Free Alternate Bars in Rivers: Key Physical Mechanisms and Simple Formation Criterion

Free alternate bars are large‐scale, downstream‐migrating bedforms characterized by an alternating sequence of three‐dimensional depositional fronts and scour holes that frequently develop in rivers as the result of an intrinsic instability of the erodible bed. Theoretical models based on two‐dimensional shallow water and Exner equations have been successfully employed to capture the bar instability phenomenon, and to estimate bar properties such as height, wavelength and migration rate. However, the mathematical complexity of the problem hampered the understanding of the key physical mechanisms that sustain bar formation. To fill this gap, we considered a simplified version of the equations, based on neglecting the deformation of the free surface, which allows us to: (a) provide the first complete explanation of the bar formation mechanism as the result of a simple bond between variations of the water weight and flow acceleration; (b) derive a simplified, physically based formula for predicting bar formation in a river reach, depending on channel width‐to‐depth ratio, Shields number and relative submergence. Comparison with an unprecedented large set of laboratory experiments reveals that our simplified formula appropriately predicts alternate bar formation in a wide range of conditions. Noteworthy, the hypothesis of negligible free surface effect also implies that bar formation is fully independent of the Froude number. We show that this intriguing property is intimately related to the three‐dimensional nature of river bars, which allows for a gentle lateral deviation of the flow without significant deformation of the water surface.

Mathematical modeling allowed for the identification of the essential processes needed to reproduce free bar formation. They revealed that three-dimensional effects such as helical motion or flow separation are of secondary importance, so that the process of bar formation can be effectively predicted by means of depth-averaged, two-dimensional shallow water and Exner equations. Specifically, linear stability analyses (Callander, 1969;Colombini et al., 1987;Fredsoe, 1978;Parker, 1976) demonstrated that even for a straight channel of constant width the basic, uniform-flow solution is inherently unstable, which leads from the spontaneous formation of long, three-dimensional bed deformations representing free alternate bars. Moreover, these theoretical analyses provided an useful criterion to determine marginal stability conditions, which are mainly controlled by the channel width-todepth ratio. Specifically, bars are expected to form when the width-to-depth ratio exceeds a critical threshold that depends on other river characteristics (primarily relative roughness and Shields number).
Nevertheless, the mathematical complexity of the problem limited the derivation of explicit, physically based expressions for the critical aspect ratio as a function of the controlling parameters, as also recently highlighted by Crosato & Mosselman (2020). As a consequence, application of the theory currently requires either to numerically solve a dispersion relation involving complex numbers, or to rely on plots made available by different authors, with limited possibility to explore the space of parameters and the effect of different transport and friction formulae. A possible alternative is based on empirical criteria proposed in the literature (e.g., Ahmari & Da Silva, 2011;Jaeggi, 1984;Muramoto & Fujita, 1978;Yalin & Da Silva, 2001). Despite being originally formulated in different ways, empirical relations can be re-expressed in terms of the threshold value of the width-to-depth ratio that needs to be exceeded to enable the formation of bars. However, the empirical nature of these criteria makes it difficult to extend predictions out of the set of conditions for which they are derived. Moreover, empirical relations do not allow for isolating the effect of the individual physical parameters, and to assimilate information that may come from site-specific estimations of hydraulic roughness or sediment transport relations.
More fundamentally, the mathematical complexity highly limited the possibility to provide a satisfactory physical explanation of the mechanism of bar instability. A first tentative explanation was proposed by Einstein & Shen (1964), who suggested that bars may form as a consequence of helical motion, possibly reinforced by the presence of rough banks. However, subsequent models have conclusively demonstrated that neither three-dimensional flow nor rough banks are essential for capturing the bar instability mechanism. More recent explanations (Nelson, 1990;Tubino et al., 1999) are given in terms of the divergence of the flow field around bars, due to a sort of topographic steering. However, the mechanism that produces this flow field has not been clarified. Therefore, a complete physical description of the instability process is essentially missing.
In this study, we consider a simplified version of the governing equations, in order to: (a) derive a simple, explicit expression for predicting bar stability conditions and validate it by means of existing laboratory experiments; (b) provide a physically based explanation of the bar formation mechanism. REDOLFI 10.1029/2021WR030617 3 of 20 The study is organized as follows: in Section 2, we define the governing equations and we specify the fundamental assumptions; in the Results Section 3, we introduce the simple criterion for predicting the formation of free alternate bars, we test it against existing laboratory data, and we provide a physical explanation of the bar formation mechanism; in Section 4, we discuss model hypotheses and associated limitations. Finally, details about the derivation of the explicit expression for the critical aspect ratio are reported in Appendix A.

