The formation of alternate bars in straightened river reaches represents a fundamental process of river morphodynamics that has received great attention in the last decades. It is well-established that migrating alternate bars arise from an autogenic instability mechanism occurring when the channel width-to-depth ratio is sufficiently large. While several empirical and theoretical relations are available for predicting how bar height and length depend on the key dimensionless parameters, there is a lack of direct, quantitative information about the dependence of bar properties on flow discharge. We performed a series of experiments in a long, mobile-bed flume with fixed and straight banks at different discharges. The self-formed bed topography was surveyed, different metrics were analyzed to obtain quantitative information about bar height and shape, and results were interpreted in the light of existing theoretical models. The analysis reveals that the shape of alternate bars highly depends on their formative discharge, with remarkable variations in the harmonic composition and a strong decreasing trend of the skewness of the bed elevation. Similarly, the height of alternate bars clearly decreases with the water discharge, in quantitative agreement with theoretical predictions. However, the disappearance of bars when discharge exceeds a critical threshold is not as sharp as expected due to the formation of so-called “diagonal bars”. This work provides basic information for modeling and interpreting short-term morphological variations during individual flood events and long-term trajectories due to alterations of the hydrological regime.

Alternate bars are large-scale bedforms characterized by a repetitive sequence of scour holes and depositional diagonal
fronts with longitudinal spacing on the order of several channel widths, which are observed in both sand and
gravel bed rivers

A number of studies

However, when the width-to-depth ratio is smaller than the threshold value, the equilibrium bed configuration is not
necessarily planar, as other bed features may result from different instability mechanisms, such as short, shallow,
and fast-migrating three-dimensional bedforms, usually called diagonal bars

Under steady flow conditions, free bars attain an equilibrium state, whereby they migrate downstream without
changing their morphology

Nevertheless, there is basically no direct, quantitative analysis regarding how equilibrium properties of alternate bars
depend on water discharge. In particular, very few data exist about the shape of alternate bars, as previous experiments
have mainly focused on bar height, wavelength, and growth rate

Laboratory experiments were conducted in a

Summary data from the laboratory experiments. Channel width, slope, and median grain size are constant and equal to

Laser surveys were processed by removing points falling outside the channel bed and by subtracting the average
longitudinal slope. This allowed for obtaining digital elevation models (DEMs) of the detrended bed elevation. The investigation of the geometric
properties of alternate bars required the identification of individual bar units. To this aim, we applied the widely
accepted definition of bar wavelength as the length between two successive troughs

To facilitate the comparison of the shape of individual bars, spatial coordinates of each bar DEM (

Alternate bars are commonly described in terms of their wavelength, height, and migration rate. Bar wavelength is the distance between consecutive, corresponding points along the flow direction. Bar height is usually defined as the vertical distance between the bottom of the pool and the top of the bar surface, with several method and metrics proposed in the literature. Finally, for freely migrating bars, the migration rate is the speed at which the bar front moves downstream. However, the geometrical properties of bars are not limited to their height and wavelength, as more detailed information about their geometrical shape can be derived by analyzing the bed morphology.

The most intuitive and widely used way to define bar height is the difference between the maximum and minimum
elevation within a bar unit, computed after removing the mean bed slope. Though
different symbols have been used in the literature, we refer to the

The above definitions have a clear physical meaning, as they directly represent the bar height from the crest to the
trough. However, being based on extreme elevation values, such metrics are sensitive to outliers and measurement errors.
Therefore, it is sometimes convenient to estimate the topographic effect of alternate bars through different metrics,
which measure the “relief” rather than the bar height.
Specifically, the bed relief can be defined through the standard deviation of the elevation distribution,

All of these metrics are first computed for each individual bar and then averaged among all bars formed at the same
discharge. It is important to note that, while

One method to characterize the shape of bars is via the skewness parameter (SK), which measures the asymmetry of the
bed elevation distribution, thus providing information on the relative proportion of high and low areas within a bar.
Riverbed elevation maps often show negative skewness, with deep, narrow channels carved into large, higher-elevation
bars

Being based on the relative frequency of the elevation values, the above metrics, however, do not to provide information
about the spatial arrangement of the bedforms. To obtain synthetic information about the spatial structure of bars, we
analyzed the bed elevation maps through the two-dimensional Fourier transform

The theory of

The model of

The bed topographies obtained under different discharges are illustrated in Fig.

Maps of detrended bed elevation, showing the equilibrium bed morphology for increasing values of discharge. Flow is from top to bottom. The longitudinal scale is compressed for clarity.

A comparison of metrics for bar height and bed relief is presented in Fig.

Mean properties of bars depending on discharge:

Bars are downstream-migrating with a speed of the order of a few millimeters per second. At the lowest discharge, the
migration rate was not measured because of the lack of easily recognizable fronts and the presence of complex patterns
of erosion and deposition. For higher flows, the migration rate gradually increases from almost zero to

The values of the equilibrium bar height predicted by the weakly nonlinear theory are reported in Fig.

