A nonlinear model for river meandering

Authors


Abstract

[1] We develop a nonlinear asymptotic theory of flow and bed topography in meandering channels able to describe finite amplitude perturbations of bottom topography and account for arbitrary, yet slow, variations of channel curvature. This approach then allows us to formulate a nonlinear bend instability theory, which predicts several characteristic features of the actual meandering process and extends results obtained by classical linear bend theories. In particular, in agreement with previous weakly nonlinear findings and consistently with field observations, the bend growth rate is found to peak at some value of the meander wave number, reminiscent of the resonant value of linear stability theory. Moreover, a feature typical of nonlinear waves arises: the selected wave number depends on the amplitude of the initial perturbation (for given values of the relevant dimensionless parameters), and in particular, larger wavelengths are associated with larger amplitudes. Meanders are found to migrate preferentially downstream, though upstream migration is found to be possible for relatively large values of the aspect ratio of the channel, a finding in agreement with the picture provided by linear theory. Meanders are found to slow down as their amplitude increases, again a feature typical of nonlinear waves, driven in the present case by flow rather than geometric nonlinearities. The model is substantiated by comparing predictions with field observations obtained for a test case. The potential use of the present approach to investigate a number of as yet unexplored aspects of meander evolution (e.g., chute cutoff) is finally discussed.

1. Introduction

[2] River meandering is a major topic in the field of morphodynamics. It has been the subject of extensive investigations in the recent past. The review paper of Seminara [2006], to which the reader is referred for a broad overview of the subject, has outlined the main steps whereby progress has been made in this field.

[3] Let us briefly recall them. The attention was initially focused on understanding the mechanism of meander formation starting from a straight channel configuration. Linear stability analyses (so-called “bend” theories) were employed, hence linear models were developed to determine flow and bed topography in weakly curved channels. The physical implications of the linearity assumption can be appreciated recalling the main ingredients of the process whereby the pattern of bed topography develops in a sinuous channel. The first feature is the establishment of a centrifugal secondary flow directed outwards close to the free surface and inward close to the bed: it arises because the lateral pressure gradient driven by the lateral slope of the free surface established in a bend is unable to provide the centripetal force required for fluid particles to move along purely longitudinal trajectories. If the bed is nonerodible, a “free vortex” effect prevails initially, longitudinal trajectories in the inner part of the bend being shorter than in the outer part. As a result, flow at the inner bend accelerates relative to the outer bend. This is a purely metric effect which is accounted for also in linear models. Proceeding downstream, a net transfer of momentum toward the outer bend is driven by the secondary flow (the outward transfer occurring in the upper layer prevailing on the inward transfer occurring in the lower layer), hence the thread of high velocity progressively moves from the inner to the outer bend. In the context of perturbation approaches where the basic state is laterally uniform, this is a second-order effect hence it is not accounted for in the context of linear models. The picture changes considerably when the bed is erodible. Under the latter conditions, secondary flow also affects the motion of grain particles: they deviate from the longitudinal direction, hence sediments are transported toward the inner bends where a sequence of so-called point bars are built up while pools develop at the outer bends. The bar-pool pattern then drives a topographical component of the secondary flow and an additional contribution to sediment transport which further modifies the bed topography. Linear theories are indeed able to generate this topographical effect, though the important role of the lateral transfer of momentum driven by the topographical secondary flow is again neglected, formally appearing still at second order. The need to relax the linear constraint was recognized in the engineering literature, where a large effort was made to construct a rational framework, amenable to numerical treatment, in order to predict flow and bed topography in meandering channels with finite curvature and arbitrary width variations. These models serve the interests of river engineering, being fairly successful when applied to relatively short reaches of alluvial rivers and fairly short events.

[4] However, a more general interest toward the construction of sound analytical nonlinear models arises in the context of the fundamental research on the subject. In fact, the availability of such a model is potentially suitable to investigate a number of important processes observed in meander evolution, which still await to be understood. Let us list some of these processes, which did motivate our present work.

[5] An inspection of the patterns of meandering rivers (e.g., Figure 1) reveals that the river width, defined as the width of the stream free surface, undergoes typically spatial oscillations which display a distinct correlation with channel curvature. The river width may peak at bend apexes, reaching a minimum at inflection points (Figure 1a) or viceversa (Figure 1b). Note that this issue bears both a conceptual and a practical relevance. In fact, we know from the seminal contributions of Parker [1978a, 1978b] that the average width of straight channels in equilibrium is ultimately controlled by requirements of bank stability. We also know that meandering does not alter such equilibrium in the mean. Why and how? An attempt at clarifying the latter points has been recently proposed by Parker and Shimizu [2008]. However, provided the river is free to erode and deposit, i.e., it is able to choose its own width, then curvature makes the stream unable to keep a constant width. Why? And to what extent channel widening at the bend apexes modifies the scour pattern typically observed at the outer banks, thus affecting the lateral migration of meanders?

Figure 1.

Meander bends showing the dependence of river width on curvature and stage. (a) Maximum widths experienced at bend apexes (Eel River, California) (from Google Earth) and (b) minimum widths at bend apexes, and local widening in straight reach (tributary of the Amazon River, Brazil) (from Google Earth).

[6] A further reason of interest is related to a second observation: it is not uncommon to detect the formation of an island at bends of meandering channels (Figure 2). The presence of the island then forces the stream to bifurcate into an outer and an inner branch. In natural settings this is not a stable configuration: sooner or later, the stream will cut through and abandon the outer branch. The latter well known process is described as chute cutoff and occurs typically in wide bends with fairly large curvatures, high discharges, poorly cohesive unvegetated banks and high slope [Howard and Knutson, 1984]. Though some recent numerical investigations [Jager, 2003, and references therein] have attempted to model the latter process, it is not unfair to state that the occurrence of chute cutoff is a problem yet awaiting to be understood. The availability of an analytical nonlinear model of river meanders would allow to approach the latter problem. In fact, the process of widening is known [Repetto et al., 2002] to promote the formation of steady central bars in straight channels: it is then natural to wonder whether the formation of bend islands is similarly related to a bottom instability driven by widening of a curved channel. The next step would then consist of modeling the tendency of the central bar to force the stream to bifurcate into an outer and an inner branch leading to the occurrence of chute cutoff.

