see for review Corenblit et al. 2011). The first land plants probably evolved from aquatic algae, with this greening of the continents underway by at least the early Ordovician (c. 480 Ma; Strother and Foster 2021). From then, plant species evolved that were particularly suited to, or competitive in, the dynamic zones along rivers (Corenblit et al. 2015; Davies et al. 2021). As plant evolution takes place on much longer timescales than meandering, the feedbacks between the river system and evolution are excluded from the model system, but are in fact acknowledged to exist through reference to the concept of ecosystem engineers. This is not to say that any present-day plant species can come to dominate the pattern formation in a biogeomorphic landscape: the positive feedbacks are most effective in a biogeomorphic feedback window between conditions dominated by physical processes that disturb the vegetation too much for it to develop the eco-engineering feedbacks and conditions where vegetation development and succession are so strong that physical processes are suppressed (Eichel et al. 2016). River meandering is precisely in this window, with braiding on one boundary and rivers fixated by vegetation (and mud) on the other.

The notion of eco-engineering species points to a relation between the size of plants and their effect depending on the size of rivers. Given the same riparian tree species, a river of, for example, 1 m water depth may be nearly fixated by the roots and above-ground biomass. Conversely, a 20 m deep river is more likely to have the pre-requisite depth to undercut outer banks and the high overbank discharge onto the floodplain to uproot standing vegetation on the inner banks (e.g. van Oorschot et al. 2016), and the largest rivers such as the Ganges might not noticeably be affected by vegetation. Mud, on the other hand, will show a different scale dependence if the thickness of the cohesive floodplain increases with the depth of the river, because the threshold for erosion or failure of a cohesive layer depends on clay composition whereas the force of the flow depends on hydraulic characteristics and river dimensions. Likewise, the characteristic timescale of meander displacement is scale dependent: in a tiny laboratory river the meanders displace within hours (van Dijk et al. 2013a), while in large rivers this may be a decadal process. Vegetation, however, has its own timescale of growth; for instance, willow and poplar species take about two decades to mature and may take years to die under inundation stress, uprooting or burial. The importance of life-stage-dependent plant traits further complicates comparisons between rivers of a range of sizes (Latrubesse 2008), and points to a great potential for differences between the tropics, temperate and colder areas where different species with different dimensions, growth rates and inundation sensitivity dominate the riparian zone (Corenblit et al. 2011). One can speculate that the effects of animal eco-engineering species may likewise depend on scale (Post 2019). In conclusion, a comparison between rivers of different sizes inevitably involves a number of scale-dependent physical and biological mechanisms, and statements about ‘the effect of absence and presence of riparian vegetation on meandering’ require elaboration of channel scale and type of vegetation, both of which depend on the climate, hinterland geology and history of the fluvial system.

The deep time record of river meandering is of interest as it potentially offers the opportunity to recognize river behaviour in the absence of causes and mechanisms that are innate in modern systems. Chief amongst such components is vegetation. Even ‘unvegetated’ modern rivers exist on a planet that has a historicity of land plants, and many such rivers were vegetated at some earlier interval of their existence (Davies et al. 2017). In contrast, rivers that operated prior to the early Paleozoic operated on a planet where vegetation did not, and never had, existed. If such rivers can be accurately reconstructed there is clear potential to understand whether vegetation is a necessary condition or an INUS condition for river meandering, whether through the biogeomorphic interactions within rivers or through the modification of boundary conditions of river systems (Fig. 2). Recognizing this fact, Schumm (1968) offered ‘Speculations concerning palaeohydrological controls of terrestrial sedimentation’, in which he speculated how Earth's rivers may have operated in deep time by attempting to envisage observations and measurements from modern rivers without input from vegetation. Schumm's (1968) conclusion was that bedload deposition would have dominated and that rivers would frequently have adopted a braided, rather than meandering, planform.