Bar height as a function of discharge according to the weakly nonlinear theory of

We note that the weakly nonlinear theory is formally valid near the critical conditions, although the comparison with
experimental data suggests its applicability within a wider range of conditions

Dimensionless bar parameters as a function of the scaled discharge from theory (lines) and experiments (markers).

The theoretical response of bar height to varying flow conditions is then compared with the laboratory data, which gives
the results illustrated in Fig.

From this comparison it is apparent that bars observed at

The analytical model reproduces both the bar height and the bed relief of alternate bars remarkably well (Fig.

In order to filter out the relatively small differences of single bar units, we computed for each discharge value an
ensemble bar shape, defined as the average topography of all the bars formed under the same flow conditions. The
resulting ensemble topographies represented in Fig.

Maps of ensemble bars representing the variation of the average bar topography for increasing values of discharge. Spatial coordinates (

Figure

Skewness of the bed elevation distribution as a function of the scaled discharge. Markers indicate the skewness of the experimental ensemble bars; lines illustrate results from the weakly nonlinear theory, with the dash–dot line referring to the solution obtained by limiting the bar growth to the fully wet condition.

The analysis of the Fourier spectral composition of bed topography provides the amplitude of each component along the
transverse and longitudinal direction. An example is shown in Fig.

Amplitude of the first

Amplitude of the main Fourier components depending on discharge.

We note that the above results are valid in general, regardless of the value of flow discharge. However, relevant
variations of the Fourier spectrum composition occur when changing

Amplitude of the main Fourier components depending on the scaled discharge from theory (lines) and experiments (markers).

To quantify the shape of the bars regardless of their absolute height, we then refer to the relative amplitude of the
Fourier modes given as a proportion of the amplitude

To investigate the overall importance of the

Experimental data reveal that bar height and relief generally decrease with increasing discharge and are therefore
inversely correlated with the sediment transport rate. This finding, which at a first sight may appear counterintuitive,
is a direct consequence of the decrease in channel aspect ratio for progressively higher flows that are typical of
single-thread rivers. This implies that the largest bars tend to develop under moderate flow conditions in which
discharge is high enough to mobilize the bed material and at the same time is sufficiently low with respect to the
critical discharge for bar formation,

Our results reveal that the weakly nonlinear model allows for reproducing the
observed bar height both qualitatively and quantitatively. Despite the calibration of the parameter

The definition of suitable metrics for quantifying variations of the bar shape allows us to highlight how the shape of
alternate bars at equilibrium changes with discharge.
The weakly nonlinear theory of

Experimental observations presented in this study provide detailed information on the relationship between bar
characteristics and discharge, while other relevant channel properties, such as grain size and slope, are kept constant.
Within the tested flow range, bars exhibit a variety of sizes and shapes and pass smoothly from one shape to the other
as discharge increases. On the basis of their geometrical properties and migration rate it is possible to identify four
main types of bars.

At low flows, when the channel aspect ratio is high, alternate bars are very irregular, and the channel tends to
switch to a more complex, wandering morphology. Sediment transport occurs on a limited portion of the bed, and the bed
evolution is not dominated by the downstream migration of bar fronts but rather by lateral erosion and cutoffs. This
kind of bar is associated with conditions in which the top of the bars emerges so that the bed is not fully wet (

At low to intermediate flows, bars are clearly delineated and relief is high. Their transverse shape is highly
asymmetric, with narrow, deep, elongated pools and high, flat bar tops occupying a large proportion of the cross
section so that the elevation along the centerline of the channel is always above the median detrended elevation. The
distribution of elevation is strongly negatively skewed, and the Fourier components

At intermediate to high flows, relief and bar wavelength decrease with increasing discharge, and bar fronts become
curved and oblique. The bed elevation distribution is less skewed, and higher-order components of the Fourier spectrum
become less relevant with respect to the fundamental harmonic

Finally, at high flows diagonal bars form. Despite preserving an alternate shape, diagonal bars are rather different
from alternate bars in terms of both geometrical properties and formation mechanism. The height of these bedforms is
low and largely independent of discharge, and their elevation distribution is almost symmetrical. Diagonal bars are
relatively short (less than five channel widths), with oblique, almost straight fronts that migrate downstream at high
speed. They are observed outside the range of alternate bar formation (i.e., for

The three-dimensional character of the flow field is fundamental for explaining the morphology of diagonal bars.
Specifically, when

From a visual inspection of the topographies illustrated in Fig.