Figure 2.

A meander showing the formation of an island close to the bend apex (Finke River, Australia, courtesy of Aberdeen University, Geoff Pickup).

[7] A third motivation to develop an analytical approach to nonlinear meanders is related to the fundamental interest of a nonlinear theory of bend instability. In fact, linear theories display the occurrence of a resonance mechanism which controls the selection of the preferred wavelength for bend instability [Blondeaux and Seminara, 1985]. Resonance is obviously damped by nonlinear effects, as shown by the weakly nonlinear theory of Seminara and Tubino [1992]. However, no fully nonlinear theory has been proposed so far, though the role of nonlinearity is known to affect the flow field, hence the selection mechanism, considerably. In the present work, we do develop a nonlinear theory of bend instability, based on the nonlinear asymptotic solution of flow and bed topography in meandering rivers developed herein.

[8] Finally an analytical approach to nonlinear meanders involving a sufficiently modest computational effort, will hopefully allow us to investigate the long-term morphodynamic evolution of meandering rivers, a topic which has attracted the attention of both geomorphologists [Sun et al., 1996] and engineers [Camporeale et al., 2007]. For such applications, numerical solutions of the full 3-D governing equations or their shallow water version [e.g., Mosselman, 1991; Shimizu, 2002] are not appropriate tools as the computational effort they require would be prohibitive. Researchers have then been forced to employ analytical linear models for flow and bed topography, allowing only for geometric nonlinearities arising from planform evolution. The present model removes the latter restriction.

[9] This idea is pursued by resorting to the use of perturbation methods. We set up an appropriate perturbation expansion for the solution of the problem of morphodynamics, valid in the general case of rivers with arbitrary distributions of channel curvature, the only constraint being that flow and bottom topography must be “slowly varying” in both longitudinal and lateral directions and channel curvature must be “sufficiently small”. The former assumption requires the channel to be “wide” with channel alignment varying on a longitudinal scale much larger than channel width, while the latter assumption is satisfied provided the radius of curvature of channel axis is large compared with channel width. Both conditions are typically met in actual rivers but, in spite of the popularity enjoyed by linear models, neither of them implies that perturbations of bottom topography are necessarily “small”. It may be useful at this stage to clarify the latter statement by pointing out the fundamental distinction between the notion of linearity and the notion of slow variation. The former is based on the assumption that the amplitude of perturbations of bed topography driven by channel curvature must be small. The latter is based on the assumption that perturbations of bed topography must vary slowly in space. In order to further clarify this concept, let us consider the simplest possible case: a meander with channel axis defined in Cartesian coordinates as follows:

equation image

with ε meander amplitude, λ(= 2π/L) meander wave number, L meander intrinsic length, ω angular frequency and c meander curvature. The above relationships show various obvious facts: (1) channel curvature c can be small with meander amplitude ε “large” as long as the meander is “long” enough (λ ≪ 1) (i.e., slowly varying), and (2) conversely, a meander bend can be “sharp” (large curvature) even if its amplitude ε is small, provided the meander is “short” enough (λ ≫ 1). In this paper we are only interested in the former case.

[10] Taking advantage of the slowly varying assumption, a suitable extension of the approach developed by Seminara and Solari [1998] to investigate bed deformations in constant curvature channels with constant width can be developed. The latter approach allows for slow, yet finite, perturbations of flow and bed topography relative to a basic state consisting of a gradually varying sequence of locally uniform flow, slowly varying in both the lateral and longitudinal directions. The only unknowns left for numerical computation are then flow depth, a slow function of longitudinal and lateral coordinates, and variation of the longitudinal free surface slope satisfying a strongly nonlinear differential equation subject to continuity constraints.

[11] The content of the paper is then organized as follows. In chapter 2 we formulate the 3D problem of flow in sinuous alluvial channels with a noncohesive bed. In the analysis, the direct effect of secondary flow on the transverse distribution of the main flow, leading to lateral transfer of longitudinal momentum is accounted for. This effect, which has been argued to be important by many authors [e.g., Nelson and Smith, 1989; Imran and Parker, 1999], appears at the first order of approximation in the present scheme. Section 3 is then devoted to ascertaining the role of flow nonlinearity on “bend instability theory” by coupling the morphodynamic model with a bank erosion law, expressing the dependence of erosion intensity on the flow field. We are then able to predict the wavelength selected by bend instability as well as the meander wave speed in a nonlinear context. The model predictions are then tested by a direct application to a test case (a reach of the Cecina River, Italy) for which accurate data obtained by recent detailed monitoring are available (section 4). Predictions of both the equilibrium configuration and of the wave number selected in the meandering process do support the soundness of the present nonlinear approach. Finally, some discussion of future developments concludes the paper.

2. Nonlinear Theory of Slowly Varying Meanders

[12] River morphodynamics deals with the turbulent free surface flow of a low-concentration two-phase mixture of water and sediment particles bounded by a granular medium consisting of still sediment particles packed at their highest concentration: in river morphodynamics one ultimately wishes to determine the configuration of the bed interface. In other words, the mathematical problem of river morphodynamics is essentially a free boundary problem. Let us formulate it.

2.1. Formulation of the Problem

[13] Let us consider a sinuous alluvial channel with a noncohesive bed and refer it to the intrinsic coordinates sketched in Figure 3 (s*, n* and z* representing the longitudinal, lateral and vertical coordinates, respectively). In the case of channels with constant width, say 2Bu*, the appropriate scaling for the intrinsic coordinates, the local mean velocity averaged over turbulence u* = (u*, v*, w*)T, the flow depth D*, the free surface elevation h*, the eddy viscosity νT* and the sediment flux per unit width (qs*, qn*)T reads:

equation image

where a star denotes dimensional quantities. Moreover, sp is the relative particle density (= ρs/ρ, with ρ and ρs water and particle density respectively), d* is the particle diameter (taken to be uniform), Cfu is the friction coefficient, βu is the aspect ratio of the channel, Fr is the Froude number and the index u refers to properties of uniform flow in a straight channel with the same flow discharge and the average channel slope S. Two parameters arise, namely:

equation image
Figure 3.

