### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. One-Dimensional Perturbations
- 3. Two-Dimensional Waves
- 4. Planform Waves
- 5. Concluding Remarks
- Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
- Acknowledgments
- References

[1] Like most media, open channel flows propagate information through waves. When the channel boundary is fixed, the vectors of information consist typically of surface gravity waves. In the less straightforward case of channels with cohesionless bed and possibly erodible banks, other types of waves arise from the erodible nature of the boundaries and the ability of the stream to transport sediments. In this paper we restrict our attention to the important case of long waves, which can be described by employing the shallow water approximation for the flow field and a quasi-equilibrium assumption for sediment transport on weakly sloping beds. We focus on a major issue: In which direction is information propagated? This is a problem raised and partially solved by de Vries in the context of one-dimensional morphological modeling as early as 1965. We review some of the available knowledge on this subject, viewed in a more general context where vectors of information can be a variety of waves: purely longitudinal one-dimensional sediment waves, two-dimensional waves driven by large-scale bed forms (bars), and plan form waves carrying information related to the planform shape of the channel. Both linear and nonlinear, migrating and stationary waves are considered. It turns out that the role played by the Froude number in determining the direction of one-dimensional perturbations of bed topography is somewhat taken by the aspect ratio of the channel when large-scale two-dimensional bed forms as well as planform waves are considered.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. One-Dimensional Perturbations
- 3. Two-Dimensional Waves
- 4. Planform Waves
- 5. Concluding Remarks
- Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
- Acknowledgments
- References

[2] Waves are essentially vectors of information. A number of ways are experienced by open channel flows to propagate information. Here we restrict ourselves to the important case of long waves, which can be described employing the shallow water approximation. The subject has been thoroughly analyzed in the case of channels with fixed beds and nonerodible banks [e.g., *Whitham*, 1974]. We wish to assess some of the knowledge available on the less straightforward case of channels with cohesionless bed and possibly erodible banks. Under these conditions, information can still be propagated through surface gravity waves, but other types of waves arise from the erodible nature of the boundaries and the ability of the stream to transport sediments. In particular, we wish to focus on a major issue: in which direction do waves propagate information? This is not a new problem in morphodynamics, as *de Vries* [1965] raised it in the context of one-dimensional modeling. However, when viewed in a more general context, the question of morphodynamic influence turns out to be much richer than one may perhaps expect and full of important implications which will deserve attention. In essence, the problem can be reduced to the following. Let us consider the flow of water in an open channel with a mobile bed and possibly erodible banks and assume the flow to be sufficiently intense to entrain sediments. For the sake of simplicity and without significant loss of generality, we will restrict our attention to uniform sediments transported as bed load. If the channel slope is constant and its width uniform, the system allows for an equilibrium state consisting of uniform flow on a flat bed (except for the possible presence of small-scale bed forms): the sediment flux will also be uniformly distributed such that no aggradation nor degradation will be experienced by the channel. Let us now impose to the system some physical perturbation of equilibrium: a number of possible initial perturbations may be readily envisaged, e.g., a variation in sediment supply, some geometrical constraint altering channel width or channel alignment, a modification of bottom topography, etc. A number of questions then arise: Will such initial perturbations be felt downstream, upstream or in both directions? Will they grow or decay? What implications follow on the choice of appropriate boundary conditions for the general problem of morphodynamic evolution of erodible channels?

[3] In the case of unstable (i.e., growing) waves the latter issues are strictly related to the nature of the instability process: an instability is classified as absolute whenever an initial small localized perturbation spreads both in the upstream and downstream directions as time grows, affecting eventually the whole flow domain (Figure 1a). Conversely, an instability turns out to be convective if an initial, nonpersistent, small perturbation localized in space is convected away (typically, though not necessarily, in the direction of the main flow) leaving, as time tends to infinity, the flow domain unperturbed (Figures 1b, 1c, and 1d). This fundamental distinction has been developed within the area of plasma physics [*Briggs*, 1964; *Bers*, 1975] and has since been extended to several other fields (see the review of *Huerre and Monkewitz* [1990]).

[4] In order to answer the above questions, we need to distinguish between various classes of perturbations: perturbations devoid of any lateral structure, which can be investigated in the context of a 1-D formulation; perturbations with lateral structure (fluvial bars), which require a 2-D formulation; perturbations of channel alignment in meandering rivers, which call for a quasi 3-D approach. A second fundamental distinction concerns the amplitude of perturbations: linear and nonlinear waves will require distinct treatments.