Subsequent to Schumm's (1968) ‘speculations’, a number of geological studies have used alluvial sedimentary rocks that were deposited prior to the evolution of land plants to elucidate the nature of ‘pre-vegetation rivers’ (e.g. Cotter 1977; Els 1998; Eriksson et al. 1998; Davies and Gibling 2010; Ielpi et al. 2018). However, all such studies have to contend with a fundamental problem: there is presently no such thing as a pre-vegetation river. There are rivers without vegetation, but these are part of a planet that contains land plants that have affected boundary conditions. This assertion appears glib, but it is essential to recognize that ‘pre-vegetation rivers’, and their planforms (even when purportedly quantitative: Lyster et al. 2022), are essentially interpretations based on shapes preserved in sedimentary rocks and geological outcrops that are divorced by huge temporal gaps from their now ceased formative processes. Thus only by dually considering the geological record from both the perspective of interpreted fluvial style and the perspective of secular geological characteristics can the singular impact of the evolution of vegetation be understood.

From the 1960s to the early 2000s, geological studies of ancient alluvium frequently attempted to interpret geomorphic planform utilizing knowledge of the differing ways in which modern ‘end-member’ fluvial styles organize sediment. The classic facies models of meandering deposition, developed by Allen (1964, 1974) to explain sedimentary motifs in Devonian-aged strata of the Welsh borders, provided the template from which other successions worldwide were broadly grouped into ‘meandering facies’ and ‘braided facies’. Cotter (1977) developed such facies models for the eastern USA and combined these with the ideas of Schumm (1968) to suggest that sedimentary characteristics ascribed to meandering deposition increased in total frequency and abundance in successions deposited after the Paleozoic evolution of land plants. Subsequently, other workers focusing primarily on Precambrian (fully pre-vegetation) aged strata noted the absence of sedimentary motifs attributable to Allen's meandering facies models and suggested that, prior to land plants, meandering deposition was near impossible (Els 1998; Eriksson et al. 1998).

Davies and Gibling (2010) revisited and expanded upon Cotter's (1977) review of published interpretations of ‘meandering facies’. They showed that there was indeed an upsurge in such interpretations in strata younger than the Silurian, but also noted that not all meandering rivers would leave behind the signature encapsulated by Allen's models, emphasizing that such models would overlook coarse-grained meandering systems, or rapidly migrating meanders where chute cut-offs eroded point-bars, both of which are common (Fig. 4). Drawing on satellite imagery of meandering channels from Mars, they indicated that some meandering rivers were likely to have existed on pre-vegetation Earth, but that rivers that left behind traditional ‘meandering facies’ did not (contentions subsequently borne out by geological observations, e.g. Ielpi et al. 2017). They distilled the components of ‘meandering facies’ into several discrete secular rock characteristics (e.g. lateral accretion sets, >10% mudrock, pedogenic strata, root fossils), which all individually also showed an uptick in abundance after the evolution of land plants.

The unidirectional shift in both interpreted ‘meandering facies’ and several secular sedimentary characteristics strongly implies that the evolution of vegetation had a major impact on the type of alluvium that was archived in the geological record, but such interpretations and characteristics capture only a fraction of records of meandering rivers. For example, architectural characteristics that can be identified as indicative of meandering must necessarily manifest at spatial scales smaller than a geological outcrop (Fig. 2, Davies and Shillito 2021). These are commonly limited to small- to medium-scale lateral accretion sets (McMahon and Davies 2018a; McMahon et al. 2022), and a geological outcrop is thus ill-suited to making a confident diagnosis of large-scale lateral accretion, or to identifying other components of meandering architecture such as the upstream and downstream components of scroll bars that may be underdetermined as such from the available evidence. Likewise, abundant mudrock or pedogenic horizons are not present in all meandering rivers, so their increase in abundance cannot rule out a pre-vegetation abundance of meandering rivers in which such phenomena were insignificant.

The geological record is likewise imperfect at archiving evidence for all potential vegetation-induced causes for meandering. For example, in explaining observed shifts in secular characteristics, Davies et al. (2011) suggested ‘By the Upper Devonian, fluvial deposits commonly contain fossil trees and large mainstem meandering channels with lateral accretion sets, indicating that rooted vegetation stabilized channels’ (Davies et al. 2011, p. 220) and Davies and Gibling (2011) suggested

In a pristine alluvial system, the high bank stability necessary to allow the preservation of steep-margined, aggradational channels with low width-to-depth ratios is promoted by one or more of the following: vegetative rooting, cohesive clays … or, potentially, permafrost. (Davies and Gibling 2011, p. 631)

These are references only to meander theories in which reduction of the channel aspect ratio by erosion-resistant banks is the cause of meandering, primarily because direct evidence for such has the greatest preservation potential (i.e. deep rooting in the outer bend). Because of the way alluvium accrues, there is limited directly preserved evidence for the inner-bend mechanism of vegetation-induced hydraulic resistance that may have been as influential as or more influential than that found in models and experiments.