Sketch illustrating the meandering channel and notations.

[14] We then consider a sinuous channel characterized by a slowly varying distribution of curvature c*(s) of the channel axis. Flow and bottom topography are then assumed to be slowly varying in both longitudinal and lateral directions. The above assumptions do not imply that perturbations of flow and bottom topography are necessarily small. It is then appropriate to rescale the longitudinal coordinate s as follows:

equation image

We also take advantage of the hydrostatic approximation, which applies when the spatial scale of the relevant hydrodynamic processes largely exceeds the flow depth. Under these circumstances the vertical component of Reynolds equations simply states that pressure is hydrostatically distributed in the vertical direction. The steady turbulent flow of water in a channel characterized by a slowly varying spatial distribution of channel curvature is then governed by the longitudinal and lateral components of Reynolds equations, along with the continuity equations for the fluid and solid phases. In dimensionless form, they read:

equation image
equation image
equation image
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where ∇ is defined by (hs−1ν0∂/∂σ, ∂/∂n, ∂/∂z). Moreover ν0 is a curvature parameter, c(σ) is dimensionless curvature and hs is a metric coefficient, such that:

equation image

where r0* is some typical radius of curvature of the channel axis. In the lateral component of the momentum equation appear the main factors controlling the intensity of secondary flow, i.e., centripetal acceleration and the lateral slope of the free surface. The equations (4)(7) must be supplemented with boundary conditions which may be written in the dimensionless form:

equation image
equation image
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The equations (9) impose no slip at the conventional reference bed level z0; the equations (10) impose the conditions of no stress at the free surface and the requirement that the latter must be a material surface; finally, the condition (11) imposes the constraint that both the water and the sediment flux must vanish at the banks. The latter conditions should be rigorously imposed requiring that the component of the flow velocity in the direction normal to the banks must vanish. However, for a wide cross section (βu ≫ 1) it is more convenient to follow a simple approximate procedure. In fact, considering the boundary layer of thickness O(Du*) adjacent to the bank and imposing a mass balance in a control volume confined laterally by the interface boundary layer-core flow and by the outer bank, longitudinally by two vertical sections of the boundary layer and vertically by the free surface, it is easy to show that the depth averaged lateral component of flow velocity at the interface boundary layer-core is O(equation image). Since the quantity (equation image) is typically small it is convenient to replace the condition at the bank by the condition (11).

[15] Closure relationships are then needed for the sediment flux per unit width q and for the eddy viscosity vT. We now take advantage of the slowly varying character of flow field and bed topography to assume that the turbulent structure is in quasi equilibrium with the local conditions, only slightly perturbed by weak curvature effects. Hence we write:

equation image

where τ* is the local value of the bottom stress, D(n, s) is the local dimensionless value of the flow depth and N(ξ) is the vertical distribution of the eddy viscosity in a plane uniform free surface flow. Note that ξ is a normalized vertical coordinate which reads:

equation image

Hence, ξ attains values in the range ξ0ξ ≤ 1, with ξ0 normalized reference level, a weakly dependent function of the longitudinal and lateral coordinates, here assumed to be constant. The distribution N (ξ) is taken to coincide with the classical parabolic distribution characteristic of uniform flows corrected by Dean's [1974] wake function:

equation image

with k the von Karman's constant.

[16] The closure for the sediment flux per unit width q derives from a well established approach of semi empirical nature. In uniform open channel flow over a homogeneous noncohesive plane bed no significant sediment transport occurs below some critical value θc of a dimensionless form θ of the average shear stress τ* acting on the bed, depending on the particle Reynolds number Rp. With vf kinematic viscosity of the fluid, the Shields stress [Shields, 1936] and Rp read:

equation image

For values of θ exceeding θc but lower than a second threshold value θs, particles are transported as bed load with a distinct dynamics driven by, but different from, the dynamics of fluid particles. Under these conditions, on pure dimensional ground, the average bed load flux per unit width on a weakly sloping bottom may be given the general form:

equation image

where η (= Fr2hD) is the dimensionless bed elevation, ∇h is (hs−1∂/∂s, ∂/∂n), Φ is a monotonically increasing function of the excess Shields stress (θ − θc) for given particle Reynolds number Rp, while G is a (2 × 2) matrix dependent on θ, θc and the angle of repose of the sediment. The function Φ can be estimated through well known empirical of semi empirical relationships: in the following we use the relation proposed by Parker [1990]. Moreover we only account for the prevailing lateral effect of gravity on the particle motion and write [Parker, 1984]:

equation image

with R a typically small parameter which reads:

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rc being an empirical constant ranging about 0.56 [Talmon et al., 1995].

[17] The reader should note that (16) fails close to sharp fronts (for the case of arbitrarily sloping beds, see Kovacs and Parker [1994] and Seminara et al. [2002]).

[18] At last, the problem formulated above is subject to two integral constraints stipulating that flow and sediment supply must be constant at any cross section, hence:

equation image
equation image

2.2. Solution for Channels With Slowly Varying Distribution of Curvature

[19] We may then expand the unknown functions in a neighborhood of the solution for uniform flow in a straight channel with an unknown shape of the cross section, described by a slowly varying function D(n, σ) of both the longitudinal and lateral coordinates. We can then expand the solution in the form:

equation image

where δ is the small parameter (v0/βuequation image). Note that, in order to account for the small variations of the longitudinal free surface slope associated with channel curvature, the free surface elevation is divided by δ. This allows the latter term to be an order one quantity.

[20] The latter expansion is then substituted into the governing differential problem (4)(7), conveniently rewritten in terms of the transformed variables σ, n and ξ using the chain rules:

equation image
equation image

[21] By substituting into the Reynolds equation the expression for the vertical component of velocity derived from integrating the continuity equation of the liquid phase in the vertical direction, we derive the final form of the integro-differential equations, which are reported in Appendix A.

