The Impact of Climate Change on River Alternate Bars

Climate change is expected to alter the distribution of flow discharge in rivers worldwide. We study the impact of climate‐driven flow changes on the shape of riverbed, and specifically on alternate bars, large deposits of gravel/sand that often form in rivers. We consider the illustrative example of the Alpine Rhine River, showing two nearby reaches with similar hydro‐morphological characteristics, but different channel width. Hydrological projections are obtained from literature, while the evolution of alternate bars is predicted through a novel, semi‐analytical model. Results show a remarkably different behavior of the two reaches: the upstream one, being wide enough for a full development of alternate bars, is resistant to flow alterations; the downstream reach, whose width is close to threshold conditions, is highly susceptible to future changes, showing a strong tendency to increase bar prominence. These findings reflect a general tendency of near‐threshold geomorphic systems to be vulnerable to anthropic stressors.

Downstream-migrating alternate bars often form in straightened river reaches as a result of an autogenic instability mechanism (Colombini et al., 1987;Parker, 1976;Redolfi, 2021), frequently resulting in bed elevation changes of the order of several meters. Therefore, they significantly impact river's navigation, produce intense erosion along banks and instream structures, and locally increase the flooding risk (Claude et al., 2014). However, they also provide important ecosystem functions, because their morphological diversity structures the physical habitat template for a number of aquatic and riparian species (Petts et al., 2000;Tockner et al., 2006), besides creating a valuable environment for recreational activities (Basak et al., 2021).
In this paper we take the Alpine stretch of the Rhine River as a case study to investigate the expected impact of climate change on the morphodynamical response of river bars. Specifically, we consider two consecutive channelized reaches with similar slope, bed material and hydrological regime, but with different channel width. The alteration of flow regime is estimated via existing hydrological projections from the Hydro-CH2018 project (Muelchi et al., 2021), while the associated variations of bar response are modeled by means of a custom-made semi-analytical morphodynamic model.

Study Site
Our study site is located in the so-called Alpine Rhine, the segment of the Rhine River flowing from the Swiss Alps to the Lake of Constance, whose subbasin covers an area of 6,123 km 2 among Liechtenstein, Austria and Switzerland. The flow regime is pluvio-nival, with significant snowmelt in spring and summer, and larger floods mainly occurring in autumn. The river segment was fully channelized in the last two centuries, by straightening the river course and reducing the width of the fluvial corridor, which unavoidably led to a drastic simplification of the original complex multi-thread morphology (Adami et al., 2016).
We specifically focus on two nearby reaches located upstream and downstream the confluence of the Ill River near Eichenwies (see Figure S2 in Supporting Information S1). The analysis of expected bar trajectories in the latter reach seems particularly relevant when considering the ongoing international project "Rhesi-Rhein, Erholung und Sicherhei" (Mähr et al., 2021), which aims at restoring a near-natural dynamics in the 26 km long stretch downstream the confluence of the Ill tributary by an extensive channel widening intervention, designed to increase the discharge capacity from the current 3,100 m 3 s −1 up to at least 4,300 m 3 s −1 .
Discharge data recorded at intervals of 10 min are available from the Swiss station of Diepoldsau from 1984 to 2010. The two reaches have similar channel slope (0.13% and 0.10%, respectively) and nearly the same median grain size (25 mm), and are subject to a similar hydrological regime, despite the flow contribution from the tributary Ill River, which increases the two-years discharge from 1,075 to 1,231 m 3 s −1 . However, the downstream reach is significantly narrower, showing a width reduction from 106 to 63 m.
The examined reaches are representative of two diverse morphological responses to channelization, the upstream one showing prominent, downstream-migrating alternate bars (Adami et al., 2016;Jaeggi, 1984), while the downstream one barely displaying low-relief bars. Based on the methodology recently proposed by Carlin et al. (2021) to predict the long-term response of bar topography to the hydrological regime, such different behavior can be explained in terms of the balance between the respective effectiveness of bar-forming and bar-suppressing flow events, which strongly depends on channel width.