It is clear that both traditional facies models and the rock record itself are an imperfect lens through which to gauge the past abundance of meandering rivers: facies models encapsulate a very specific subset of smaller meandering planforms and geological outcrops are biased towards recording the outer banks of smaller rivers. Some recent work has taken these natural imperfections to imply that there is little value in the empirical evidence that they host. For example, Ganti et al. (2019) erected a straw man argument that geological interpretations suggested that all pre-Silurian rivers were characterized by steep palaeoslopes and braided river attributes, and then offered a numerical model to contradict this. They reinterpreted a classic case study of pre-vegetation ‘braided facies’ (Precambrian Applecross Formation, Scotland, e.g. Nicholson et al. 1993; Ielpi and Ghinassi 2015; McMahon and Davies 2020) as a sinuous single channel, and thus inferred uniformity in meandering rivers before and after land plants, by estimating unknowable facets related to the studied geological outcrop (catchment area contributing to streamflow, flow velocity, the preserved fraction of cross-strata, bedform stability fields lacking biocohesion, mean discharge, constant drag coefficient, Precambrian precipitation volume). Despite being apparently quantitative, the potential inaccuracy of the estimates probably resulted in an order of magnitude bias in their estimated palaeoslopes (Long 2021) and significantly overlooked the value in the imperfect empirical evidence of the rock record.

Rather than offering no insights into controls on river meandering, when the innate limitations of the geological record are fully appreciated, the Silurian–Devonian shift in alluvial rock properties has persistent significance: the rise in the abundance of ‘meandering facies’ may suggest that the relative importance of vegetation as a cause was magnified in smaller channels and, as Silurian to early Devonian vegetation types developed very little rooting yet still raised the probability of meandering, it is likely that the role of inner-bend hydraulic resistance was particularly significant, if not directly archived (McMahon and Davies 2018a). The geological record thus implies that vegetation is not a necessary condition but an INUS condition for meandering (Long 2011; Lapôtre et al. 2019; McMahon et al. 2022, 2023), but that the significance of that role may be amplified depending on the type of vegetation and the size of the meandering channel.

As the scale experiments and numerical models with inner-bend floodplain development have demonstrated, it is possible that primitive vegetation types caused meandering by the inner-bend mechanism of hydraulic resistance and surface protection alone (Kleinhans et al. 2018). Moreover, the larger the system is, the less likely it is that rooting penetrates the entire height of the outer-bend bank, which is especially the case in estuaries that may have channels tens of metres deep and are flanked by low saltmarsh vegetation. Nevertheless, the coastal zone is the gateway from which the greening of the continents commenced (Strother and Foster 2021), and numerical models of Paleozoic vegetation types both with and without mud caused considerable changes to estuarine channel behaviour (Brückner et al. 2021). The alternative inner-bend surface protection mechanism by vegetation and/or mud has been identified and does raise the probability of meandering in agreement with the observations.

Scale experiments (Fig. 5, Schumm and Khan 1972; Smith 1998; van Dijk et al. 2013a) and numerical models (Fig. 3, Kleinhans et al. 2018) have shown that mud on the inner bend alone may induce meandering.

The above means that by the Paleozoic there were at least two causal relations between vegetation and mud that were already relevant for meandering channel belts. The first is that Silurian–Early Devonian rootless vegetation introduced novel above-ground baffling effects which considerably increased mud accumulation during the flooding of typically lower energy environments (e.g. Kleinhans et al. 2018; Brückner et al. 2021). Whilst this is not a necessary condition for meandering, it does increase the probability and can help explain the shift in sedimentary properties of the alluvial stratigraphic record, especially when those properties are viewed through the prism of frequency distribution curves (e.g. Davies et al. 2020). The second is that vegetation may have enhanced mud production across sedimentary landscapes by heightening biochemical weathering (Davies et al. 2011; McMahon and Davies 2018b). A more active mud factory may serve as an allogenic forcing, with increased cohesive sediments residing in sediment floodplains (Lewin and Macklin 2003), making a sufficient difference in channel pattern (van Dijk et al. 2013a; Kleinhans et al. 2018). The evolution of land plants has been shown to correlate with an up to 25-fold increase in alluvial mudrock abundance (McMahon and Davies 2018b).