[22] We then equate likewise powers of δ to obtain a sequence of differential problems, to be solved in terms of the unknown functions D and h,σ.

equation image

[23] At the leading order of approximation, the longitudinal component of the Reynolds equations reduces to a uniform balance between gravity and friction in a channel with unknown distribution of flow depth D0(n, σ) and free surface slope (−h0,σ(σ)) with relative boundary conditions:

equation image

After setting:

equation image
equation image

with R0 = 1 − h0,σ/equation image, one finds

equation image

The solution for F0 is the classical logarithmic distribution corrected by a wake function

equation image

where ξ0 is the normalized conventional reference level, here assumed to be constant.

equation image

[24] At first order, the lateral component of the Reynolds equations reduces to a balance between lateral component of gravity, centripetal inertia and lateral friction in a channel with unknown, yet slowly varying, distributions of flow depth D0(n, σ) and free surface slope (−h0,σ) as well as given slowly varying distribution of channel curvature c(σ). We find:

equation image
equation image

[25] We then set

equation image
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where G1 is the solution of the following ordinary differential problem:

equation image

[26] Let us write the solution for G1 in the form:

equation image

where

equation image

and gj(j = 0, 1, 2) solutions of ordinary differential problems identical with those solved by Seminara and Solari [1998, equations 26 and 27]. Note that system (32) has been solved analytically but the solution is not reported here for the sake of brevity. We may then proceed to determine the function a1(n, σ) first integrating the continuity equation (4) over the flow depth, at O(δ), with the use of the kinematic boundary condition (10), and secondly integrating over the cross section, with the use of the boundary condition (11) at the sidewall to get

equation image

where

equation image

and If (f = F0, G11, G12) is the integral (∫ξ01fdξ).

[27] Let us finally come to the sediment continuity equation (7). At O(δ), in the rescaled coordinates, it reads:

equation image

With the help of the closure relationship for q (equations (16)(18)) rewritten in terms of the rescaled coordinates and expanded in powers of δ and with boundary conditions forcing the normal component of sediment flux to vanish at the sidewalls (11), the equation (37) can be reduced to a nonlinear partial differential equation for the unknown functions D0(n,σ) and h0,σ (σ). We find:

equation image

Note that the slowly varying character of the lateral distribution of flow depth ensures that the value assumed by the ratio R/δ appearing in equation (38) is an O(1) parameter. The reader will easily check that this is verified provided the Shields stress attains values smaller than a few units. Equation (38) is to be solved with integral constraints (19) whereby the flow and sediment discharges must keep constant in the longitudinal direction. The above differential problem can be solved numerically, marching in n for every single cross section. In particular, at each cross section j, we start with a set of trial values of flow depth at the bank D0j,0 and free surface slope correction h0,σj. We then solve the equations (38) numerically in the whole domain. The differences between the values of the liquid and the solid discharges associated with the computed solution and the assigned values are then computed and the trial initial values are modified correspondingly until residual errors reduce below some chosen value. This trial and error procedure allows us to determine the unknown functions D0 (n, σ) and h0,σ(σ). We have also checked whether distinct solutions of the above system of equations might exist, possibly depending on the inability of part of the cross section to transport sediments. The numerical tests we have performed suggest that this is not the case, provided the entire cross section transports sediments and the channel width is given.

[28] Proceeding at the first order of approximation, convective terms appear in the longitudinal component of the equations of motion (5). After setting:

equation image

some algebra allows us to derive the problem for F1 which takes the form:

equation image

with Rj(j = 1, 5) coefficients depending on σ and n reported in Appendix B. The solution for F1 can be given the form:

equation image

where:

equation image

and fj (j = 0, 5) are solutions of the following ordinary differential systems:

equation image

where:

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Again system (43) has been solved analytically and the solution is not reported here for the sake of brevity.

equation image

[29] The lateral component of the Reynolds equations at order O(δ2) includes many second-order effects which correct the secondary flow. After setting

equation image

some algebra allows us to derive the differential system for G2 which takes the form:

equation image

with Rj(j = 1, 8) coefficients depending on σ and n and reported in the appendix. The system 46 has been solved analytically but the solution is not reported here for the sake of brevity.

[30] Following a procedure similar to the one reported for the previous order the sediment continuity equation at O(δ2) leads to the following differential equation:

equation image

to be solved with similar boundary conditions as for equation (38). Again, the above differential problem is solved numerically, marching in n for every cross section and it allows us to determine the unknown functions D1(n, σ) and h1,σ(n, σ) by a trial and error procedure.

3. Results

[31] Results are reported in Figure 4, for two periodic sequences of sine generated meanders such that c(s) = cos(λs) [Langbein and Leopold, 1966], characterized by different dimensionless wave numbers λ (quite small in Figure 4a, fairly large in Figure 4b). The latter measures the degree to which the longitudinal variations of the flow field may be considered as slowly varying. In fact, for very long meanders the spatial scale over which variations occur, namely the meander wavelength, become much greater than the adaptation length required for the flow to adjust to the varying curvature. As expected, the phase lag of bed topography relative to channel curvature is fairly small when convective effects play a negligible role, i.e., for small wave numbers. As the latter increases, the location of maximum scour moves from downstream to upstream of the bend apex and the pattern of scour and deposits displays oscillations larger than those found for smaller wave numbers. Note that hereinafter results represent our solution truncated at second order of approximation, i.e., O(δ) for the Reynolds equation along the longitudinal direction and O(δ2) for the remaining equations.

Figure 4.

(a, b) Dimensionless bed elevation relative to the undisturbed bed and (c, d) dimensionless value of the vertically averaged longitudinal velocity predicted by the present theory for two periodic sequences of sine generated meanders, characterized by different dimensionless wave numbers (λ = 0.07 (Figure 4a), λ = 0.185 (Figure 4b), λ = 0.07 (Figure 4c), and λ = 0.185 (Figure 4d)). The values of the relevant dimensionless parameters are ds = 5 · 10−3, ν0 = 0.04, ϑu = 0.1, and βu = 7. Green arrows show the locations of maximum scour and maximum velocity.