Hydrological Effect of Climate Change
Hydrological scenarios are based on the projections made available by the project Hydro-CH2018, which provides data of the expected monthly discharge in the entire Swiss fluvial network for a reference period and for three future scenarios, depending on the representative concentration pathways of greenhouse gases RCP2.6, RCP4.5, RCP8.5 defined by IPCC (Moss et al., 2010;van Vuuren et al., 2011). These projections are based on the semi-distributed hydrological model PREVAH (Viviroli et al., 2009), which includes various sub-models to account for hydrologically relevant processes such as snowmelt, glacimelt, soil moisture, evapotranspiration, runoff, and baseflow generation (see Muelchi et al., 2021).
Monthly averaged runoff for the periods 2020-2049, 2045-2074, 2070-2099 are then compared to simulated values during the reference period (1981-2010) to compute monthly change factors for each representative concentration pathway. To avoid unphysical discontinuities between different months, the change factors c f (t) are then calculated at the sub-daily scale through linear interpolation, obtaining the annual trends reported in Figure  S1 in Supporting Information S1. In general, the model suggests a clear net increase of flowing discharge in winter and a reduction in summer. This is in line with the projections obtained by Middelkoop et al. (2001) and Brunner et al. (2019), who associated the above effects to the expected decrease in snow accumulation in Alpine catchments.
A classic multiplicative delta change method (Hay et al., 2000;Räty et al., 2014) is then applied to compute representative future flow series from historical records. Specifically, flow measurements from the reference period 1984-2010 are multiplied by the change factors to obtain projected hydrographs. Application of this method can be questionable for projecting extreme events statistics, as their variations can differ significantly from those of the monthly averaged values. However, this method seems appropriate for our specific purpose, considering that bar response mainly results from the cumulative work of moderate flow events, rather than from rare extreme floods (Carlin et al., 2021).

The Semi-Analytical Bar Evolution Model
Different modeling approaches are possible to simulate the adaptation of alternate bars to climate change scenarios. However, numerical morphodynamical models (e.g., Cordier et al., 2019;Nelson et al., 2015;Nicholas, 2010) remain computationally expensive, therefore unsuitable for modeling riverbed evolution on the scale of decades, especially when considering multiple scenarios. By comparison, existing analytical approaches are meant to model the bed response to individual flow events (Tubino, 1991) or to periodic flow variations Hall, 2004), rather than the long-term bar response to the complex hydrological regime. Therefore, in this work we adopt a novel model that is based on a semi-analytical implementation of the weakly nonlinear solution by Colombini et al. (1987) (hereinafter CST).
The CST model allows for computing the evolution in time of the bar amplitude through the following differential equation: where t is time and γ 1 , γ 2 are coefficients that depend on flow discharge and channel characteristics. The amplitude A is technically defined as the difference between the maximum and the mean value of the first, double-sinusoidal Fourier component of bed topography (see Redolfi et al., 2020). A direct relationship between the amplitude A and the bar height ( Figure S3 in Supporting Information S1) can be obtained from the CST model through the computation of the higher-order Fourier components that are needed to fully reconstruct the bed topography.
The original CST formulation was meant to model bar evolution and equilibrium bar characteristics for a given discharge value (steady flow conditions), in which case the γ coefficients are constant and Equation 1 can be analytically solved. The main implication is that bar development crucially depends on whether the flow discharge Q is larger or smaller than a critical value (Q cr ): in the former case the coefficient γ 1 , which represents the linear growth rate, is negative and bar amplitude decays in time (bar-suppressing conditions); conversely when Q < Q cr the coefficient γ 1 is positive and the amplitude grows until it reaches an equilibrium value (bar-forming conditions). The ability of CST model to predict the formation of migrating bars and to provide an estimate of their equilibrium height has been tested based on comparison with a large number of flume experiments (Colombini et al., 1987;Redolfi, 2021).
In the present work we seek an extension of the model to unsteady flow conditions by simply allowing the γ coefficients to vary in time, depending on current discharge stage. From a computational point of view, Equation 1 is therefore integrated by discretizing the hydrograph at equally spaced time steps, and computing the amplitude evolution by a simple explicit Euler method. Based on experimental observations , our model also assumes that bars are morphologically inactive at flow stages that are not sufficient to completely submerge bar topography, when the CST assumption of fully wet domain is no longer valid.

