Water Resources Research

Simple physics-based predictor for the number of river bars and the transition between meandering and braiding

Authors


Abstract

[1] The number of bars that form in an alluvial channel cross section can be determined from a physics-based linear model for alluvial bed topography. The classical approach defines separators between ranges in which river planform styles with certain numbers of bars are linearly stable and linearly unstable. We propose an alternative method that is easier to apply. Instead of defining separators between stable and unstable conditions for certain river planform styles, the method directly estimates the most likely number of bars. It is based on a demonstration that conditions of zero spatial damping in a linear model for steady bars are representative for the bar mode that develops inside a river channel. We argue that a method based on steady bars is more appropriate for real rivers than a method based on free migrating bars. We verified the method by applying it to several existing rivers at bankfull conditions. The results are good for width-to-depth ratios up to 100 but deteriorate for higher width-to-depth ratios. We explain the deficiencies for large width-to-depth ratios from the linearity of the model. The results show that our method can be used as a reliable predictor for whether reducing or enlarging the width of a river will lead to a meandering, transition, or braided planform.

1. Introduction

[2] River improvement often involves changes in channel width, such as constriction to improve navigability and widening to reduce flooding risk. Parks and restoration projects are often designed with single-thread sinuous rivers [Kondolf and Railsback, 2001; Parker, 2004; Piégay et al., 2006], which is obtained by imposing a new channel width and alignment to the river. Models are developed for “remeandering” of canalized streams [Abad and Garcia, 2006], which consists of removing the existing bank protection works, allowing the river to widen and migrate, albeit with limited freedom.

[3] In all these cases the morphology of the river is likely to change further, at all spatial scales. At the cross-sectional scale, river widening may lead to the formation of alternate or multiple bars, river narrowing to their disappearance. This may produce changes of river planform style at larger scales, as steady bars influence bank erosion and accretion, with consequences for the planimetric development of the river [Olesen, 1984]. Their presence is therefore important for the design of river corridors [Malavoi et al., 2002], because freely meandering rivers wind over a wider area than braided channels.

[4] The ecological condition of a river depends partly on the morphological state. One of the important quantitative indicators of the river morphological state is the number of bars that form in the cross section [Toffolon and Crosato, 2007]. Bars increase morphological diversity and create specific aquatic habitats. The number of bars thus gives ecologists an indication of the size of these habitats as well as of the emergence of islands and beaches during low flows. Bars also affect the size of the navigable channel. All these examples show the importance of being able to assess the effects of river training or restoration works on the number of bars in a river cross section.

[5] The classical approach to determine the number of bars in a channel cross section defines a separator between ranges in which river planform styles with certain numbers of bars are linearly stable or unstable. The stability analyses performed by, among others, Hansen [1967] and Callander [1969] define the conditions that govern the development of free bars in a river channel. They show that multiple bars form at larger width-to-depth ratios than alternate bars. Free bars, either migrating or steady, are channel bed oscillations that occur as a system response to perturbations. They should be distinguished from forced bars, i.e., confined sediment deposits induced by bank line geometry over some distance, such as point bars. Parker [1976] and Fredsøe [1978] related the presence or absence of free bars to the channel planform style.

[6] The linear theory by Seminara and Tubino [1989] defines a marginal stability curve which separates the conditions in which a certain number of bars per cross section (defined by the bar mode, m,Figure 1) grows from the conditions in which the same bar mode decays. The river is supposed to select the bar mode with the fastest growth rate, which is a function of the width-to-depth ratio, the Shields parameter, the sediment grain size, and the particle Reynolds number.

Figure 1.

Typical bed and flow deformation due to the presence of bars: (left) first bar mode (alternate bars) and (right) second bar mode (central bars).

[7] Seminara and Tubino's method to assess the number of bars in the channel cross section is relatively laborious, because each individual river requires computation of its corresponding marginal stability curves. The minima of subsequent marginal stability curves define the width-to-depth ratio ranges pertaining to different bar modes. Our alternative method leads to a single formula for all rivers. We derive it from a reformulation of the linear model for steady bars by Struiksma et al. [1985], which underlies physics-based meander migration models [Crosato, 1987, 2007]. A model for steady bars is more appropriate than a model for free migrating bars, because most real rivers outside the laboratory have curved or otherwise nonuniform bank lines. Such bank lines force the bed topography into a steady pattern. Moreover, steady bars dominate in meandering rivers, since bar migration speed decreases with increasing channel sinuosity until migrating bars transform into steady bars at a certain value of sinuosity [Kinoshita and Miwa, 1994; Tubino and Seminara, 1990]. Bars in the central portion of braided rivers, away from the banks, may be freely migrating, but close to the banks they are steady too. They migrate only if discharges vary or if bank lines move by erosion or accretion. Bar migration inferred from low-flow satellite images of the braided Brahmaputra-Jamuna river in Bangladesh can even be spurious, because the bar does not exist during the intermediate flood when the bed is reworked. The bar is recreated when the flood recedes, forced by the new planform after bank erosion [Delft Hydraulics, 1996].

[8] The novelty of our method is that it directly defines an estimator of the most likely number of bars, instead of a separator between stable and unstable conditions for a certain river and a certain bar mode. The method estimates the number of steady bars that form in a river cross section as a function of river width-to-depth ratio, longitudinal slope, bed roughness and sediment characteristics at bankfull conditions. A particular application is that it predicts whether a river is meandering or braided by assuming that meandering is characterized by at most one bar per cross section (bar mode m ≤ 1.5) and braiding by at least two bars (bar mode m ≥ 2.5). Rivers at the transition between meandering and braiding are characterized by the intermediate range (1.5 < m < 2.5).