The Governing Equations
We consider an infinitely long channel, with straight, fixed banks and rectangular cross-section of width W, whose bottom is formed by cohesionless particles with representative (e.g., median) grain size d. We adopt a two-dimensional, mobile-bed, depth-averaged shallow water model (e.g., Colombini et al., 1987;Parker, 1976;Siviglia et al., 2013), which can be written as a nonlinear differential system of four equations in the four dependent variables U, V, D, and H (longitudinal and transverse velocity, water depth and water surface elevation), in the independent variables x, y (planimetric coordinates) and t (time). As sketched in Figure 2, the origin of the Cartesian system of reference is positioned at the right bank, and elevations are calculated with respect to a sloping plane having longitudinal gradient S 0 . Under quasi-steady approximation (i.e., assuming that the flow field adapts instantaneously to variations of the bed topography), the depth-averaged equations that express the conservation of momentum, liquid and solid mass read: where p is the sediment porosity, g is the gravitational acceleration, η = H − D is the bed elevation, and the couples {τ x , τ y } and {qs x , qs y } indicate the components of the shear stress and the sediment transport, respectively.
The set of four differential equations is then completed by specifying closure relationships. Specifically, the two components of bed shear stress are estimated as follows: where c is the dimensionless Chézy coefficient (a function of the relative submergence D/d) and γ q is the angle of the velocity vector ⃖⃖ ⃗ . The components of sediment transport are expressed as: where Δ is the relative submerged weight of the sediment and Φ is the dimensionless sediment flux, which is considered to be a function of the Shields number θ (e.g., Meyer-Peter & Muller, 1948;Parker, 1990). The angle of the sediment transport vector, γ s , is computed by taking into account the deflection exerted by the lateral bed slope, by means of the following expression (e.g., Blondeaux & Seminara, 1985;Engelund, 1981): where r is a dimensionless empirical coefficient (see Baar et al., 2018).
Despite neglecting three-dimensional flow structures, this model has been proven to be able to capture the essential characteristics of river alternate bars, at least in conditions where most of the sediment is transported as bedload (e.g., Blondeaux & Seminara, 1985). Specifically, linear solutions allow for calculating bar formation conditions, while weakly nonlinear and fully nonlinear theories enable for reproducing bar height and to estimate other bar properties.

The Key Hypothesis
The present manuscript is founded on the key hypothesis that the deformation of the free surface due to the incipient formation of bars is negligible. More precisely, we assume that: (a) the pressure term g ∂H/∂x in the Equation of longitudinal momentum 1a, and (b) the variation of H when computing the bed elevation as η = H − D in Equations 1d and 4, are both negligible. It is worth noticing, however, that variations of the free surface elevation are still considered in the Equation of transverse momentum 1b, in which the pressure term g ∂H/∂y can not be disregarded.
This hypothesis has been used to model the formation of forced steady bars in rivers (e.g. Crosato & Mosselman, 2009;Struiksma et al., 1985), and constitutes the basis for the so-called second order models for the evolution of meandering channels (see Camporeale et al., 2007). The appropriateness of adopting this hypothesis for modelling the evolution of free migrating bars is suggested by visual inspection of experimental data, where fluctuations are usually small, even at relatively high values of the Froude number (García & Niño, 1993). Moreover, it is indirectly indicated by the weak dependence of alternate bars on Froude number (Wilkinson et al., 2008), as characteristic of processes where the influence of free surface variations is small.
In the following Section 3, the comparison with the complete model and the validation against experimental data are used to demonstrate the suitability of this key hypothesis for predicting bar stability conditions. Moreover, in Section 4, we will discuss about the physical reasoning of why water surface deformation is negligible for typical hydrodynamic conditions on river bars.