Results
The time evolution of bar amplitude in the two study reaches predicted by the model for the reference period 1984-2010 is illustrated in Figure 1, which highlights the different behavior of the two consecutive reaches.
Specifically, for the upstream reach the model predicts well-developed alternate bars with a mean amplitude of 1.4 m, which slightly varies during flood events. Large flows tend to reduce bar height, which then recovers during the receding phase of the same flood, or during successive smaller floods. Conversely, the downstream reach manifests frequent shifts between a state of well developed bars and flat-bed conditions, which results in a reduced mean amplitude of 0.7 m. This is symptomatic of a near-critical situation, in which river's conditions frequently shift from bar-forming to bar-suppressing, mainly depending on water discharge. Critical discharge is indeed equal to Q cr = 467 m 3 s −1 , a value which is easily exceeded during moderate floods.
The near-critical situation of the downstream reach can be assessed by evaluating the parameter P form introduced by Carlin et al. (2021). The definition of this parameter is founded on the long-established idea (Andrews, 1980;Wolman & Miller, 1960) that the relevance of the different flow stages in determining the long-term river evolution depends on the product between their frequency of occurrence and their effectiveness to produce morphological alterations. For the specific case of alternate bars a suitable measure of the effectiveness is the bar growth rate γ 1 (Q), which is then multiplied by the flow probability density function to obtain a weighted frequency f(Q). Integrating over all bar-forming (Q < Q cr , shaded area of Figure 1) and bar-suppressing (Q > Q cr ) stages gives the cumulative weighted frequencies f BF and f BS , which are finally combined to obtain the relative probability of bar-forming conditions: Therefore, P form = 50% represents a threshold condition in which bar-forming and bar-suppressing stages balance each other. This is almost the case of the downstream reach, for which a value P form = 62% is found, indicating a near-threshold condition. By comparison, in the upstream reach the critical discharge (Q cr = 1,493 m 3 s −1 ) is rarely exceeded, which makes bar-suppressing conditions very unlikely (P form = 99.9%).
To explore the possible future bar trajectories we first consider the intermediate scenario set by RCP4.5 and the medium-term period 2045-2074, and we compute the bar evolution in the two reaches using the projected flow series. Changes in the frequency distribution of flow discharge are illustrated in Figure 2 (upper panels), which reveals a similar effect on the upstream and the downstream reaches, both showing an increased frequency of intermediate discharge values, a reduced occurrence of moderately high flows, and a net decrease of the mean discharge of about 8%. Conversely, changes in the simulated bar amplitude distribution (lower panels) are markedly different in the two reaches: in the upstream one the distribution does not show any systematic variation, so that the mean amplitude remains essentially unchanged, while in the downstream reach the frequency of flat-bar states strongly reduces, and periods when bars are well developed (A > 1 m) become more frequent. As a result in the upstream reach all bar metrics remain basically unchanged, whereas in the downstream part significant alterations are expected. Specifically, for the downstream reach our analysis suggests a 35% increase of the mean bar amplitude (from 0.70 to 0.92 m) and a substantial reduction of its variability, as highlighted by the 45% decrease of the coefficient of variation. Moreover, the reduction of flat-bed states is revealed by the increase of the bar presence metric from 51.6% to 72.8%. Such higher propensity to form bars can be again explained in terms of the parameter P form , which is expected to increase up to 79.1%.
A more complete overview of the expected bar trajectory depending on the emission scenario is illustrated in Figure 3, which confirms the general increase of mean bar amplitude, bar presence, and P form , and the opposite trend of the coefficient of variation c v . For the emission scenario RCP4.5 these changes progressively develop in time, though the metrics tend to reach a plateau toward the end of the century (period 2070-2099). By comparison, the pathway RCP2.6 leads to similar changes for the period 2020-2049, while a tendency to recover is expected for the second part of the century. Conversely, the scenario RCP8.5 leads to a strong, continuous morphological change, eventually leading to well developed bars (markedly increased bar amplitude, bar presence and P form , strongly reduced c v ) for the period 2070-2099. Figure 3 also provides an estimate of the degree of uncertainty of bar response, computed by considering the extreme outcomes of the hydrological projections. This preliminary assessment highlights once again the differing response of the two reaches: while a significant degree of uncertainty appears in future bar trajectories for the downstream reach, especially when considering the most optimistic scenario RCP2.6, the future trend of the wider upstream reach seems much more predictable, being essentially unaffected by hydrological alterations.