[9] We tested the method by applying it to several existing rivers. The method was a good predictor of whether a river is meandering or braided. It was found to perform better than Parker's [1976] method to distinguish between meandering and braiding. Application to real rivers showed that braiding starts at width-to-depth ratios of approximately 50, which lies between Fredsøe's [1978] value of 60 and Rosgen's [1994] value of 40.

[10] The number of bars per cross section was estimated well for width-to-depth ratios up to 100, as the absolute value of the difference between computed and observed bar mode did not exceed one. The performance of the method deteriorated at width-to-depth ratios larger than 100, for which the computed number of bars significantly exceeded the observed one. This can be ascribed to the linearity of the model. Nonlinear terms would have the effect of reducing the number of bars, as multiple bars tend to merge [Fujita, 1989]. Width-to-depth ratios on the order of thousand pertain to anabranched rather than braided rivers.

2. Theory

2.1. Linear Model

[11] Our starting point is the physics-based second-order linear model for river morphology derived by Struiksma et al. [1985]. This model is also at the heart of Crosato's [1987, 2007] physics-based meander migration model, but does not contain the corresponding formulations for channel curvature and bank erosion. The linear equations describe the longitudinal variation of the near-bank deviations of flow velocity and bed topography from the normal flow conditions as a result of upstream disturbances. They are obtained from the steady state 2-D depth-averaged continuity and momentum equations for water motion, a sediment balance equation, a sediment transport formula and an equation for the direction of sediment transport. If the water and sediment motion in a straight channel is perturbed by, for instance, an obstacle, the effects of the perturbation appear as extra terms in the equations. Assuming that all the variables in the equations are given by the sum of a reach-averaged value plus a perturbation term, the main flow velocity and water depth can be written as: u = u0 + u′ and h = h0 + h′, in which u0 and h0 are their reach-averaged values and u′ and h′ the perturbation terms. The perturbations are assumed to have a transverse periodic shape so that they represent the situation of a river channel with bars, e.g.,

equation image
equation image

in which n is the transverse coordinate (positive when pointing to the left bank); U and H are the near-bank values of the main flow velocity and water depth perturbations as a function of the downstream distance, s: U = U(s) and H = H(s). The term kB = equation image is their transverse wave number, with B being the channel width and m being the transverse perturbation mode. The latter is a function of the number of bars in the channel cross section (m = 1 characterizes rivers with alternate bars, m = 2 characterizes rivers with central bars, Figure 1).

[12] After linearization of the momentum and continuity equations for small u′ and h′ with respect to u0 and h0, respectively, the equation for the longitudinal profile of the near-bank velocity excess becomes

equation image

where s is the streamwise coordinate and λW is the streamwise adaptation length for perturbations in the transverse profile of depth-averaged streamwise flow velocity, given by

equation image

in which Cf is the friction factor defined by Cf = equation image, with C = Chézy coefficient for hydraulic resistance and g = acceleration due to gravity. The adaptation length λW characterizes the longitudinal distance needed for the decay of perturbations in the transverse distribution of streamwise flow velocity due to an upstream disturbance, as for instance an asymmetrical constriction or a change in channel curvature.

[13] The equation for the near-bank water depth perturbation is derived from the sediment transport and sediment balance equations:

equation image

in which b is the degree of nonlinearity in the dependence of sediment transport on the flow velocity. It is defined as b = equation image, in which qs0 is the cross-sectionally averaged value of the sediment transport per unit of channel width. The degree of nonlinearity is equal to the exponent in case of a power law dependence qs0u0b and is always larger than 3 [e.g., Mosselman, 2005]. Furthermore, λS is the streamwise adaptation length for perturbations in the cross-sectional river bed profile. It is given by

equation image

where f0) accounts for the effect of gravity on the direction of sediment transport over transverse bed slopes. It satisfies an empirical relation [Talmon et al., 1995]

equation image

where E is a calibration coefficient and θ0 is the reach-averaged value of the Shields parameter. The adaptation length λS characterizes the longitudinal distance needed for the decay of perturbations in the cross-sectional water depth profile (bed topography) due to an upstream disturbance.

[14] Equations (3) and (5) represent first-order relaxation equations for flow and bed topography, respectively. Their right-hand terms show that they are linked. Combination results in a single second-order equation which may exhibit an oscillatory behavior in downstream direction. This equation can be expressed in either H or U (near-bank perturbations of water depth and flow velocity, respectively). The equations in H and U are identical. The equation in H reads

equation image

Depending on the value of the factor in brackets in the second term, the solution is either harmonic or purely exponential. The harmonic solution yields the spatial eigenoscillation of the system, called overshoot [Struiksma et al., 1985] or overdeepening [Parker and Johannesson, 1989]. It represents an oscillating channel bed with non propagating bars [Seminara and Tubino, 1989]. The harmonic solution has the following form:

equation image

where H(0) is the left bank water depth perturbation at the upstream boundary; H(s) is the left bank water depth perturbation at a distance s from the upstream boundary; sP is the spatial lag; 2π/LP is the streamwise wave number and 1/LD is the damping coefficient of the streamwise oscillation. The latter can be derived from substitution of equation (9) into equation (8):

equation image
equation image

The equations having been linearized, the eigenoscillation represents incipient free steady bars rather than fully developed bars. Nonlinear effects could change the number of bars during further morphological development. Therefore the mode m of the eigenoscillation does not necessarily coincide with the mode of fully developed free steady bars in a river cross section (cf. discussion in section 5.3).