For the first causal relation, it is clear that vegetation is the prior cause of meandering, while the trapping of mud results in a positive feedback on both surface stabilization and plant survival. Regardless of how it was produced, mud transported to areas without ubiquitous vegetation also makes a difference to the channel pattern (Brückner et al. 2021). The second causal relation, of plants producing mud through biochemical weathering, requires that the system be expanded to include the hinterland. Not only is the mud in the river subsystem coming from the hinterland, but also the vegetation must have spread from the river onto the slopes as roots evolved further and soil formation caused more nutrient and water retention. This, and the links to the coastal zone, are a further illustration of how open, complex and multilevel in spatiotemporal scales meandering river systems are, and how many causes and feedback mechanisms are involved depending on the scale of interest.

A number of interwoven causal pathways in river systems lead to various possible forms of meandering (Fig. 7) and various planetary histories. The pluralistic notion of meandering that emerged from the synthesis in this chapter is here summarized from the lowest level (of moving fluid) upwards, where the cause of the periodicity lies, but river meandering would not exist without higher-level processes of floodplain formation. As such, various alternative channel and floodplain mechanisms are INUS conditions for meandering (Fig. 7).

A meandering channel pattern can arise from the growth of instabilities along an initially narrow and deep channel bed and along the channel banks, under the condition that the bed and banks are deformable. Here, the momentum and friction of the flow affect the characteristic wavelength. Momentum depends on the channel width and depth, which, in turn, depend on the upstream discharge, and on the land surface slope, which is a result of physiography and longer-term history of tectonics, erosion and base level. Channel deformation can happen where the threshold for erosion is exceeded by the flow, and where solid matter can deposit in the inner bends. With unhindered bank deformation, bends grow in amplitude and migrate laterally until the water surface slope and flow force have reduced with increasing sinuosity, or until bend cutoff takes place. In the latter, dynamic cases the variability in the horizontal direction of the flow ignites renewed bend growth in the downstream direction. In this mechanism, a periodic pattern arises with a wavelength partly determined by flow momentum. To sustain dynamic meandering, continued perturbations are required, suggesting that meander bend expansion and migration depend not only on floodplain properties but also on upstream channel dynamics (or perhaps other dynamics such as that of vegetation).

So far, these mechanisms are valid for alluvial rivers as well as gullies on mud flats, where inner-bend deposition of mud or continued incision avoids widening, and are similarly valid for streams on ice, where freezing on the inner bend or continued incision could lead to constant stream width. Given enough time, a meandering channel can likewise form by incision in bedrock channels. Without incision, deposition on the inner bends is a necessary condition to maintain a steady channel width, where unimpeded widening would lead to straight and braided patterns. In an initially wide and shallow river, multiple channels and bars would form that together lead to a braided pattern, even if some channels are temporarily curved and migrate by outer-bank erosion. On the other hand, barely or merely locally deformable banks can lead to local, aperiodic bends of which the length can be consistent with the momentum and the width, as in very cohesive settings such as those found in estuaries, peat-filled valleys and small streams with self-formed muddy floodplain or dense vegetation. In these situations, horizontal flow separation or recirculation is often found, which can produce positive feedback on sharp bend growth and counter-point bar formation (Fig. 7).

A meandering river pattern including bars can arise in granular material (of greater density than water) from the aforementioned mechanisms. River meandering can initiate from alternate bars that are hindered in migration and protected against cutoff by mud, vegetation or both (Fig. 7). The wavelength is determined by flow momentum and by gravitational effects on sediment transport, so that bar dimensions depend most strongly on channel width and depth, but are hardly affected by curved flows. Bar formation can also be forced by meander bends of sufficient curvature, in which case alternate bar mechanisms and bar growth by bend flow are combined. Bedforms and sediment properties, notably grain size and sorting, make only a modest difference to bars and bends. Sediment properties depend in part on hinterland lithology and sediment production mechanisms, and bedforms depend on channel dimensions and sediment properties, and on flow intensity and recent history thereof (Fig. 2).