[32] The corresponding values of the vertically averaged longitudinal velocity predicted in the previous meander configurations are reported in Figures 4c and 4d. In both cases, the high-velocity core (greater for smaller wavelengths), which shifts from one side to the other side of the channel with distance through the meander, displays a peak just downstream to the bend apex.

[33] The complete flow field calculated in four cross sections located at different positions along the shorter meander (λ = 0.185) are also reported in Figure 5. Here the contour lines represent the values of the dimensionless downstream velocity and are plotted together with the vectors showing the secondary flow and the projection of some streamlines on the cross section. At the inflection point (Figure 5a) the secondary flow is nearly uniform in the cross section and is directed from the left to the right bank except for regions close to the sidewalls where a secondary flow cell is found. However, note that, at the banks, the transverse flow rate has vanishing depth average as required by the boundary conditions. Moving downstream, the secondary flow driven by streamline curvature and topographic effects is initially enhanced near the bottom and close to the outer bank (Figure 5b). Further downstream it occupies the outer part of the cross section (Figure 5c). In the shallower portion of the cross section, flow is toward the left bank except for the region close to the sidewall where the secondary flow with vanishing depth average does again prevail. Downstream of the bend apex (Figure 5d) the bed elevation is nearly uniform in the transverse direction and the secondary flow is driven by convective effects. Also note that the values of the secondary flow velocity are typically one order of magnitude smaller than those of the longitudinal motion. On the online version of the paper, the complete flow field calculated in 144 cross sections located at different positions along the shorter meander (λ = 0.185) is also reported (Animation S1). Note that in our computations we have always employed sufficiently wide channels in order to avoid the formation of “turbulence-driven secondary flow” which are clearly not accounted for by our simple turbulence closures. Following Callander [1978], a single helix in each bend is formed provided βu > 5.

Figure 5.

Isocontours of dimensionless downstream velocity at different cross sections of the sine-generated meander of Figure 4b (λ = 0.185). Vectors showing the secondary motion and the projection of some streamlines on the cross section are also represented ((a) /π = 0.5, (b) /π = 0.75, (c) /π = 1, and (d) /π = 1.25). The values of the relevant dimensionless parameters are ds = 5 · 10−3, ν0 = 0.04, ϑu = 0.1, and βu = 7.

[34] The Figure 6 reports the peak value (Figure 6a) and relative phase lag (Figure 6b) of the minimum bed elevation relative to the undisturbed bed calculated in a sine generated meander characterized by a given amplitude parameter v0 for different values of βu. Note that, for increasing values of βu, the curvature parameter v0 being kept constant, the perturbation parameter δ decreases, hence smaller values of the maximum scour (−ηmin) are experienced.

Figure 6.

(a) Peak value and (b) relative phase lag (λs/π) of the minimum bed elevation relative to the undisturbed bed for different values of βu. (c) Peak values and (d) relative phase lag of vertically averaged velocity for different values of βu (ds = 5 · 10−3, ν0 = 0.04, and ϑu = 0.1).

[35] Moreover, for small values of βu, the location of the cross section where the maximum scour is experienced moves from downstream to upstream the bend apex as the wave number increases. For larger values of βu the trend is similar but the maximum scour is located upstream the bend apex even for small wave numbers. The same tendency is shown by the phase lag of the peak value of velocity calculated along the meander (Figure 6d). Also note that, for given value of βu, the curves representing the maximum value of velocity (Figure 6c) exhibit a peak. In the case of βu = 5, the curve is interrupted because the value of the Shields stress falls below the threshold of motion somewhere along the meander.

[36] A comparison with the solution truncated at leading order is reported in Figure 7, where it appears that the first-order correction generally contributes to increasing the values of maximum scour and maximum velocity and to shifting the latter peaks downstream. These corrections increase their importance as βu decreases.

Figure 7.

Same as Figure 6; solution is truncated at leading order.

[37] Also note that the ratio between the free surface slope if, averaged along the entire meander, and the reference uniform flow slope ifu, is slightly different and invariably smaller than one at leading order δ0 (Figure 8a). On the contrary, at the first order of approximation δ1, convective terms appear and, for small values of βu, they lead to an increased value of the mean free surface slope (Figure 8b). The value of the correction decreases as the aspect ratio βu increases. The latter effect is strictly related to the lateral transfer of longitudinal momentum.

Figure 8.

Ratio between the mean free surface slope if and the reference uniform flow slope ifu (a) at leading order and (b) at first order of approximation.

4. Nonlinear Bend Stability Theory for River Meanders

[38] The model presented above is suitable for the formulation of a non linear bend instability theory. In order to pursue the latter goal one needs to associate a bank erosion equation to the governing equation for flow and bed topography.

[39] The detailed mechanics of bank erosion, both the continuous process of particle removal of small particles from the bank surface and the intermittent process of bank collapse occurring typically during the decaying stage of flood events, depends on several factors, namely scour at the bank toe, bank cohesion, wetting and drying of banks, its rate being ultimately controlled by the ability of the stream to remove sediments accumulated at the bank foot. However, for long-term investigations, rather than attempting to investigate in detail the mechanics of single events, it is more appropriate to resort to some integrated formulation: in other words, one simply locates the region of the outer bank where erosion is expected to occur on the basis of the knowledge of the hydrodynamic field and simply models the actual intermittent mechanism as effectively continuous and such to reproduce the averaged effects of the actual process. Long ago Ikeda et al. [1981] proposed that a simple rule accomplishing the latter task is to assume that bank erosion is linearly proportional to an excess flow speed at the outer bank while bank deposition is conversely linearly related to a defect of flow speed at the inner bank. In other words, Ikeda et al. [1981] proposed the following expression for the lateral migration speed:

equation image

where both the lateral migration speed ζ(s) and the depth averaged longitudinal velocity U are scaled by some reference speed Uu* and E is a dimensionless long-term erosion coefficient. The above linear rule has been employed by virtually all researchers who have investigated the planform development of meandering rivers [Sun et al., 2001a, 2001b; Edwards and Smith, 2002; Camporeale et al., 2007; Lanzoni and Seminara, 2006; Lanzoni et al., 2006]. Its actual suitability has so far been tested [Pizzuto and Meckelnburg, 1989] by field observations on rivers with fairly uniform cohesive banks. Note that the rule (48) is such that channel width is preserved throughout the process of meander development.