[15] Wave number and longitudinal damping of the eigenoscillation (equations (10) and (11)) depend on the ratio between the two adaptation lengths: α = equation image. Substitution of equations (4) and (6) shows this ratio to be a function of the mode, m, and the width-to-depth ratio, β = equation image:

equation image

In rivers with fixed width-to-depth ratio, the damping coefficient of the spatial eigenoscillation increases with increasing mode m, which means that mode m + 1 is damped more strongly (in downstream direction) than mode m, as is readily shown by substituting the expression for α = equation image (equation (12)) into equation (11):

equation image

2.2. Derivation of the Predictor

[16] Incipient bars characterized by mode m can form in the river channel only if the solution of equation (8) falls within the harmonic range, which occurs if

equation image

Assuming that the degree of nonlinearity of the sediment transport formula, b, the friction factor, Cf, and the Shields parameter, θ0, are independent from the bar mode, the harmonic range of incipient bars with higher modes (larger values of m) requires larger with-to-depth ratios. This becomes clear when substituting the expression for α (equation (12)) into equation (14):

equation image

Outside the harmonic range, the damping coefficient can be either positive or negative. For positive damping, the linear river response to disturbances is an exponential decay of the disturbance without periodic bars forming in downstream direction, such as in Figure 2. For negative damping, the linear river response is an exponential growth of the disturbance. Non linear effects limit this growth in the complete model. Mosselman et al. [2006] review physical interpretations of situations with negative damping.

Figure 2.

Exponential downstream decay of perturbation.

[17] Figure 3 shows that, for every bar mode, the point of zero damping (λW/LD = 0) always falls within the harmonic range. For this reason, the condition λW/LD = 0 can be taken to be representative for the formation of m mode incipient bars. In this case,

equation image

Substitution of equation (16) in equation (12) yields the following expression for the characteristic width-to-depth ratio, βm, for m mode bars:

equation image

The characteristic width-to-depth ratio of the second-mode incipient bars (central bars) is twice the characteristic width-to-depth ratio of incipient alternate bars if Cf and θ0 are constant. In reality, both Cf and θ0 depend on the type of river configuration and in particular Cf increases whereas θ0 decreases for increasing m. However, for sake of simplicity, we assume that the factor f0)Cf does not change appreciably if the bar mode changes.

Figure 3.

Upper and lower limits of the harmonic range and zero damping as a function of the degree of nonlinearity of the dependence of sediment transport capacity on flow velocity.

[18] For a given river with width-to-depth ratio β, the mode m of the incipient bars can be determined by inverting equation (17):

equation image

By substituting f0) with 1.7equation image (obtained by imposing E = 0.5 in equation (7) [Struiksma and Crosato, 1989]), considering that Cf = equation image, θ0 = equation image (with Δ = equation image = relative sediment density under water and D50 = sediment median grain size) and applying the Chézy equation u0 = Cequation image (with i being the longitudinal gradient) as well as the continuity equation for water QW = u0h0B, equation (18) can be rewritten as

equation image

The validity of equation (19) is formally restricted to rivers with uniform flow.

[19] We recall that m denotes the mode of nonmigrating free incipient bars. They are sometimes called “forced bars,” despite their origin as an eigenoscillation of the system, because they occur when they are forced by a local perturbation of the flow. Local bars forced by bank line geometry over some distance, such as point bars inside channel bends, do not belong to this category. Point bars can also form outside the harmonic range of solutions.

[20] Assuming that b, θ0 and Cf are independent from bar mode, Figure 4 shows the upper and lower limits of the harmonic range for different bar modes as a function of b and the width-to-depth ratio (on the vertical axis the dimensionless parameter, function of the width-to-depth ratio, F(β) = βequation image, derived from equation (15)). The harmonic range of the m + 1 bar mode overlaps with the harmonic range of the m bar mode. This means that at certain river conditions more than one mode may develop at incipient bar formation. For increasing b the overlap intervals are increasingly smaller.

Figure 4.

Upper and lower limits of the harmonic range for different bar modes as a function of b. The vertical axis gives the function of the width-to-depth ratio F(β) = βequation image.

3. Method of Verification

[21] We tested the predictor against the data from several rivers that are provided by Struiksma and Klaasssen [1988], MacDonald et al. [1992], Delft Hydraulics [1995], Blom [1997], Pitlick and Cress [2002], Pitlick [2002], and Miguel Alfaro [2006]. We used values at bankfull conditions, i.e., at flow conditions with a return period of 1 to 2 years [Leopold and Wolman, 1957; Parker et al., 2007]. The width-to-depth ratio of the morphologically active river bed actually changes with flow stage. It can be larger at low-flow stages than at high-flow stages, which means that higher-mode bars may form at low flows. However, in most rivers the largest amounts of sediment are transported by the highest flows, whereas the sediment transport rates at low-flow stages are marginal. This means that at low flows the channel bed needs a longer time to reach its final configuration and in most cases high-flow conditions interfere before this has occurred. This justifies the use of data at bankfull conditions rather than data at low-flow conditions.