Floodplain formation is not simply a necessary and sufficient condition for river meandering. Meandering river systems, which include floodplains, require not only that the bed and banks of the channel are deformable, but also that the flow is sufficiently focused into the channel and has sufficiently lost the power to incise elsewhere, lest braiding commences. River meandering can initiate from more complex initial conditions such as a braided river, wherein the selection of one channel occurs as others are gradually blocked by floodplain formation processes. The focusing of flow can be caused by filling of the overbank area with sediment fine enough to be transported there, by cohesion caused by mud or another additive, by hydraulic resistance from vegetation, and by combinations of the above. These different causes have meandering as the common effect (Fig. 7). Other phenomena that resemble meandering rivers in aspects of form and process may not need floodplain formation, such as meandering turbidites (Corney et al. 2006; Deptuck et al. 2007) and channels on Titan (Gilliam and Lerman 2016).

Meandering occurs for a certain range of pacing and intensity of floodplain formation relative to the flow strength and some characteristic timescale of channel meandering. The balance of floodplain formation and floodplain destruction determines the difference that the floodplain makes to the channel pattern. For weaker or slower-forming floodplains, braided rivers may develop, whereas for stronger or faster-forming floodplains, tortuous or straight channels with localized bends may form, in addition to anabranching patterns (Makaske 2001; not included in Fig. 7). For sufficient floodplain to prevent chute cutoffs, meander growth is limited mainly by neck cutoffs, in addition to lateral constraints in erodibility. Cutoff through inner bends, which leads to incipient braiding, can already be hindered by thin surface covers.

With thick cohesive floodplain sediments and/or deep rooting relative to channel depth, erosion of outer banks can be more substantially reduced. These causal relations depend on the scale of the rivers and the vegetation species, as well as the climate and geology of the hinterland sourcing the sediments. Animal species can change the effects of floodplains significantly in several directions: bioturbation, sediment compaction through trampling and vegetation removal reduce the floodplain strength and tendency to focus flow into channels, while channel blocking, notably by beavers, may force the flow out of the channel and onto floodplains. All of the above causes and feedbacks, and other factors not mentioned here, such as bidirectional tidal flow, determine not only whether a river can meander, but also determine meander shape (Leuven et al. 2018; Finotello et al. 2020), cutoff dynamics (Zinger et al. 2011; van Dijk et al. 2014) and scroll bar dimension (Strick et al. 2018) (among others).

A wealth of theories, models and empirical generalizations have been developed to explain how and why river meandering differs from other patterns. The platitude that all models are simplifications and idealizations here means that literature from different disciplines and based on different methods describes different kinds of meandering and has other biases and unarticulated background assumptions. For instance, mechanistic theories for meandering and alternate bar patterns are dimensionless so as to be valid for rivers of all sizes, but have to ignore the complex dynamics of river floodplains that render the physical processes in the channel sensitive to scale and ecological aspects. Channel mechanisms can cause, for instance, skewed meanders, but mask the processes of enhanced riparian vegetation settling and mud deposition in the downstream half of bends. Empirical relations between putative causes and their effects on meandering can be confounded by a great many other causal variables that counteract or act in the same direction. Geological reconstructions from patterns such as inclined heterolithic stratification (Fig. 6) exclude the largest rivers (McMahon and Davies 2018a) and may underrepresent floodplain deposits. Laboratory experiments, like numerical models, demonstrate that certain combinations of causes can lead to meandering patterns and associated stratification, but cannot demonstrate that these combinations indeed were the cause of meandering in specific systems and, in the case of experiments, at much larger scales. Contrary to common practice, alternative models of meandering do not need to be contradicted or ignored, but are complementary and can be used to clarify the aspects and background assumptions for which a certain model is valid, and where it is of limited explanatory or predictive value.

A useful conceptual apparatus to consider multiple meandering models and expand the perspective beyond a few putative causes is that of complex open systems. The main argument for considering a meandering river as a complex system is that this allows explicit consideration of scale-dependent causal pathways with multiple causal relations and feedback mechanisms at and between several scale levels of organization, such as the level of flow and sediment transport, the level of bars and large-scale flow structure, the level of channel-floodplain pattern and the biological levels of eco-engineering species. Considering something to be a system means that one can focus on the main mechanisms and boundary conditions, or forcings, that make a difference to the pattern at characteristic, or representative, timescales of its dynamics, while excluding mechanisms operating at much smaller and much larger spatiotemporal scales than the pattern and its dynamics of interest. When a process is changing roughly at the same pace as the bend migration of a river with a certain scale, we include it in the system. Much slower and much faster processes can be backgrounded but not excluded (Wimsatt 1994).