Bar Sensitivity and River Characteristics
The relevance of the above results in a broader context can be explored by considering cases with different channel slope, width, grain size, subject to a synthetic flow series that contains the essential information about mean flow and its variability. Specifically, we consider a Compound Poisson Process (CPP) (e.g., Bertagni et al., 2018;Weiss, 1977), according to which the streamflow is modeled as a sequence of exponentially distributed random jumps, followed by an exponential flow decay. Such stochastic process is fully characterized by three independent parameters, namely the mean discharge , the coefficient of variation of the flow series c v and the characteristic time of flow decay τ, as detailed in Text S2 in Supporting Information S1.
For each combination of parameters, we consider a reference and a modified scenario, the latter constituted by the simplest possible hydrological alteration given by a constant change factor (Q cc = 0.9 Q ref ). Modeling longterm bar evolution in the two scenarios allows for computing bar sensitivity, which is measured through the parameter: where and indicate the mean bar amplitude in the reference and the modified scenario, respectively.
This enables for analyzing how bar sensitivity depends on the distance between the channel width W and the threshold value W th , which is defined as minimum width for which alternate bars are expected to form, corresponding to the condition P form = 50%. Such threshold width can be uniquely defined as a function of flow probability density function, channel slope and grain size (Carlin et al., 2021), as illustrated in Figure S5 in Supporting Information S1. Analysis of bar sensitivity reveals the primary role played by the threshold width in controlling the bar response. Specifically, results illustrated in Figure 4 show that for any combination of river parameters the bar sensitivity ΔA increases when channel width approaches W th , with the other parameters playing a secondary role.

Discussion
Taking advantage of state-of-the art hydrological projections, we propose a semi-analytical morphodynamic model for investigating the possible effects of climate change on development of large alternating bars in channelized rivers. Results reveal a markedly different response of alternate bars to similar climate forcings, depending on the distance between the average hydro-morphological conditions and the critical discharge Q cr that discriminates favorable conditions for bar formation (Q < Q cr ) from those in which bars tend to be suppressed (Q > Q cr ).
The sharp dependence of the critical discharge on channel width (Carlin et al., 2021), which results from the fact that the suppressing effect of gravitational pull on bedload transport is much more effective in relatively narrow channels (Redolfi, 2021), implies that the response of alternate bars to climate change strongly depends on the width of the channelization. In relatively wide reaches, where the critical discharge is so high that it is rarely exceeded, bars are currently well-developed and are expected to easily absorb climatic alterations. Conversely, in reaches that are close to the geomorphic threshold set by the channel width W th , bar-forming and bar-suppressing flow events are nearly balanced, so that relatively small hydrological changes are sufficient to produce long-term alterations of bar dynamics.
The semi-analytical model we propose, based on existing weakly nonlinear solutions of shallow water and sediment continuity equations, contains all the key ingredients needed to capture the overall evolution of river alternate bars in channelized reaches, including the sharp variations of bar properties in near-critical conditions and the dependence of bar growth rate on flow discharge (Carlin et al., 2021). If compared with alternative numerical approaches, based on fully nonlinear solutions (e.g., Crosato et al., 2012;Defina, 2003;Qian et al., 2017), our model provides a parsimonious, computationally efficient strategy to explore reach-scale bar trajectories driven by long-term hydrological changes.
This work constitutes a precise example of application of the concept of river sensitivity, expressed as the likelihood that a given change in the key controls will produce a sensible, recognizable and persistent response (Brunsden & Thornes, 1979;Fryirs, 2017). In this perspective, our preliminary uncertainty analysis also suggests that highly sensitive cases are not only characterized by more intense morphological changes, but also by a stronger dependence on the unavoidable uncertainties in the modeling chain. The high sensitivity of river bars dynamics when channel width is close to W th represents a noteworthy example of a more general characteristic of near-threshold geomorphic systems to be fragile and susceptible to external disturbances, potentially manifesting rapid shifts and transitions, in contrast to systems that being far from thresholds are likely to maintain their physical attributes in the long term (Phillips, 2006(Phillips, , 2009Schumm, 1973). Notably, the role of the threshold in controlling the average, long-term bed response is rather sharp, despite the fact that critical conditions are frequently crossed due to flow variability. This is because bar evolution depends on the cumulative effect of flow events, so that an even marginal dominance of bar-forming or bar-suppressing conditions can result in markedly different long-term trajectories. In real conditions, different factors (e.g., channel curvature, width and bed-texture variations) can contribute to a more gradual bar response. However, the distance from threshold conditions is still expected to play a primary role, as it ultimately represents the key parameter that controls the long-term riverbed stability.