[22] We used the following methods to deduce parameter values from data sets if direct information on these values was missing. First, Pitlick and Cress [2002] do not give values of bankfull discharges or Chézy coefficients, but they do give information on longitudinal slope, channel width, water depth, median sediment grain size and Shields stress at bankfull conditions for several reaches of the Colorado River. For the reach of the Colorado River between rkm 274 and rkm 153, where rkm is kilometers measured along the river axis, the bankfull discharge was derived from Pitlick [2002]. For the other reaches, assuming normal flow, we derived Chézy coefficient, flow velocity and bankfull discharge for individual cross sections using the Chézy equation u0 = Cequation image, the continuity equation for water QW = u0h0B and the definition of Shields stress τ0 = ρgequation image = ρgΔD50θ0. Second, data from Delft Hydraulics [1995] do not always include the value of the Chézy coefficient, but sometimes express hydraulic roughness in terms of Manning's coefficient, n. In those cases we derived the Chézy coefficient using C = equation image. Third, we used satellite images from Google Earth (as available on the Internet from January to March 2008) to assess the planform characteristics, in addition to planform information reported by previous researchers. In this way, we verified and corrected the number of channels for rivers in Pakistan that Delft Hydraulics [1995] had derived originally from low-resolution satellite images. The use of Google Earth was restricted to cases in which the satellite image had been made at bankfull or slightly lower-discharge conditions. This was checked by comparing the (averaged) river width in Google Earth to the known bankfull width. An exact count of channels and bars in braided rivers remains difficult, as it depends on image resolution and stage. Nikora's [1991] and Sapozhnikov and Foufoula-Georgiou's [1996] suggestion that braided rivers are fractal would even render bar counts meaningless. We find, however, that the Google Earth images allowed us to determine the main planform style features in a meaningful way.

[23] We compared the observed bar mode with the bar mode computed using equation (18). Our definition of the bar mode, m, coincides with the definition by Engelund and Skovgaard [1973] and Parker [1976]. However, most of the empirical studies use a “braiding index” (recent summary by Egozi and Ashmore [2008]), often defined as the number of channels at a cross section [e.g., Howard et al., 1970] or the mean number of bars [Germanoski and Schumm, 1993]. In general, the braiding index can be converted easily to m, since the number of channels or bars increases by 0.5 if m increases by 1 (Table 1).

Table 1. Observed Bar Mode and Number of Bars or Channels per River Cross Section
Number of Bars/ChannelsObserved Bar Mode
11
1–22
23
2–34
35
3–46
47
4–58
59
5–610
611
6–712
713
7–814
815
8–916
917

[24] The observed bar mode, m, was determined by counting the number of bars or channels in the cross section, as for instance in Figure 5, and by translating this into the observed bar mode according to Table 1. The observed bar mode was assumed to be less than 1 if only point bars were recognizable.

Figure 5.

Examples of bar modes and corresponding river bed configurations.

[25] The selected river reaches had either rather uniform width or were subdivided into subreaches of uniform width. In particular, the Sutlej River reach from the Islam Barrage to the Chenab River was subdivided into 4 subreaches and the Upper Indus reach from the Chasma Barrage to the Taunsa Barrage into 3 subreaches. The anabranched Upper Indus was studied in two ways, by considering either the width of the entire river belt or the main anabranch alone. In the latter case, the full flow was assumed to be conveyed by the main anabranch as we had no information on the distribution of discharges over the anabranches. The fact that the main anabranch was by far the largest one justified this assumption.

[26] As a first estimate, the degree of nonlinearity of the dependence of sediment transport on flow velocity, b, was assumed to be equal to 5. This is equivalent to assuming that the sediment transport capacity can be computed using Engelund and Hansen's [1967] formula. However, many sand bed rivers have a lower value of b at bankfull conditions and many gravel bed rivers have a higher value. Therefore, we also computed the bar mode by assuming b = 4 for sand bed rivers and b = 10 for gravel bed rivers. The effect of gravity on the direction of sediment transport over transversely sloping beds was computed imposing E = 0.5 [Crosato, 2007].

[27] As a general criterion we assumed that the agreement can be considered good if the predicted bar mode deviates less than one from the observed mode. Furthermore, we investigated whether the computed bar mode, m, can be used to predict whether a river is meandering, braided or in transition. We define a transition river here as a river with clear meanders, but locally more than one conveying channel. A typical meandering river was assumed to present either only point bars or point bars and alternate bars (m ≤ 1.5). A typical braided river was assumed to present several bars (m ≥ 2.5). A transition river was assumed to have one to two bars per cross section (1.5 < m < 2.5). We computed the bar mode using b = 4 for sand bed rivers and b = 10 for gravel bed rivers. For the distinction between meandering, transition and braiding, we also used data from the work by MacDonald et al. [1992], in which river planform styles were given or detectable from figures, but information on the braiding index or bar mode was lacking.

[28] The exercise was repeated estimating b as a function of the Shields parameter, using the Meyer-Peter and Müller [1948] transport formula. This formula is strictly valid if the sediment has diameters larger than 0.4 mm and is transported as bed load. It can be written as

equation image

with the following parameters. Transport parameter

equation image

Flow parameter

equation image

Ripple factor

equation image

in which qs0 is the cross-sectionally averaged value of sediment transport per unit of channel width (m2/s); Dm is the mean sediment grain size (m); C90 = 18log10equation image with D90 being the grain size for which 90% of the sediment particles is smaller (m). The formula can be used if the following condition is satisfied: μθ < 0.2.

[29] The degree of non linearity, b, in the dependence of sediment transport on flow velocity is defined as b = equation image. For the Meyer-Peter and Müller transport formula this results in the following expression:

equation image

Finally, we compared the performance of our method with the performance of Parker's [1976] method, which is another simple method based on stability analysis. Parker defined the parameter ɛ*:

equation image

with F = equation image (Froude number for shallow flow).

[30] He proposed ɛ* = O(1) for the division between meandering and braiding. This corresponds to the following criteria. Meandering

equation image

Transition

equation image

Braiding

equation image

Parker also suggested that rivers have the tendency to remain straight if equation image.