[40] We then follow a classical normal mode analysis and assume that the channel axis follows a sinusoidal curve in the (x, y) plane, denoting by k its Cartesian wave number and by ε its amplitude, having normalized both quantities by half the channel width Bu*. One then readily finds the following relation:

equation image

In the present nonlinear context it is then convenient to define an average measure of the migration vector (equation imagex, equation imagey) integrating the local values of ζ(s) given by the equation (48) along the intrinsic coordinate s, between two consecutive inflection points. The x and y Cartesian components of the above vector provide measures of the meander wave speed and meander growth rate, respectively: a positive (negative) value of equation imagex corresponding to downstream (upstream) meander migration while positive (negative) values of equation imagey corresponding to meander amplification (attenuation). In fact, from the average migration vector one readily derives the following forms for the meander migration speed c and meander growth rate (ε,t/ε):

equation image

[41] The Figures 9a and 9b show the growth rate and wave speed, respectively, as functions of the wave number λ for different values of the amplitude ε. Results of linear theory are also reported and it is evident that for small wave numbers, i.e., for meanders characterized by very slow longitudinal variations, nonlinear terms are negligible, hence all the curves tend to collapse. On the contrary, as the wave number increases, the solution strongly depends on the amplitude of the perturbation ε due to the increasing importance of convective terms. It is worth noting that O(equation imagex/E) ∼ v02 = ε2λ4 and O(equation imagey/E) ∼ v0 = ελ2 hence the quantities c and {ε,t/ε} keep bounded even in the limits of ε → 0 and λ → 0.

Figure 9.

(a) Meander amplification rate ε,t/ε and (b) wave speed c (scaled by the erosion coefficient E) are reported versus the wave number in the case of a meander following a sinusoidal planimetric pattern with different values of amplitude ε. Values corresponding to the linear theory are also reported (ds = 5 · 10−3, ϑu = 0.1, and βu = 10).

[42] The curves obtained by the present nonlinear model are interrupted when the value of the Shields stress falls below the threshold of motion somewhere along the meander. Clearly, within the linear context, neither amplification nor migration rates are affected by the amplitude of the initial perturbation ε. However, note that the linear solution for flow field and bed topography depends on the value attained by the small parameter v0, related to both ε and λ through (49). Hence, for example, in the case of ε = 15 the bed emerges in the linear case for a value of λ = 0.079, the latter value increasing as ε decreases.

[43] In Figure 10 we plot the nonlinear (Figure 10a) and linear (Figure 10b) values of the meander amplification rate ε,t/ε (scaled by the erosion coefficient E) as a function of the meander wave number λ for a given value of the amplitude ε. Note that, for different values of βu, the nonlinear solution is characterized by a peak which corresponds to the value of the critical wave number selected in the meandering process. The location of the peak, i.e., the wave number selected, increases monotonically as βu increases. On the other hand, the linear solution is highly influenced by the fact that the amplification rate tends to infinity for values of βu and λ close to the resonant values (βR = 18.25, λR = 0.123). Close to resonance the linear solution is plotted with dotted lines when the bed emerges (Figure 10b, βu = 20).

Figure 10.

(a) Nonlinear and (b) linear solutions for the meander amplification rate ε,t/ε (scaled by the erosion coefficient E) are plotted versus the meander wave number for different values of βu in the case of a meander following a sinusoidal pattern with amplitude ε = 10 (ds = 5 · 10−3 and ϑu = 0.1).

[44] The wave numbers corresponding to the maximum bend amplification are plotted versus βu in Figure 11 for different values of the amplitude ε. Note that the selected wave number depends on the amplitude of the initial perturbation and in particular larger wavelengths (smaller wave numbers) are associated with larger amplitudes ε. Decreasing the relative roughness (compare Figures 11b and 11a) also has a minor influence, which leads to a decrease of the wave number selected. A similar trend has been observed increasing the reference Shield stress. The values corresponding to the linear theory of Blondeaux and Seminara [1985] are also reported in Figures 11a and 11b and show a peak close to the resonant values (βR, λR). It turns out that, for small values of βu, the nonlinear model typically selects a larger wave number, i.e., shorter wavelengths. The situation is reversed increasing bed friction (Figure 11b), independently of the value attained by the aspect ratio βu. Also note that the perturbation parameter δ in these cases attains values that are always smaller than 0.15.

Figure 11.

The selected wave numbers for bend instability are reported for different values of the amplitude ε and compared with the linear theory for (a) ds = 5 · 10−3, ϑu = 0.1 and (b) ds = 0.1, ϑu = 0.1.

[45] In Figure 12 the values of the meander wave speed c selected by bend instability (scaled by the erosion coefficient E) are plotted versus the aspect ratio βu for different values of the amplitude ε. The wave speed corresponding to the linear theory is also reported and shows the well-known feature of linear resonators [Kevorkian and Cole, 1981], whereby the phase of the response changes quadrant on crossing the resonant conditions. Note that, in the linear case, the wave speed is strongly affected by resonance for a wide range of values of βu leading to results markedly different from those obtained in the nonlinear case. However, an important feature of linear theory is preserved: the resonant value still distinguishes between upstream and downstream migration (Figures 12a and 12b). Note also that the meander wave speed grows as the meander amplitude decreases, the Shield stress increases and friction decreases (compare Figures 12b and 12a). Moreover, for each given meander amplitude ε, a threshold value of the aspect ratio βu exists above which the model predicts upstream migration, a finding which confirms the picture obtained in the context of the linear model of Zolezzi and Seminara [2001].

Figure 12.

The values of meander wave speed c (scaled by the erosion coefficient E) selected by bend instability are plotted versus the aspect ratio βu for different values of the amplitude ε and compared with the values obtained by the linear theory: (a) ds = 5 · 10−3, ϑu = 0.1 and (b) ds = 0.1, ϑu = 0.1.