On the basis of theoretical results, it is possible to define an additional threshold value of discharge corresponding
to conditions in which the channel aspect ratio

The checkerboard pattern of the Fourier spectra indicates that both alternate and diagonal bars are purely
alternate in the sense that the elevation map of the upstream half-wavelength is nearly identical to the downstream
half but mirrored along the channel centerline. Note that this does not imply a point symmetry with respect to the
center of the bar (

This particular pattern is intrinsically linked to bar formation mechanisms. To some extent, both alternate and diagonal bars can be considered free bars in the sense that they both arise from an autogenic, three-dimensional instability of the erodible bed. This kind of instability does not break the overall symmetry of the problem; therefore, if a deposition patch tends to form near one bank, a similar feature should appear somewhere else but on the opposite side of the channel. This suggests that if periodic, three-dimensional bedforms develop, they should follow an alternate pattern, at least in an average statistical sense.

From a mathematical point of view, the checkerboard pattern can be explained by considering the fact that free bars tend to
initially appear as a bed deformation having a double sinusoidal shape (

Finally, it is worth noting that the dominance of the even–even and odd–odd modes has an experimental significance, as it indicates that (i) there are no systematic trends associated with channel asymmetries (e.g., product of initial bed leveling), and (ii) random effects resulting from measurement errors, experimental imperfections, or the intrinsic stochasticity of sediment transport processes are not significantly affecting the shape of the ensemble bar.

The laboratory dataset used for this work allowed for the comparison of a number of methods and metrics to characterize
bar height and relief.
Historically, interest in the quantification of bar height arose from the influence of bars on human activities and
interaction with artificial structures

Comparatively, SD and BRI are robust indices that do not depend on extreme values of elevation but on the entire
bed elevation distribution. Moreover, these bed relief metrics can be applied to a range of different morphologies, thus
allowing for comparisons between bars and other bedforms. Since SD and BRI show the same trend observed for

It is also important to note that metrics based on the comparison of elevation values at different longitudinal
positions (i.e.,

We explored how the equilibrium properties of free migrating alternate bars depend on water discharge through a series
of laboratory experiments, wherein width, channel slope, and bed material were kept constant. A proper definition of the
most suitable metrics, the analysis of the experimental results, and the comparison with existing theoretical models
allow us to draw the following conclusions.

The equilibrium bar height generally decreases with increasing discharge. However, at low flows, when bars start emerging from the water surface, an opposite trend is observed, which implies that moderate flows are mainly responsible for the formation of large alternate bedforms.

The shape of alternate bars significantly changes with discharge; relatively low flow conditions are characterized by a high negative skewness of the bed elevation distribution and an important contribution of the higher-order Fourier modes with respect to the fundamental harmonic.

At low discharge, when the width-to-depth ratio is relatively high, the mode-2 Fourier components become increasingly important. However, the channel does not tend to develop regular central bars but rather a bell-shaped distortion of the average cross section, with deposition preferentially occurring near the center of the channel (mid-channel bars) and scour pools mainly located near the banks.

The significant variations of the bar morphology and the associated metrics allow for identifying four main types of bars, which are associated with different flow conditions with respect to the relevant morphodynamic thresholds.

The weakly nonlinear theory allows for a satisfactory prediction of bar height and migration speed, while its capability to reproduce the bar shape is limited to a qualitative analysis. Moreover, limiting the bar growth to the fully wet condition allows for correcting the theoretical predictions at low values of discharge, at which alternate bars tend to emerge from the water surface.

The transition from alternate bar morphology to a plane-bed configuration that is expected when discharge exceeds
the critical threshold

The definition of ensemble bars that represent the average bar topography clearly highlights the purely alternate character of both alternate bars and diagonal bars, which manifests itself as a checkerboard pattern of the Fourier spectrum. In general, our definition of ensemble topography can be used for analyzing any quasi-periodic morphological pattern, such as curvature-driven point bars forming in meandering rivers.

Here we detail the procedure needed to expand the signal in the form of Eq. (

Considering the

Illustration of the grid used to discretize a domain of size

Equation (

Equation (

The Fourier coefficients

A MATLAB code for the computation of the critical and resonant conditions

MR analyzed the data, wrote the paper, and responded to the referee comments. MW designed and performed the experiments, processed data, and wrote part of the paper. MC performed the calculations by means of the weakly nonlinear theory. MT supervised the work. WB designed the experiments and revised the paper. All authors contributed to the interpretation of the results.

The authors declare that they have no conflict of interest.

We thank Chris Paola and Eric Prokocki for the stimulating comments and Marco Colombini for the interesting discussion.

This research has been supported by the Autonomous Province of Bozen-Bolzano (project no. 42, GLORI – Glaciers-to-Rivers Sediment Transfer in Alpine Basins), the Italian Ministry of Education, University and Research (MIUR) in the framework of the “Departments of Excellence” (grant no. L. 232/2016), and the “Agenzia Provinciale per le Risorse Idriche e l'Energia” (APRIE) of the Province of Trento (Italy).

This paper was edited by Paola Passalacqua and reviewed by Christopher Paola and Eric Prokocki.