The concept of system is an approximation that simplifies reality, but less so than singular causal relations. A system can be controlled and manipulated after its construction at the laboratory scale and in analytical and numerical models. Sometimes control is offered if a large change triggers a cascade of effects through the system, in the downstream or upstream direction, or across the physical and biological levels of the system. The mechanisms and boundary conditions that are backgrounded are causes that may be important. The intricacies of small-scale turbulence might be ignored as noise but are underlying friction and flow structures that need to be accounted for in models, at least in parameterized form. The large-scale effects of hinterland characteristics on the boundary conditions, including discharge variations, sediment production and co-evolution of species and landforms, might be considered constant on the characteristic timescales of a river pattern, but cascade through the river system when ongoing or externally imposed changes occur.

There is a cyclic aspect to system construction: manipulation allows the tracing of causal pathways of mechanisms and the identification of which factors make a difference to the pattern, but the construction of a system already entails a choice of foregrounding and backgrounding of causal factors. This choice is in part determined by which aspects of what system are of interest. Consequently, the application of some limited notion of one kind of meandering to other situations poses the risk of ignoring backgrounded causal relations that make a difference in other situations, especially where different disciplines implicitly put different causes in the foreground and background. This is particularly problematic for conditions that no longer exist on Earth or Mars. Whether the same physical processes were in operation back then, as the simplistic uniformitarian stance goes, is irrelevant in view of the importance of higher-level mechanisms of floodplain formation. The same ecological processes and interactions with the physical processes were clearly not in operation as species evolved and became more effective eco-engineers operating at larger scales.

• River meandering is not covered by a single theory but can arise through several parallel causal pathways and from INUS conditions (alternative sets of causes producing similar meandering patterns).

• Bar and bend theories based on the interaction between flow and channel sediment provide the mechanistic cause–effect relations that effectively explain channel- and bar-related aspects of meandering and their deposits. Data on rivers confirm that the theoretical causes make a difference to the meandering pattern, from the size of experimental meandering rivers to the largest meandering rivers on Earth and also in tidal environments.

• Meandering arises from multiple (alternative) interactions between meandering channels and floodplains with cohesive sediment and/or vegetation. The importance of these interactions depends on the scale of a system in relation to the resistance of floodplain deposits and of the vegetation to the flow, and depends on the scale of floodplain thickness and vegetation rooting relative to the dimensions of the river.

• The biogeomorphic river system is open and complex in the sense that its dynamics and temporary state are sensitive to initial and boundary conditions. On the timescale of bend growth and cutoff, the supply of mud and/or distribution of vegetation can make a significant difference to the rate and pattern of floodplain erosion and formation. Furthermore, the frequency and amplitude of (upstream) perturbations make a difference in the meandering dynamics. On the timescale of tectonic activity, hinterland erosion and plant evolution, the interactions between vegetation and rock are part of the complex system.

• Traditional facies models and the rock record provide an imperfect window through which to study ancient meanders. When utilized in conjunction with mechanisms identified in geomorphology and experiments, even if poorly preservable, studies of ancient geological strata provide a great opportunity to assess river functioning in the absence of particular causes and mechanisms perhaps ubiquitous in modern systems.

• Overgeneralization or overnarrowing of the term ‘meandering’ can only be avoided by elucidating the aspects of a river meandering notion in terms of the assumed cause–effect relations, INUS conditions, systemic feedbacks, backgrounded relations and historicity.

• Explicating the concepts, methodologies and underlying notions of causality and complex systems clarifies how different approaches and perspectives are complementary and leads to a comprehensive understanding of river patterns.

Gary Parker and an anonymous reviewer are gratefully acknowledged for critical reviews.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

MGK: conceptualization (lead), funding acquisition (lead), visualization (equal), writing – original draft (lead); WJM: visualization (equal), writing – review & editing (equal); NSD: writing – review & editing (supporting).

MK and WM were funded by the European Research Council Consolidator grant 647570 to MK. MK developed part of the manuscript in a 2019–20 fellowship at the Netherlands Institute for Advanced Study in the Humanities and Social Sciences (NIAS-KNAW), Amsterdam, The Netherlands.

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.