4. Results

4.1. Bar Mode

[31] Table 2 presents river data at bankfull conditions and Table 3 presents the computed and observed bar modes, along with the computed river planform style, for the rivers listed in Table 2. Figure 6 shows the absolute value of the difference between computed and observed bar modes for b = 5 as a function of river width-to-depth ratio. Figure 6 shows that predictions using equation (18) deteriorate for width-to-depth ratios larger than 100.

Figure 6.

Absolute value of the difference between computed and observed m as a function of the width-to-depth ratio. The values have been computed for b = 5 (Engelund and Hansen sediment transport formula).

Table 2. Data of Rivers at Bankfull Stage, Including Observed Planform Style and Bar Modes
River ReachaiQW (m3/s)B (m)h0 (m)D50 (mm)Chézy (m1/2/s)B/h0u (m/s)FObserved Bar ModePlanformb
Allier River [1] upstream of Moulins0.00083325652.4547272.100.432T
Geul River [2] at Mechelen0.0024228.02.0252041.390.31<1M
Colorado River [3] (rkm 274–246)0.00130979c1753.05429.5d581.840.342–3T
Colorado River [3] (rkm 245–206)0.00100979c1293.64435d352.100.351–2M
Colorado River [3] (rkm 180–153)0.00066979c1474.53827.5d331.500.232T
Colorado River [3] (rkm 151–140)0.000471200e1325.13535d261.710.241–2M
Colorado River [3] (rkm 138–126)0.001491200e2034.66930d442.480.372T
Colorado River [3] (rkm 111–105)0.000341200e1515.12535d291.460.211–2M
Sutlej River [4] Islam Barrage-Chenab 10.0002321001249.50.16638131.780.181M
Sutlej River [4] Islam Barrage-Chenab 20.0002321002655.70.16638461.380.182T
Sutlej River [4] Islam Barrage-Chenab 30.0002321003354.90.16638681.280.183T
Sutlej River [4] Islam Barrage-Chenab 40.0002321004074.30.16638951.200.184B
Upper Indus [4] Chasma- Taunsa 1(B)0.00025710037481.80.19248.120821.020.24about 17B
Upper Indus [4] Chasma- Taunsa 2(B)0.00025710028632.20.19249.413011.160.25about 17B
Upper Indus [4] Chasma- Taunsa 3(B)0.00025710045241.60.19247.228270.940.24about 17B
Upper Indus [4] Chasma- Taunsa 1(MC)0.00025710013353.40.19253.33931.550.274B
Upper Indus [4] Chasma- Taunsa 2(MC)0.0002571009584.20.19255.12281.790.283B
Upper Indus [4] Chasma- Taunsa 3(MC)0.00025710017082.90.19252.15891.400.264B
Table 3. Computed and Observed Bar Modes as Well as Planform Styles for the Rivers Listed in Table 2a
River ReachbComputed m b = 4Computed m b = 5Computed m b = 10Observed Bar ModePredicted Planform, New MethodcPredicted Planform, Parker [1976]Observed Planform
  • a

    Planform styles have also been computed using Parker's [1976] method.

  • b

    B, entire braid belt; MC, only main channel.

  • c

    Using b = 4 for sand bed rivers and b = 10 for gravel bed rivers.

Allier River upstream of Moulins-0.71.42MMT
Geul River at Mechelen-0.20.4< 1MMM
Colorado River (rkm 274–246)-1.73.12–3BMT
Colorado River (rkm 245–206)-0.91.71–2TMM
Colorado River (rkm 180–153)-1.01.92TMT
Colorado River (rkm 151–140)-0.61.11–2MMM
Colorado River (rkm 138–126)-1.32.52TMT
Colorado River (rkm 111–105)-0.71.31–2MMM
Sutlej River–Islam Barrage–Chenab 10.81.1-1MMM
Sutlej River–Islam Barrage–Chenab 22.43.3-2TMT
Sutlej River–Islam Barrage–Chenab 33.34.7-3BMT
Sutlej River–Islam Barrage–Chenab 44.56.3-4BMB
Upper Indus Chasma–Taunsa 1(B)87.1123.2-about 17BBB
Upper Indus Chasma–Taunsa 2(B)5375-about 17BTB
Upper Indus Chasma–Taunsa 3(B)120.5170.5-about 17BBB
Upper Indus Chasma–Taunsa 1(MC)16.723.6-4BTB
Upper Indus Chasma–Taunsa 2(MC)9.413.3-3BMB
Upper Indus Chasma–Taunsa 3(MC)25.636.2-4BTB

[32] Figure 7 compares observed bar modes with computed bar modes for b = 5 and width-to-depth ratios smaller than 100. The solid line indicates perfect match, whereas the dashed lines delimit the zone in which the error is less than 1. The errors sometimes exceed unity when adopting b = 5 for all rivers, without distinguishing between sand bed rivers and gravel bed rivers. Figures 8 and 9 compare computed and observed bar modes by using b = 4 for sand bed rivers and b = 10 for gravel bed rivers. Figure 8 holds for width-to-depth ratios smaller than 50 and Figure 9 for width-to-depth ratios between 50 and 100. Here the agreement between computed and observed bar modes is good: the error is at most equal to 1.

Figure 7.

Computed versus observed m for width-to-depth ratios smaller than 100. The computed values assume b = 5 (Engelund and Hansen sediment transport formula) as a first estimate for all rivers. Solid line, perfect match; dashed lines, error ± 1.

Figure 8.

Computed versus observed m in rivers with width-to-depth ratios smaller than 50. The computed values assume b = 4 for sand bed rivers and b = 10 for gravel bed rivers. Solid line, perfect match; dashed lines, error ± 1.