Expression for the Critical Width-to-Depth Ratio
Neglecting the deformation of the free surface elevation allows for deriving an explicit formula for determining the possibility of migrating alternate bars to form, depending on channel characteristics and flow conditions. To this aim, we first need to specify a reference depth D 0 and the associated reference Shields number θ 0 , which is given by the following uniform-flow relationship: Bars formation primarily depends on the channel aspect ratio, which for historical reasons is here defined as half the width-to-depth ratio, namely: Specifically, when the aspect ratio exceeds a critical threshold value (β C ) the initial, plane-bed configuration is unstable, and alternate bars are expected to spontaneously form (Colombini et al., 1987).
A very simple formula for this critical aspect ratio can be obtained by: (a) linearizing the governing equations, (b) considering the first mode of the Fourier expansion of the solution, (c) analyzing the time development of an initially small bed perturbation, (d) determining the set of parameters for which this initial perturbation tends to grow, eventually leading to finite-amplitude alternate bars. Considering that these mathematical procedure is rather standard and straightforward, we prefer avoid cluttering this section with a large number of equations. Therefore, we reported all the mathematical details in Appendix A, here providing only the final result of the linear stability analysis, which gives the following expression: where the empirical coefficient r can be assumed equal to 0.3 and λ is the dimensionless wavenumber, defined as λ = πW/L, with L indicating the bar wavelength. Differently from the complete model of Colombini et al. (1987) our approach does not allow for estimating the bar wavenumber, which needs to be given as an input. However, considering a constant value λ = 0.45 is sufficient to produce accurate results for a range of conditions. The symbol ξ indicates a function on the reference value of the Shields number (see Equation A14b), which depends on the choice of the sediment transport formula. Specifically, considering the sediment transport formula of Parker (1978) it reads: Similarly, the reference dimensionless Chézy coefficient c 0 and the associated c D coefficient (Equation A6b) depend on the choice of the friction formula. Adopting the widely used logarithmic expression (Engelund & Hansen, 1967) gives: where the ratio D 0 /d represents the relative submergence. Alternatively, the friction coefficients can be calculated from the Manning formula as follows: where the Manning coefficient n needs to be estimated on the basis of the bed roughness.

Why Do Free Bars Form? A Physical Explanation
The hypothesis of negligible variations of the water surface elevation allows for a great simplification of the problem, as needed to physically understand the mechanisms that drive the formation and suppression of free alternate bars.

The Bar-Forming Mechanism
We consider the depth-averaged Equation of the streamwise momentum 1a, where we neglect the transverse flux of longitudinal momentum (second term), as appropriate when studying the initial stages of bar development (see Appendix A): By discarding the term related to the water surface deformation (i.e., according to our fundamental hypothesis), the above Equation 11, once multiplied by ρD, reads: which simply states that any imbalance between the longitudinal component of the water weight and the bottom friction necessarily produces a flow acceleration or deceleration.
In plane-bed conditions the flow is uniform, weight and friction keep in balance (i.e., τ x = ρgDS 0 ) and no acceleration/deceleration occur. In this case, the sediment transport is also uniform, so that neither erosion nor deposition appear. Conversely, if a three-dimensional perturbation of the bed is introduced, the flow is no longer uniform. Let us consider for example a deposition bump at one side of the channel (i.e., a three-dimensional bed disturbance), having a length of several times the channel width and an initially small height ( Figure 3). Since the free surface deformation is negligible, the depth over the deposition bump does clearly reduce, and the weight 6 of 20 of the water column decreases. Considering that the friction term does not substantially change until the flow velocity varies (it is actually possible to assume τ x to be constant, as discussed later), the decrease of weight does necessarily produce a flow deceleration (∂U/∂x < 0). This implies a spatial decrease in the sediment flux and an associated deposition, which increases the height of the initial bed disturbance. Similarly, a scour hole would produce an increase of depth and water weight, which in turn would lead to flow acceleration, spatial increase of sediment flux and further erosion. This represents a self-sustained instability mechanism, which ultimately leads to the formation of the large-scale, finite-amplitude bedforms called free bars.
It is worth highlighting that the above-described mechanism is only valid for a three-dimensional bed perturbation, where the flow has enough space to move laterally around the obstacle without significant deformation of the free surface. Conversely, if the bed perturbation was purely two-dimensional, the flow would be obliged to entirely transit over the bedform, and the momentum balance would be affected by the pressure terms associated with the variations of the free surface. In these conditions, the shallow-water-Exner model invariably gives a suppression of the perturbation, which indicates that the basic uniform flow is always stable.