[46] The meander amplification rates are finally reported in Figure 13 for the values of meander wave numbers selected by bend instability and different values of the amplitude ε. Unlike predictions by the linear model, amplification rates are only slightly affected by the aspect ratio βu for a given amplitude ε. Similarly to wave speed, the meander amplification rate grows as the meander amplitude decreases, the Shield stress increases and friction decreases (compare Figures 13b and 13a).

Figure 13.

The values of the meander amplification rate ε,t/ε (scaled by the erosion coefficient E) corresponding to the wave numbers selected by bend instability are plotted versus the aspect ratio βu for different values of the amplitude ε and compared with the values obtained in the context of the linear theory: (a) ds = 5 · 10−3, ϑu = 0.1 and (b) ds = 0.1, ϑu = 0.1.

5. Comparison With Field Observations for a Test Case

[47] We now attempt to substantiate the soundness of the present model by applying it to a short reach of the Cecina River (Tuscany, Italy), a gravel bed river with actively migrating outer banks [Rossi Romanelli et al., 2004].

[48] The Cecina River Basin is located in central Italy and comprises a basin surface area of approximately 900 km2 with a total length of about 79 km. The study site is located a few kilometers upstream from the confluence between the tributary Sterza and the main course. The criteria guiding the selection of this site were the availability of aerial photographs taken at different years (1954, 1978, 1986, and 2004) showing the formation of a meander from a nearly straight reach (Figure 14). Data for the flow discharge were also available from a gauging station, located just downstream of the study site, at Ponte di Monterufoli. Grain size distributions were measured at the site and made available to the Authors (M. Rinaldi, personal communication, 2008). The study reach is about 1000 m long and is characterized by an average slope of about 0.002.

Figure 14.

The reach of the Cecina River showing the formation of a meander from a nearly straight configuration. Flow is from right to left.

[49] The first group of numerical simulations was performed for a set of data obtained by extrapolating the planform shape of the channel axis from a 1978 aerial picture. The shape turned out to follow closely a sine generated curve [Langbein and Leopold, 1966], characterized by a minimum radius of curvature r0* ≃ 325 m, an intrinsic wavelength Ls* ≃ 970 m and a channel width 2Bu* ≃ 40 m, taken to keep constant throughout the reach. The water discharge with a return period of 1 year, Qu* ≃ 140 m3/s, was used in the simulations, corresponding to a uniform flow depth of Du* ≃ 1.3 m. Sediments were modeled as uniform and characterized by d* = 7.4 mm.

[50] The values of the relevant dimensionless parameters could then be calculated to give:

equation image

With the latter values, the model was then run to predict the equilibrium bed topography and the associated flow field. The former is plotted in Figure 15a and shows that the value of the maximum scour depth relative to the mean bed elevation is slightly larger than the uniform flow depth and is roughly located at the bend apex. On the contrary, the position of the forced bars is upstream the bend apex and shows a value of bed elevation which does not lead to bar emergence. The corresponding values of the vertically averaged longitudinal velocity predicted for this configuration are reported in Figure 15b. Note that the high-velocity core shifts from one side to the other side of the channel with distance along the meander, displaying a peak just downstream the bend apex. The value of the maximum velocity is slightly smaller than 4 m/s.

Figure 15.

(a) The equilibrium configuration of the bed topography and (b) longitudinal velocity simulated in the Cecina River extrapolating the planform shape of the channel axis from a 1978 aerial picture. The bed elevations (in meters) represent the deviation from the mean longitudinal slope. Velocities are expressed in meters/second. Green arrows show the locations of maximum scour and maximum velocity. The values of the relevant dimensionless parameter are ν0 ≃ 0.062, λ ≃ 0.129, βu ≃ 15, ϑu ≃ 0.210, and ds ≃ 0.005.

[51] We next investigated whether the nonlinear bend instability theory is able to predict the wavelength selected in the meandering process. From the first aerial picture available (see Figure 14, year 1954) it can be easily noted that two parallel straight reaches, probably rectified by human intervention, are joined together by an oblique stretch. The distance between the two parallel straight reaches is roughly equal to two channel widths. The aerial picture of 1978 already shows the evidence of a meandering process taking place downstream to the connection between the oblique and straight reaches. The intrinsic wavelength Ls* of the incipient meander can be estimated at 970 m (corresponding to an intrinsic dimensionless wave number λ = 0.129). The most recent pictures then clearly reveal the processes of meander amplification and downstream migration occurred in the following years.

[52] In Figure 16a the meander amplification rate ε,t/ε (scaled by the erosion coefficient E) is plotted versus the meander wave number for the values of the dimensionless parameters corresponding to the reach of the Cecina River considered as test case. The predicted value of the wave number selected by bend instability turns out to be λ ≃ 0.133 and corresponds to an intrinsic wavelength Ls* ≃ 945 m, very close to the value estimated from the aerial pictures.

Figure 16.

(a) Meander amplification rate ε,t/ε and (b) wave speed divided by the erosion coefficient E are plotted versus the wave number for the values of the dimensionless parameters corresponding to the reach of the Cecina River. Values corresponding to the linear theory are also reported. The selected wave number for bend instability turns out to be λ ≃ 0.133 and λ ≃ 0.084 for the nonlinear and the linear theory, respectively (ds ≃ 5 · 10−3, βu ≃ 15.2, ϑu ≃ 0.210, and ε = 4).

[53] In order to investigate the sensitivity of the wave number selected to the value chosen for the formative discharge, we evaluated the meander amplification rate by varying the latter within a range of ±50%. Results are reported in Figure 17, where the ranges of selected wave numbers for bend instability corresponding to the linear and nonlinear theory are also evidenced and compared with the value observed in the field.

Figure 17.

Meander amplification rate ε,t/ε divided by the erosion coefficient E is plotted versus the wave number for different values of formative discharge relative to the reach of the Cecina River. Values corresponding to the linear theory are also reported. The ranges of selected wave numbers for bend instability corresponding to the linear and nonlinear theory are also evidenced (ε = 4).