Figure 9.

Computed versus observed m in rivers with width-to-depth ratios between 50 and 100. The computed values assume b = 4 for sand bed rivers and b = 10 for gravel bed rivers. Solid line, perfect match; dashed lines, error ± 1.

4.2. Planform Style

[33] Table 3 presents the predicted and observed planform styles using our new method as well as Parker's [1976] method for the rivers listed in Table 2. Table 4 presents the data at bankfull conditions, including the observed planform styles, for a number of rivers from literature. For these rivers no information was given on either braiding index or bar mode. Table 5 lists the planform styles predicted by our new method and by Parker's method for the rivers listed in Table 4.

Table 4. River Data at Bankfull Stage Including Planform Style
RiverQW (m3/s)B (m)h0 (m)iD50 (mm)equation image0u0 (m/s)FObserved Planforma
  • a

    M, meandering; B, braiding; T, transition (clear meanders but more than one channel at certain locations).

Ranoli (India) [Struiksma and Klaassen, 1988]4002870.90.000780.113.801.540.521B
Beaver Creek (USA) [Struiksma and Klaassen, 1988]27612800.30.006629.60.040.710.418B
Ohua River (New Zealand) [Struiksma and Klaassen, 1988]3784500.60.0065200.111.40.577B
Savannah (USA) [Struiksma and Klaassen, 1988]8601075.20.000110.80.431.540.216M
Jamuna (Bangladesh) [Struiksma and Klaassen, 1988]4000050006.00.000060.220.991.330.173B
Big Fork River at Koochiching County (USA) [MacDonald et al., 1992]155552.70.000636.50.101.040.202T
Minnesota River at Nicollet and Blue Earth Counties (USA) [MacDonald et al., 1992]314434.70.000240.51.361.550.228M
Rice Creek at Anoka County (USA) [MacDonald et al., 1992]12.913.40.50.001750.90.6591.920.869M
Table 5. Planform Styles Determined With the New Method and Parker's [1976] Method for the Rivers Listed in Table 4a
RiverParker [1976]New Method
i/Fh0/B(i/F)/(h0/B)Predictedm (b = 4)m (b = 10)PredictedObserved
  • a

    M, meandering; B, braiding; T, transition (clear meanders but more than one channel at certain locations).

Ranoli (India)0.0014970.003130.47725T9.8-BB
Beaver Creek (USA)0.0157530.0002367.21248B-375.8BB
Ohua River (NZ)0.0112640.001338.44805B-63.9BB
Savannah (USA)0.0005080.0485980.01046M0.3-MM
Jamuna (Bangladesh)0.0003450.00120.28770T15.4-BB
Big Fork River at Kooch. County (USA)0.0031060.049090.06327M-2.1TT
Minnesota River at Nic. and Blue Earth Co (USA)0.0010490.109300.00959M0.3-MM
Rice Creek at Anoka County (USA)0.0020130.037310.05394M0.5-MM

[34] Equation (18) is found to be a good predictor of whether a river is meandering or braided. Braiding appears in rivers having width-to-depth ratios larger than 50, which lies between Fredsøe's [1978] threshold value of 60 and Rosgen's [1994] value of 40.

[35] The Geul is the only river in our data set with a small width-to-depth ratio at bankfull stage (β = 4, Table 2, third row). This is a clearly meandering river, with sinuosity equal to 1.67 [De Moor et al., 2007]. Neither Parker's [1976] criterion nor Fredsøe's [1978] criterion is found to work for this river, because both criteria assign such small width-to-depth ratios to straight rivers. Our new method does not distinguish “straight” as a separate planform style.

[36] By comparison with data (Tables 3 and 5), equation (18) is found to provide better estimates than Parker's method. Equation (18) includes the effects of channel geometry, flow, sediment size and sediment transport formula, whereas Parker's planform style parameter (equation (25)) includes only the effects of channel geometry and flow characteristics. Accounting for sediment transport characteristics seems to provide an important improvement with respect to earlier simple river planform style prediction methods based on stability analyses.

[37] The computation of b using the Meyer-Peter and Müller transport formula (equation (24) assuming Dm = D50) gave unsatisfactory results: for most gravel bed rivers the formula predicted zero sediment transport at bankfull conditions. These conditions cannot be considered representative. Spatial variations in grain size may still make part of the river bed mobile. Discharges above bankfull may actually transport most of the sediment. However, such refinements are beyond the scope of this study. We wanted to present a practical method based on bankfull conditions. Therefore we advise to use b = 10 for gravel bed rivers and b = 4 for sand bed rivers.