The Bar-Suppressing Mechanism
The main contrasting mechanism is due to the gravitational effect on the direction of the bedload transport: the sediment tends to be deviated by an angle γ s that depends on the lateral slope according to Equation 4. As illustrated in Figure 4 this deviation produces a transverse sediment flux toward the lower part of the cross-section (bar pools). This mechanism tends to suppress three-dimensional bedforms, eventually leading to flat-bed conditions if no other, constructive forces exist.
Specifically, the transverse flux of sediment (qs y ) predicted by Equation 4 is proportional to the lateral slope ∂η/∂y. This represents the characteristic relation of diffusive processes, where the mass flux depends on the gradient, and is directed in the opposite direction (e.g., Crank, 1979). As any diffusive process, the bed adaptation follows a timescale that is proportional to the square of the domain size (i.e., T ∝ W 2 ). For example, considering a purely transverse bed deformation (no variations along the longitudinal direction -no constructive forces) the time needed to attain flat-bed conditions is proportional to the square of the channel width. This indicates that the bar-suppressing mechanism is more intense in relatively narrow channels, which justifies the presence of a Figure 3. Illustration of the physical mechanism that sustains the bar growth. The generic, initially small, three-dimensional deposition bump located near the right bank (see contour lines in the plan view) produces a decrease of the local water depth (D) and an associated reduction of tracting force due to the weight of the water column (ρgDS 0 ). The imbalance between the reduced water weight and the bed friction τ x necessarily produces a flow deceleration (U OUT < U IN ), which induces further deposition, thus producing a self-sustained bar growth.
7 of 20 lower threshold of the channel aspect ratio (β C ). More precisely, this explains why an exponent −1/2 appears in the expression for the critical aspect ratio 7 (i.e., the stability condition depends on the square of β, see also Equation A14a).
In physical terms, this quadratic dependence can be easily understood by considering that channel width has a twofold effect. First, transverse bed gradient and the associated transverse flux of sediment are inversely proportional to the channel width. Second, the volume of sediments that needs to be laterally transferred is proportional to the width itself. Consequently, bed flattening in wider channel needs a larger mass transfer with a lower flux, therefore requiring a much longer time.

When Do Free Bars Form? Results From the Simplified Criterion
The explicit expression for the critical aspect ratio 7 derived above provides a simple criterion for bar formation. Specifically, migrating alternate bars are predicted to form when the aspect ratio β exceeds the critical threshold β C , while in the opposite case plane-bed conditions are expected, despite the possible development of low-relief oblique dunes (e.g., Redolfi et al., 2020) or other kind of small-scale bedform.
As illustrated in Figure 5, the critical aspect ratio initially increases with the Shields number while it tends to slightly decrease when θ 0 exceeds 0.21, value at which the function ξ(θ 0 ) is minimum. Moreover, β C significantly increases for higher values of the relative submergence, which according to Equation 9a are associated with higher values of the Chézy coefficient c 0 . In general, predictions by our simplified expression are very similar to those resulting from the complete model of Colombini et al. (1987). Specifically, the critical aspect ratio shows a maximum relative error of 2.8% (for relatively high θ 0 and the low D 0 /d), which seems acceptable for most applications.
A further simplification can be obtained by neglecting the term 1∕( 2 0 2 ) in Equation 7, which gives a wavelength-independent stability condition. From a physical point-of-view, this means discarding the effect of velocity variations on the bed shear stress τ x . If compared with the complete model of Colombini et al. (1987) this further reduced model leads to a maximum relative error of 4.5% within the range of parameters of Figure 5, which reduces to 3.2% when focusing on Fr < 1 cases only. Ultimately, a maximal simplification arises when considering also c D = 0, which implies assuming spatially invariant bed shear stress (i.e., τ x = const). Though this may appear as an extreme hypothesis, it actually leads to a maximum relative error of about 16% with respect to the complete model, which indicates that variations of the shear stress play a secondary role in the above-described bar instability mechanism. Effect of the gravitational pull on a laterally sloping bed, which produces a downward deviation of the sediment flux ⃖⃖⃖ ⃗ with respect to the flow velocity vector ̂ as illustrated in the plan view (a). As a result, the sediment flux tends to laterally move toward the most depressed areas, as illustrated in the cross-sectional view (b), which tends to flatten the bed. The timescale of the bed adaptation (T) is proportional to the square of the channel width (W) as typical of diffusive processes, which makes the bar-suppressing mechanism much more effective in relatively narrow channels. Migrating alternate bars are expected to form when the channel aspect ratio β = W/(2D 0 ) exceeds the critical threshold β C . The maximum relative error of the simplified model is 2.8%, which reduces to 1.6% when limiting the space of parameters to cases where the Froude number is lower than 1.
As illustrated in Figure 6 our simple formula is able to correctly classify most of the experimental outcomes, as most of the experiments with alternate bars fall in the region β > β C while the remaining cases are often characterized by β < β C . More specifically, 364 experiments (87.5%) are correctly classified, 35 (8.4%) can be designated as "false negative" (bars are observed to form, despite β < β C ) and 17 (4.1%) "false positive" (bars do not develop, despite β > β C ). It is worth highlighting that this result is obtained without any specific calibration of the empirical coefficient r or distinct choice of the sediment transport formula. In this sense, additional information about the sediment transport (e.g., measured transport rate) would enable for specifically calibrating the model parameters for each set of experiments, which is expected to improve the overall accuracy of the predictions.
The capability of Equation 7 to reproduce experimental results is then compared with analogue results from the application of the complete model and of the empirical criteria by Muramoto & Fujita (1978), Jaeggi (1984), Figure 6. Comparison between our bar formation criterion and the data set of laboratory experimental observations. Red triangles indicate conditions at which alternate bars were observed, while blue circles refers to other bed configurations, including plane-bed, diagonal bars, dunes and antidunes. Free alternate bars are expected to form when points fall above the dashed line that indicates the critical aspect ratio β C . The histograms represent the frequency distribution of the experiments depending on Shields number (lower plot) and scaled width-to-depth ratio β/β C (right plot).