[54] Finally, a comparison was performed between the bed topography predicted by the present model and the values obtained during a topographic survey of the site of interest undertaken in July 2007 (Figure 14). In particular our attention was focused on the second bend, proceeding downstream, where a deep scour was located close to the outer bank, just upstream of the apex. In order to analyze the results, it proved appropriate to reinterpret data in terms of variations of bottom elevation relative to an inclined plane characterized by the average channel slope. The latter slope was computed through best fitting of the acquired data by a plane inclined in the down valley direction. The local longitudinal Cartesian slope of the river corresponding to the slope of the floodplain turned out to be ix = 0.0023. The average width of the reach was estimated by making use of both the topographical survey and the aerial picture of 2007. In particular, in the straight reach just upstream the site of interest we estimated an average width of 20 m, slightly smaller than the value we best estimated in the curved reach, being roughly equal to 2Bu* = 25 m. We then used the latter value for the simulation. The correct elevation of the inclined plane was evaluated by making sure that the intersection of the latter with measured data generated an average free surface width of the channel equal to 2Bu* and, consequently, the point bar nearly emerged. The planform shape of the reach did not strictly follow a sine generated curve, but was approximately close to such a shape, with a dimensionless intrinsic wave number λ ≃ 0.1085 and a curvature parameter v0 ≃ 0.12. To perform a simulation of flow and bed topography in the surveyed bend, an estimate of the formative discharge Qu* was also needed. This was chosen such as to generate in the model a nearly emerging point bar and turned out to be Qu* ≃ 45 m3/s. The following values of the relevant dimensionless parameters were obtained:

equation image

In Figure 18 we show the comparison between the observed deviations of bed elevations relative to the inclined plane and the results obtained by the nonlinear and the linear models, respectively. We find that, consistently with measured data, the nonlinear theory predicts a maximum (minimum) scour at the outer bank (inner bank) located just downstream (upstream) the bend apex. On the contrary, in the linear case, although the intensity is quite similar, the location of both the maximum and minimum scour is shifted far downstream relative to the bend apex. Also note that the nonlinear model is able to predict the presence of a chute channel in the downstream part of the bend, a feature clearly observed in the field data. Finally, as shown in Figure 19, the thread of high velocity is consistently located at the outer bank just downstream the bend apex. On the contrary, the maximum velocity predicted by the linear model is located close to the inflection point and the thread of high velocity is very elongated, and shifts from the inner to the outer bank quite sharply close to the apex.

Figure 18.

Comparison between (a) measured values of bottom elevation (in meters) in the Cecina River and the values obtained using (b) the present nonlinear and (c) linear models for the 2007 planimetric configuration. Flow is from right to left. The green arrow shows the location of maximum scour.

Figure 19.

Comparison between the values of longitudinal average velocity (in meters/second) obtained using the (a) present nonlinear and (b) linear models for the 2007 planimetric configuration. Flow is from right to left. The green arrow shows the location of maximum near-bank velocity.

6. Discussion and Conclusions

[55] A nonlinear asymptotic theory of flow and bed topography in meandering channels able to describe finite amplitude perturbations of bottom topography and account for arbitrary, yet slow, variations of channel curvature has been developed. This model appears to be a potentially useful and powerful tool for many purposes. In the present paper we have been able to formulate a nonlinear bend instability theory, which predicts several characteristic features of the actual meandering process and extends results obtained by classical linear bend theories. In particular, we have found the following: (1) for given values of the relevant physical parameters, the bend growth rate peaks at some value of the meander wavelength, reminiscent of (but typically smaller than) the resonant value of linear stability theory, a result confirming the weakly nonlinear results of Seminara and Tubino [1992], consistent with field observations that show that the wavelength selected by traditional linear theories is typically slightly larger than observed values; (2) the selected wavelength depends on the amplitude of the initial perturbation (for a given value of the relevant dimensionless parameters) and, in particular, larger wavelengths (smaller wave numbers) are associated with larger amplitudes ε, a feature typical of nonlinear waves; (3) the infinite peak in the linear response at resonance is damped by nonlinearity, a result again confirming its weakly nonlinear counterpart; (4) meanders are found to migrate preferentially downstream, though upstream migration is also possible, at least in principle, for relatively large values of the aspect ratio of the channel, a finding in agreement with the picture provided by the linear theory of Zolezzi and Seminara [2001]; and (5) meanders slow down as their amplitude increases (for a given value of the relevant dimensionless parameters), again a feature typical of nonlinear waves, driven in the present case by flow rather than geometric nonlinearities.

[56] In conclusion, the picture offered by results obtained through the present theory seems satisfactory and consistent with field observations as well as previous theoretical findings. Further substantiation of the model has been achieved by comparing predictions obtained for a test case (a reach of the Cecina River, Italy) with field observations, though admittedly, the nonuniform bank erodibility associated with the growth of vegetation as well as other anthropogenic effects make the significance of the latter validation only qualitative.

[57] A number of interesting future developments of the present model are called for as discussed in section 1. These include also the need to allow for slow temporal variations of flow and sediment supply such that the morphological response of the channel to a sequence of flood events. Such an investigation will help to provide a rational interpretation of the as yet loosely defined notion of “formative discharge of an alluvial river”.

Appendix A:: Equations

[58] Here we report the final form of the Reynolds equations and of the continuity equation for the liquid and solid phases written in coordinate (σ, n, ξ):

equation image
equation image
equation image
equation image
equation image

Appendix B:: Coefficients

[59] Here we report the coefficients Rj(j = 1, 8) appearing in equations (40) and (46):

equation image

Acknowledgments

[60] M. Rinaldi is kindly acknowledged for providing the aerial picture and the data of the Cecina River. The present work has been funded by Cariverona (Progetto MODITE). Partial support has also come from the Italian Ministry of University and of Scientific and Technological Research in the framework of the National Project “Evoluzione morfodinamica di ambienti lagunari” (PRIN 2006) cofunded by the University of Genova.

Ancillary

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