Relevance for River Widening Interventions
There is nowadays an increasing tendency to promote ecosystem-based river restoration practices, aimed at restoring river functionality and mitigating the effects of climate change by enhancing instream habitat diversity (Rohde et al., 2005;Wohl et al., 2015). This is often achieved by giving more space to the river, by dismantling artificial embankments or by increasing their distance. In this perspective, an assessment of the river response in a context of changing climate is essential for a successful and cost-effective implementation of this kind of intervention (Belletti et al., 2015). Our model can be regarded as an exploratory tool for assessing the possible geomorphic trajectories of rivers depending on hydro-morphological conditions and on the degree of channel widening, providing a support for the design of more detailed and sophisticated numerical or physical models (e.g., Duró et al., 2016;Rachelly et al., 2021). For example, our results for the Alpine Rhine River suggest that the above-mentioned river widening project "Rhesi" is likely to decrease the sensitivity of the river morphology to climate-driven alterations of the flow regime.

Limitations and Future Perspectives
Despite being derived for the specific case of alternate bars, our approach can be easily extended to model the dynamics of central or multiple-row bars, by considering higher-order modes in the Fourier representation of the bed topography (e.g., Tubino et al., 1999). Nonetheless, more investigation would be needed to confidently apply this approach, considering possible uncertainties in: (a) assessing the dominant bar mode (Crosato & Mosselman, 2009 under conditions of unsteady flow, when transitions among different regimes are possible (e.g., Rodrigues et al., 2015); (b) modeling the amplitude evolution of central and multiple-row bars. In this perspective, specifically designed numerical simulations would allow for directly addressing these points in future studies.
It is worth highlighting that the morphodynamic evolution scenario described in this work may change when the river is not channelized but laterally unconfined, which often generates complex multi-thread patterns (Ashmore, 2013;Garcia et al., 2015). In these cases, the river morphology is expected to be less sensitive to discharge variations, since part of them can be absorbed by a lateral flow expansion during floods (Egozi & Ashmore, 2009;Redolfi et al., 2016). Nevertheless, climate-driven flow changes may still significantly impact the geomorphic response of these river systems, as suggested by Guerrero et al. (2013). Therefore, further research is needed to assess future trajectories, especially considering the interaction between hydro-morphological processes and vegetation (e.g., Calvani et al., 2022).
Finally, our model neglects the effect of the alteration of the sediment supply, relying on the fact that readjusting river slope and bed composition involves moving large amounts of sediment over long distances and typically requires long geomorphic time. To fill this gap, the present approach could be hierarchically nested on large scale one-dimensional models (e.g., Gomez et al., 2009), as needed to estimate climatic-driven variations of slope and bed texture.

Conclusions
The development of a novel, semi-analytical model for the long-term evolution of river alternate bars provides a tool for assessing the morphological response of channelized rivers to projected hydrological alterations due to climate change. Application to two nearby reaches of the Alpine Rhine River with similar hydrological and sedimentological characteristics, but distinct channel width, reveals a markedly different response, with the wider reach appearing resistant to flow alterations, and the narrower one showing a significant increase of bar height magnitude and variability. Analysis of bar sensitivity in a wide range of parameters reveals that this contrasting behavior depends on the distance from the key threshold represented by the minimum width that allows the formation of bars. From a river management perspective, this implies that river channels that are sufficiently wide are almost unaffected by climatic alterations, being capable of preserving their physical characteristics in the long term. Finally, our results can be regarded as a noteworthy example of a more general property of near-threshold geomorphic systems to be highly sensitive to external disturbances.

Data Availability Statement
MATLAB simulation code and input data are made publicly available at https://doi.org/10.5281/zenodo.7422372.