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Yalin & Da Silva (2001) and Ahmari & Da Silva (2011), whose expressions are reported in Appendix B. To this aim, we first consider classic indicators of classification performance, namely the accuracy (ACC) and the balanced accuracy (BA) (see Tharwat, 2018), which are defined as follows: where N indicates the number of true positive (TP), true negative (TN), false positive (FP), and false negative (FN) cases, whose sum equals the total number of cases N tot . We considered all the experimental data, except those having a severely limited bed mobility, due to low Shields number (θ 0 < 0.03) or bed armoring. Results reported in Table 1 suggest that our formula gives essentially the same performance as the complete model of Colombini et al. (1987), which is overall better with respect to the other empirical criteria.
The above accuracy indicators are merely based on a binary (bars-no bars) classification but do not take into account the "degree of stability" predicted by the different cases. For example, experiments that are very close to the threshold are expected to be easily misclassified, so that an error in this case is less important that an error occurring in highly stable or unstable conditions. To overcome this limitation, we propose an indicator that accounts for the (logarithmic) distance of the incorrectly classified measurements from the critical threshold: where FP and FN indicate the set of false positive and false negative results, so that the parameter Dev ranges from zero to one, with lower values indicating a good prediction. The observed values reported in Table 1 show that our formula provides similar results as the complete model of Colombini et al. (1987), with significantly less deviation than the other existing criteria.

Discussion
In this study, we provide a novel explanation of the physical mechanism that leads to the spontaneous formation of free alternate bars in rivers. Surprisingly, this mechanism turns out to be extremely simple, to the point that it can be described as an imbalance between water weight and bottom friction, which causes deceleration near the top of bars and consequent further deposition. Specifically, the analysis of the two-dimensional solution of the shallow water model is to some extent simpler than its one-dimensional counterpart, as in the latter pressure terms due to the water surface deformation are rarely negligible. The bar formation is clearly counteracted by the effect of the lateral slope on the sediment transport, which tends to suppress bars (Fredsoe, 1978;Seminara, 2010). In this perspective, our analysis highlights the strong (i.e., quadratic) dependence of this effect on the channel width, which represents the hallmark of diffusive processes.
Neglecting variations of the free surface elevation allows for obtaining an explicit expression for calculating the critical width-to-depth ratio with an error of a few percent with respect to the complete model of Colombini et al. (1987). Comparison with an unprecedented number of laboratory experiments, encompassing more than 400 experimental runs from the existing literature, reveals that our explicit formula enables prediction of the bar formation in the vast majority of cases. Specifically, the resulting accuracy is comparable to that of the complete model and better with respect to existing empirical criteria. It is not our intention here to discuss what is the best criteria, as the answer is likely to depend on the specific objective of the analysis, on the availability of data and on the field of application. However, we find relevant to here highlight the main strengths of physically based expressions, which are directly derived from the equations of Newtonian mechanics through well-defined and testable assumptions. Following this reductionist approach (see Seminara & Bolla Pittaluga, 2012), the effect of all the essential parameters, including those that are normally fixed (e.g., the gravitational acceleration) is  embodied, and can be directly associated with the underlying physical processes. In particular, our derivation allows for clarifying the following effects: • the decrease of the critical aspect ratio for lower values of the relative submergence D 0 /d (i.e., for low values of the Chézy parameter c 0 ) can be mechanically explained by considering that to maintain comparable values of Shields number and water depth on a rougher bed, weight and bottom friction need to be higher (i.e., the slope must be higher). In such conditions, any imbalance between the two terms on the right hand side of Equation 12 is expected to produce a stronger accelerations/decelerations, which reinforces the bar-forming mechanism; • the Shields number shows two distinct and opposite effects. On the one side, increasing θ 0 makes the bar-forming mechanism less effective, as it reduces the sensitivity of the sediment transport to variations of velocity (i.e., the coefficient Φ T , see Appendix A). On the other side, increasing θ 0 weakens the bar-suppressing mechanism, as it reduces the deflection of the sediment transport predicted by Equation 4. While the former effect dominates at moderate values of the Shields number, the latter prevails when θ 0 > 0.21, which explains the nonmonotonic trend of β C appearing in Figure 5; • higher values of the empirical parameter r enhance the bar-suppressing mechanism, as they are associated with a stronger deflection of the sediment transport (see again Equation 4). Therefore, the critical aspect ratio clearly increases with r.
Knowing the effect of all the individual parameters allows for adapting the formula to the specific sediment transport and flow friction conditions, by assimilating information from measurements or antecedent studies. For example, if field calibrated values of the Manning coefficient are available it is possible to bypass Equation 9a, and to directly compute c 0 from the Manning coefficient. This may be particularly important for the design and the interpretation of numerical simulations, as in this case our formula can be adapted to consider the same friction and sediment transport formulas, and exactly the same value of the parameter r.