5. Discussion

5.1. Spatial Analysis Versus Temporal Analysis

[38] A review of the first manuscript of this paper raised the objection that prediction of the number of bars would require a “temporal analysis,” based on neglecting the spatial amplification rate of bar perturbations, instead of our “spatial analysis,” based on neglecting the temporal amplification rate. This is a distant echo of an old debate between what Parker and Johannesson [1989] term the “Delft School” and the “Genova School.” Both schools start with equivalent linear equations, but apply different simplifications. Both schools agree that the two resulting different analyses apply to different aspects of river morphology. The temporal analysis describes free migrating bars. The spatial analysis represents steady bars generated by constant local perturbations and the phenomenon of overdeepening. The debate is about whether certain other phenomena are best explained by the temporal or the spatial analysis. This reflects two different paradigms of interpretation. The debate arose first after Blondeaux and Seminara [1985] and Struiksma et al. [1985] contemporarily identified the importance of steady bars for the initiation of meandering. The International Summer School on Stability of River and Coastal Forms in Perugia, organized by Prof. Giovanni Seminara in 1990, was the main stage of vivid discussions on the question as to whether the temporal analysis or the spatial analysis offers the most fundamental explanation for the steady bars that set off meandering. The Genova School argued that the temporal analysis shows how free migrating bars slow down and evolve into steady bars as a channel widens because of bank erosion. The corresponding width-to-depth ratio corresponds to the condition of resonance. The Delft School argued that the spatial analysis shows that any constant local perturbation gives rise to the formation of steady bars. Experimentally, both mechanisms have been shown to produce the steady bars that are needed to explain the onset of river meandering. The wavelengths of the two types of steady bars are equal at the point of resonance. The debate thus may seem rather academic. Nonetheless, the Genova School maintained the explanation from the temporal analysis to be more fundamental, as it represents an intrinsic behavior without the requirement of external forcings such as the Delft School's constant local perturbation. The Delft School maintained that the explanation from the temporal analysis represented only a peculiar case, as most meandering rivers do not have a resonant width-to-depth ratio. A single constant local perturbation is a less stringent prerequisite than the widening till achieving resonant conditions.

[39] The number of river bars and the transition between meandering and braiding form a new issue in this old debate. Our main argument in favor of the spatial analysis is that most real rivers outside the laboratory have curved or otherwise nonuniform bank lines. At moderate width-to-depth ratios, such bank lines force the bed topography into a steady pattern for which the spatial analysis is the most appropriate. Bars move only if discharges vary or if bank lines move by erosion or accretion. River engineers routinely calibrate two-dimensional morphological models by using the spatial analysis, not the temporal analysis. Even the Genova meander model, which Zolezzi [1999] developed along the lines of Crosato [1987, 1989], is based on the spatial analysis, not the temporal analysis. In our experience, purely free migrating alternate bars occur only in artificially straightened rivers, such as the Rhine along Liechtenstein, and in alluvial canals large enough to exceed the critical width-to-depth ratio for free migrating bars, such as the interriver link canals in the Punjab in Pakistan. At larger width-to-depth ratios, far from the transition between meandering and braiding, we do expect that bars in the central portion of the corresponding braid pattern can be freely migrating irrespective of bank line forcing, but a linear analysis, no matter whether spatial or temporal, performs less well for those conditions anyway.

[40] Interestingly, Marra [2008] demonstrates that, at a given width-to-depth ratio, linear predictions of migrating-bar modes and steady-bar modes are similar (Figure 10). This implies that the selection of either a spatial analysis or a temporal analysis does not present a critical issue. Our results support this finding. The ability to predict bar modes for width-to-depth ratios up to 100 implies that the method works also for a range of rivers for which some of the bars observed may have been migrating.

Figure 10.

Stability of bar modes 1 to 4 for both “free” (migrating) and “forced” (steady) bars computed for ks = 0.15 m (with C = 18logequation image), i = 1 × 10−4, QW = 2500 m3/s, D50 = 2 mm, and b = 5. The critical width-to-depth ratios for migrating bars, β, are comparable to the width-to-depth ratios at the border between the overdamped and underdamped regimes of forced bars (start of the harmonic range of the solution). Courtesy of Wouter Marra and Maarten Kleinhans.

[41] A final issue needs to be addressed. Why is the spatial analysis, based on assuming zero growth rate rather than maximum growth rate, so successful in correctly predicting real river bed topographies? One possible answer is that the corresponding linear equations describe steady bed topographies consisting of small-amplitude bars that neither grow nor decay. Steady bed topographies with larger bar amplitudes, for which linear equations are not valid, should then have the same pattern as the ones with small amplitudes. This is usually true if width-to-depth ratios are not too large. We think that this answer is sufficient. However, a stricter question might be to explain how these steady bed topographies can evolve at zero growth rate from an initially flat bed when forced by a constant local perturbation. This question remains open, but we would like to give two arguments why this does not undermine the validity of our analysis. First, similar strict questions are not posed for other well established theories for equilibrium patterns such as the classical axisymmetrical bed profiles in river bends. Second, the wavelength of free migrating alternate bars can be estimated reasonably well from the minimum of the marginal stability curve in the temporal analysis, which also corresponds to a condition of zero growth rate. This is because, in the temporal analysis, the bar wavelengths of maximum growth rate at arbitrary width-to-depth ratios do not differ much from the critical bar wavelength at the lowest with-to-depth ratio for which free migrating bars become unstable.

5.2. Dependence on Grain Size

[42] The decreasing m for increasing D50 predicted by equation (19) seems to contradict the common wisdom that in general piedmont gravel bed rivers have higher bar modes (more bars) than low-land sand bed rivers. The dominant effect, however, is the steeper gradient of gravel bed rivers compared to sand bed rivers. Besides, the bar mode increases monotonically as b increases. The latter has a higher value for larger grain sizes, as can be demonstrated by evaluating b as a function of the Shields parameter, as in equation (24). The median grain size appears in equation (19) only because of the relation for transverse bed slopes, not because of any relation for sediment transport rates.