The Key Hypothesis: Physical Reasoning and Limitations
The appropriateness of neglecting free surface deformation is evident from the comparison between results of the simplified and the complete model illustrated above. However, here we would like to analyze the reason for which this hypothesis can be accepted, depending on the characteristic scales of the problem. A reader who is not interested to deepen this topic can directly jump to Section 4.2.
The validity of this hypothesis for modeling forced bars has been justified by Struiksma et al. (1985) by considering that when the Froude number is small, variations of the free surface elevation are small with respect to variations of the bed elevation. In these conditions, it is possible to introduce the so-called rigid-lid assumption, which allows for computing variations of water depth on the basis of variations of the bed topography. However, this does not explain why variations of the free surface elevation can be neglected from the longitudinal momentum balance 1a, as the term g ∂H/∂x generally remains finite when Fr → 0, representing the pressure gradient that appears under the rigid-lid assumption. For this reason, we found important to further discuss the possibility to neglect this term when modeling both free migrating bars (present paper) and forced (or hybrid) bars (Camporeale et al., 2007;Crosato & Mosselman, 2009).
Here, we show that this simplification is generally valid for the case of three-dimensional bed deformations having a longitudinal scale of several channel widths, as typically the case of all river alternate bars. Though this can be demonstrated by a mathematically rigorous perturbation approach, an analogous result can be found by simply evaluating the order of magnitude of the main terms of the fundamental conservation equations. Specifically, if we denote with ̃ and (̃̃) the order of magnitude of the depth and velocity components, the continuity Equation 1c gives:̃Δ̃∼̃Δ̃, where Δ̃ and Δ̃ are the longitudinal and the transverse scales of variation. Indicating with Λ the ratio Δx/Δy, Equation 15 can be expressed as: which reveals that the magnitude of the transverse velocity decreases with the longitudinal scale.
The Equation of transverse momentum 1b suggests that transverse acceleration and lateral inclination of the free surface have the same order magnitude, namely: where Δ̃ indicates the order of magnitude of the free surface variations. Combining Equations 17 and 16 gives: which implies that the gravitational term in the Equation of longitudinal momentum 1a is negligible when Λ is sufficiently large. The above condition (18) can be equivalently expressed in the Froude number as follows: To test this conclusion we consider a periodic, double-sinusoidal bed deformation of amplitude A and wavelength L (Figure 7a). In this case, the longitudinal and transverse scales of variations (Δ̃ and Δ̃ ) can be quantified as the distance between wave crest and trough (L/2 and W), so that their ratio Λ is given by L/(2W) = π/(2λ). We then computed the two-dimensional flow field by analytically solving Equations 1a-c under the hypothesis of small perturbations (linear analysis), by varying the wavelength of the bed oscillation (L) and the Froude number Fr, keeping all the other flow parameters invariant. Results illustrated in Figure 7 confirm that when increasing the value L/W (i.e., the value of Λ) transverse velocity and variations of the water surface elevation decrease, as predicted by Equations 16 and 19a. Moreover, for characteristic wavelengths of free migrating bars (L/W from 5 to 12, with typical value around 7, corresponding to λ = 0.45) and forced bars (L/W > 12) the complete solution is nearly independent of the Froude number, and is correctly reproduced by the simplified model, which corroborates the hypothesis of negligible free surface deformation.
This explains why alternate bars are essentially independent of the Froude number, to the point that they are weakly sensitive to the transition from sub-to super-critical flow regimes (Wilkinson et al., 2008). In this perspective, it is interesting to notice that this property has been recently observed by Ragno et al. (2021) for bifurcation-confluence loops, where the river splits in two anabranches than then rejoin downstream. This suggests that the weak dependence on the Froude number may represent a rather general, remarkable property of three-dimensional morphodynamic systems, such as multi-thread braided rivers, where the water flow is free to laterally move across bars and among different anabranches.
Similarly, the present analysis justifies why shorter three-dimensional bedforms like oblique dunes or diagonal bars (see Colombini & Stocchino, 2012) are instead significantly influenced by the Froude number. This is also the case of two-dimensional bed deformations, for which the independence of the Froude number is achieved only when the length scale of the bed slope variations is longer than the length of the backwater profiles (i.e., the so-called backwater length), so that the flow inertia is negligible and the morphological evolution is essentially diffusive (e.g., Paola, 2000;Redolfi & Tubino, 2014;Shaw & McElroy, 2016).
Ultimately, this analysis reveals that the model simplification adopted in this manuscript is possible thanks to the peculiar characteristic of bars being long, three dimensional bedforms, which allows the flow to deflect around bars without producing significant deformation of the water surface, even at moderate Froude numbers. For this reason, this hypothesis is usually not satisfied for two-dimensional bed deformations, for which the flow is obliged to surmount the bedforms, thus producing mechanically significant variations of the free surface.

Limitations and Future Perspectives
This study demonstrates that neglecting variations of the free surface elevation allows for a satisfactory prediction of the formation of free bars. However, it is worth highlighting that, differently from the complete linear theories (e.g., Colombini et al., 1987), our model does not enable to determine the bar wavelength, because it predicts a monotonically increasing instability for decreasing bar wavelength. This is clearly related to the fact that, as demonstrated in Section 4.1, the key assumption is not valid for relatively short wavelengths. However, this limitation does not prevent for an accurate prediction of the critical aspect ratio, for two reasons: (a) the wavenumber resulting from complete theories is relatively constant, so that its average value λ = 0.45 can be considered representative (b) the critical aspect ratio resulting from Equation 7 is weakly sensitive to variations of the wavenumber, to the point that even setting λ → ∞ (i.e., neglecting the term 1∕( 2 0 2 ) ) gives an error of a few percent only.
Our expression for the marginal stability condition is meant for predicting the formation of alternate (i.e. first mode) bars only, and does not provide indications about the transition to higher bar modes (i.e., central or multiple-row bars) that is expected in wider channels (Crosato & Mosselman, 2009;Fredsoe, 1978). Our approach could be easily extended to predict the growth rate of higher modes, which would allow us to determine the most unstable bar mode depending on conditions (see Tubino et al., 1999). However, this clearly goes beyond the purpose of the paper.
When applying our formula to rivers, the following question arises: "how to select an appropriate value of dominant, formative discharge that can be adopted to represent the bar response?". Previous works usually rely on either the bankfull discharge (e.g., Ahmari & Da Silva, 2011;Crosato & Mosselman, 2009 or on the discharge with 2-year return period (e.g., Adami et al., 2016), as commonly suggested for reproducing river morphodynamic processes. However, a specific methodology to derive formative conditions for migrating bars has been recently proposed by Carlin et al. (2021), who suggested that free bars are expected to form when the average growth rate, calculated over all the possible discharge states, is positive, namely: where f Q indicates the probability density function of the flow events. In this perspective, our analysis provides all the necessary information for directly computing the bar growth rate Ω as a function of discharge by means of Equation A13.
Finally, the present analysis is limited to conditions where most of the sediment is transported as bedload, as reproducing the effect of suspended load on bar stability requires a more sophisticated model, based on either a non-equilibrium stress-transport relation Federici & Seminara, 2006) or on a fully three-dimensional approach . However, our model allows for qualitatively explaining the increase of bar instability observed in suspension-dominated channels: since the suspended load is substantially not affected by the gravitational pull predicted by Equation 4, the bar-suppressing mechanism is expected to be weaker, which promotes bar formation.