5.3. Performance for High Width-to-Depth Ratios

[43] The overestimation of the number of bars for width-to-depth ratios larger than 100 can be ascribed to the linearity of the underlying model. The linearity of the model implies that equation (18) gives the number of incipient bars. With further morphological development, multiple bars tend to merge, as demonstrated experimentally by Fujita [1989] and numerically by Enggrob and Tjerry [1999] (Figure 11). This is an effect of the neglected nonlinear terms [Seminara and Tubino, 1989]. The rivers in our data set with the largest width-to-depth ratios, such as the Upper Indus (Table 2), are anabranched rather than braided. For this type of river, equation (18) systematically predicts a braided planform with too many bars if the entire river belt width is considered. The consequence is that equation (18) cannot be applied to anabranched rivers and cannot distinguish anabranching from braiding. Application to individual anabranches would seem more appropriate, but in this case it did not produce good results either, because even for the individual anabranches the width-to-depth ratios were too high. We know from application to the braided-anabranched Brahmaputra-Jamuna River in Bangladesh, not reported here, that the method of Bettess and White [1983] gives better results for anabranched rivers.

Figure 11.

(left) Nonlinear interactions reduce the number of channels as braiding evolves in a numerical computation to simulate (right) the 1987 planform of the Jamuna River in Bangladesh [Enggrob and Tjerry, 1999].

5.4. General Prediction of River Planform Styles

[44] Unlike some existing empirical predictors, equation (18) cannot be used as a general planform style predictor on the basis of hydrological data and terrain information such as valley slope, vegetation and sediment, because it requires the river width to be known in advance. Application as a general predictor of river planform style requires combination with an appropriate width predictor, e.g., the one developed by Parker et al. [2007]. The method shares this limitation with other physics-based predictors [Ferguson, 1987]. However, the method is suitable for assessments of the effect of width changes imposed by river improvement, such as constriction to improve navigability and widening to reduce flooding risk.

6. Conclusions

[45] We have derived and tested a simple physics-based method to predict the number of bars per cross section and the planform style of a river. Similarly to previous methods, the number of bars is primarily a function of the channel width-to-depth ratio. Application of the method to a number of existing rivers at bankfull conditions was satisfactory. The method gives fair results when using the Engelund and Hansen sediment transport formula irrespective of bed sediment caliber, and good results when distinguishing between sand bed rivers and gravel bed rivers. The method proves to be a good predictor for rivers with a known width-to-depth ratio below 100. In particular, it correctly predicts the absence of free bars (only point bars: m < 1) as well as the presence of alternate bars (m = 1), central bars (m = 2) or multiple bars (2 < m < 5). As a consequence, the method appears to be a good predictor for the transition between meandering and braiding.

[46] The method systematically overpredicts the number of multiple bars for width-to-depth ratios above 100. This is ascribed to the neglect of non linear effects by which the number of bars decreases through coalescence as bars grow to a finite amplitude.

[47] The method can be useful in particular for assessments of the effect of width changes imposed by river improvement, such as constriction to improve navigability and widening to reduce flooding risk. It needs to be combined with an appropriate width predictor if the river width is not known in advance.

Notation
b

degree of nonlinearity of sediment transport versus depth-averaged flow velocity, dimensionless.

B

river width, m.

C

Chézy coefficient, m1/2/s.

Cf

friction factor, dimensionless.

C90

Chézy coefficient for grain roughness of D90, m1/2/s.

D50

median sediment grain size, m.

D90

size of sediment for which 90% of the sediment particles is smaller, m.

Dm

mean sediment grain size, m.

E

calibration coefficient for gravity effects on sediment transport directly over transverse bed slopes, dimensionless.

f = f(θ)

function for gravity effects on sediment transport direction over transverse bed slopes,

F

Froude number for shallow flow, dimensionless,

g

acceleration due to gravity, m/s2.

H

near-bank water depth perturbation, m.

h

water depth, m.

h0

reach-averaged water depth, m.

h

water depth perturbation, m.

i

longitudinal river gradient, dimensionless.

kB

transverse wave number, 1/m.

ks

equivalent bed roughness for Colebrook-White equation, m.

LD

downstream damping length, m.

LP

streamwise wavelength, m.

m

transverse perturbation mode or number of bars in a cross section, dimensionless.

n

transverse coordinate, m.

n

Manning's coefficient, s/m1/3

qs0

reach-averaged sediment transport per unit of river width, m2/s.

QW

water discharge, m3/s.

s

streamwise coordinate, m.

sP

spatial lag, m.

U

near-bank flow velocity perturbation, m/s.

u

flow velocity, m/s.

u0

reach-averaged flow velocity, m/s.

u

flow velocity perturbation, m/s.

α

interaction parameter, defined as α = λS/λW, dimensionless.

β

width-to-depth ratio, dimensionless.

βm

characteristic width-to-depth ratio, dimensionless.

Δ

relative sediment density under water, dimensionless.

ɛ*

planform style parameter according to Parker [1976], dimensionless.

Φ

sediment transport parameter, dimensionless.

θ0

reach-averaged Shields parameter, dimensionless.

λS

streamwise adaptation length of cross-sectional bed profile, m.

λW

streamwise adaptation length of transverse distribution of streamwise flow velocity, m.

μ

ripple factor, dimensionless.

ρ

mass density of water, kg/m3.

ρs

mass density of sediment, kg/m3.

τ0

reach-averaged bed shear stress, Pa.

ψ

flow parameter, dimensionless.

Acknowledgments

[48] This research has been carried out with funding from the Water Research Centre Delft. The second author collected the data from the Sutlej River and the Upper Indus in the years 1994–1995 as a part of the Indus Basin River Regime Study within the Flood Protection Sector Project of the Ministry of Water and Power, Pakistan, financed by the Asian Development Bank. Our work has benefited from discussions with Giampaolo Di Silvio, which we gratefully acknowledge. We thank Huib de Vriend and the three reviewers Rob Ferguson, Chris Paola, and Giovanni Seminara for their valuable comments. Special thanks are due to Wouter Marra and Maarten Kleinhans for their permission to use their unpublished material.

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