Conclusions
Neglecting the deformation of the water surface in the classic two-dimensional shallow water and Exner model allowed for a considerable simplification of the mathematical description of the process of bar formation, which facilitated the physical understanding of the phenomenon. This led to the following conclusions: • The physical mechanism that leads to a self-sustained development of free migrating bars is surprisingly simple, as it results from an imbalance between water weight and bottom friction. Specifically, if a relatively long, three-dimensional, deposition bump is introduced, water depth and associated weight reduce, which produces a flow deceleration and further deposition. The same but reversed mechanism occurs in three-dimensional pools, where the increase of water depth produces acceleration and further scour. This bar-forming instability process tends to be counteracted by the effect of the gravitational pull on the bed particle transported as bed load. The importance of this bar-suppressing effect increases quadratically when reducing the channel width, which explains why bar formation is strongly discouraged when the channel width-to-depth ratio is low. • An explicit, physically based formula for predicting conditions for the formation of free alternate bars can be derived. Testing based on a very large number of laboratory experiments, suggests that the formula we propose is on average more accurate than existing empirical predictors. Moreover, the physically based derivation of the formula allows for assessing the effect of all the essential parameters that concur in determining bar stability, and it is therefore suitable to be adapted and extended to a wide range of conditions. • The hypothesis of negligible deformation of the water surface is intimately related to two essential characteristics of bars: (a) the three-dimensional structure; (b) the long longitudinal extension, which allow for a gentle deviation of the flow, without significant variations of the water surface elevation. For this reason it does not apply to two-dimensional or comparatively short three-dimensional bedforms, such as two-dimensional dunes or oblique dunes. Ultimately, this hypothesis implies a that the Froude number plays a negligible role in the formation of bars. This suggests that a substantial independence of the Froude parameter may be a general, remarkable property of all morphodynamics systems characterized by a three-dimensional bed topography, such as multi-thread braided rivers.
Considering the simplified shallow water equations, we look for a wavelike solution where spatial variations assume the form of a double sinusoid as illustrated in Figure 7a. Specifically: * 1 =̂ exp[ * + (Ω − ) * ]cos( * ∕2) + . ., where and ̂ are complex coefficients, = √ −1 denotes the imaginary unit, c.c. Indicates the complex conjugate. The real coefficients Ω and ω represent the dimensionless growth rate and angular frequency, while λ is the dimensionless longitudinal wavenumber, defined as λ = πW/L, where L is the bar wavelength.
Substituting Equations A8a,b into the system of linear Equations A7a,b leads to a system of algebraic equations in the unknowns and ̂ that can be expressed in the following matrix form: where the "a" coefficients are defined as in Camporeale et al. (2007), namely: A non-trivial solution of the homogeneous linear system A9 exists when the determinant of the matrix of coefficients vanishes, which gives: whose real part reads: which, substituting the coefficients A10, provides an expression for the bar growth rate Ω, namely: As illustrated in Figure A1, the resulting growth rate monotonically increases with the bar wavenumber λ. This prevents the possibility of determining the most unstable wavenumber, which needs to be provides as an input parameter. Considering the typical value λ = 0.45 the simplified model gives a growth rate that is similar to the maximum growth rate given by the complete model, which suggests its capability to correctly reproduce the formation of free alternate bars.
Marginal stability conditions are found by setting zero growth rate (Ω = 0) in Equation A13, which gives: from which it is easy to derive an explicit expression for the critical aspect ratio β C .

Appendix B: Critical Aspect Ratio According to Empirical Free Bars Predictors
In this section, we re-express existing empirical criteria in terms of the critical width-to-depth ratio, as needed for a direct comparison with our formula 7.

B1. The Criterion of Muramoto & Fujita (1978)
This criterion for the formation of free alternate bars can be written as (see Jaeggi, 1984): Once expressed in terms of the channel aspect ratio, Equation B1 reads: which depends on the relative submergence D 0 /d as illustrated in Figure B1.

B2. The Criterion of Jaeggi (1984)
The bar formation criterion provided Jaeggi (1984) (see their Equation 8), translated in our notation, reads: which can be also rewritten in terms of the channel aspect ratio β as: Despite not allowing for deriving an explicit expression, Equation B4 can be numerically solved to obtain the critical aspect ratio β C .
However, it is worth highlighting that a critical aspect ratio does not always exist. This can be noticed by analyzing the left hand side of the inequality B4, which does not increase monotonically with β but it shows a maximum when: which can be expressed in terms of the relative submergence as follows: For Shields numbers in the range 1 − 6 times θ i , as usually the case of gravel bed rivers at bankfull conditions (Parker, 1978), Equations B7a,b give minimum values of D 0 /d between 181 and 274. For higher relative submergence D/d, Equation B4 is never satisfied, which implies that bars are not expected to form regardless of the value of β (see Figure B1). It is worth noticing, however, that this prediction seems essentially a mathematical artifact, as the empirical formula was derived from observations in conditions of relatively low submergence (D 0 /d < 30).

B3. The Criterion of Yalin & Da Silva (2001)
This criterion is based on the empirical definition of a threshold value of the channel aspect ratio that only depends on the relative submergence (D 0 /d). Specifically, it can be expressed by means on the following piecewise-linear function: with alternate bars expected to form when β > β C .

B4. The Criterion of Ahmari & Da Silva (2011)
This criterion can be regarded as an updated version of Yalin & Da Silva (2001), where the constant aspect ratio for high values of D 0 /d is slightly reduced, and where a third branch of the solution is introduce to consider a decrease of the critical width-to-depth ratio for small values of the relative submergence. Specifically, the authors proposed the following piecewise-linear function: A comparison among the different expressions is illustrated in Figure B1.