[5] The paper will then be organized as follows. In section 2, we consider one-dimensional waves. We start investigating the behavior of the so-called normal modes, namely the growth and migration of single harmonic components of an arbitrary small amplitude perturbation. We derive a dispersion relationship between frequency and wave number of the perturbations, which displays various important features. In particular, no morphological instability is detected if morphodynamics is decoupled from hydrodynamics; very long waves are weakly damped and nearly nonmigrating; that is, they display the diffusive behavior pointed out by *Lisle et al.* [2001]. Failure of the decoupling approach emerges when the Froude number is close to criticality and the perturbation wave number is large. A perturbation approach is then employed to deal with coupling and reveals the existence of a weak morphological instability in the large wave number limit. We then proceed to analyze nonlinear effects by means of numerical solutions of the fully nonlinear equations in a quasi-conservative, coupled form, recently obtained by A. Siviglia et al. (The role of quasi-conservative form in morphodynamic modelling for river flow computations, paper to be presented at International Conference on Fluvial Hydraulics, River Flow 2006, 6–8 September 2006, Lisbon, Portugal, hereinafter referred to as Siviglia et al., paper to be presented, 2006). It turns out that nonlinearity damps the large wave number instability. Moreover, we show that the interaction between the propagation of hydrodynamic fronts and sediment transport is weak until propagation slows down enough to allow for stronger interactions which eventually let the fronts disappear. In section 3 we investigate the behavior of long sediment waves with lateral structure, i.e., bars. We distinguish between migrating and stationary bars and show that the role of Froude number for one-dimensional waves is “somewhat” taken by the aspect ratio of the channel which will allow us to distinguish between perturbations able to carry their information only downstream (downstream influence) and perturbations which are able to extend their influence also upstream. Section 4 revisits some recent and less recent knowledge on planform waves, concentrating on the case of meandering channels. Finally, section 5 concludes the paper with some discussion of open questions which will deserve attention in the near future.

### 2. One-Dimensional Perturbations

- Top of page
- Abstract
- 1. Introduction
- 2. One-Dimensional Perturbations
- 3. Two-Dimensional Waves
- 4. Planform Waves
- 5. Concluding Remarks
- Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
- Acknowledgments
- References

[6] Let us start considering a straight rectangular channel, with cohesionless bottom of uniform grain size *d**_{s}, width 2*B** and constant slope *S* (see Figure 2). A given constant discharge *Q** flows under uniform conditions with flow depth *D**_{0} and velocity *U**_{0}. Let the sediment be transported as bed load and denote by *Q**_{s0} the sediment flux per unit width associated with the uniform flow. Note that hereafter a star denotes dimensional quantities.

[7] In the context of a one-dimensional framework the governing equations, imposing mass conservation of the fluid phase, mass conservation of the solid phase and the momentum principle for the fluid phase, may be written in the following form:

with *H** free surface elevation, *D** and *U** cross sectionally averaged flow depth and velocity, *C* flow conductance, η* average bed elevation, *g* gravity, and *p* porosity of the granular medium. For the sake of simplicity, we model bed load transport in a simplest form and write the sediment flux per unit width as follows:

with *a* and *m* constant parameters.

[8] Note that the above formulation of the governing equations includes one term in the continuity equation for the fluid phase (i.e., the second term in equation (1)), which is most often (and appropriately) neglected in the scientific literature on the subject. A formal derivation of the mass conservation equations for the fluid and for the solid phase, (i.e., equations (1) and (3)), is reported in Appendix A. Let us make the formulation dimensionless by scaling the dimensional quantities as follows:

[9] Substituting (5) and (6) into (1), (2), and (3) we find

The following parameters emerge:

where *F*_{0} is the Froude number of the basic flow, while γ is a typical ratio between the flux of sediments and the water discharge.

#### 2.1. Linear One-Dimensional Waves

[10] Let us first seek solutions of dimensionless equations (7), (8), and (9) in the form

with ε a small (strictly infinitesimal) parameter. We then linearize the equations and reduce them to obtain

[11] A Fourier analysis of (12) can be performed expressing its solutions in the form of normal modes, hence we write

where is a *O*(1) quantity, λ = 2π/*L* is the perturbation wave number, ω is the complex angular frequency, and *i* is the imaginary unit. Substituting from (13) into (12), we end up with the following general form of the dispersion relationship:

where

[12] Note that the relationship (14) defines, in general, three eigenmodes ω_{j} (*j* = 1, 2, 3) depending on the wave number λ and on the values attained by the relevant physical parameters *F*_{0} and γ. The imaginary part of ω_{j} determines the growth rate of the perturbation, (ω_{j})_{i}. The perturbation is unstable if (ω_{j})_{i} > 0, otherwise it decays. Moreover, the perturbation wave speed *a*_{j} is given by the ratio (ω_{j})_{r}/λ, with (ω_{j})_{r} the real part of ω_{j}.

[13] One may get advantage of the typically small value attained by the parameter γ: this circumstance is classically assumed as the basis of computations of bed evolution decoupled from the computation of the flow hydrodynamics. It will appear below that as pointed out by various researchers [*Lyn*, 1987; *Lyn and Altinakar*, 2002] this procedure is not uniformly valid.

##### 2.1.1. Decoupled Theory: A Few Relevant Limits

[14] Typically, the transport parameter may be estimated as small as γ ∼ *O* (10^{−3}–10^{−4}); hence, assuming λ and *F*_{0} to be finite, it seems reasonable to expand ω in powers of γ as follows:

[15] Such an expansion is based on the physical fact that the information propagated through sediment transport, i.e., sediment waves, migrates at a speed which is typically smaller than the hydrodynamic information. We then substitute from (16) into the dispersion relationship (14) and equate likewise powers of γ.

[16] At the leading order *O*(γ^{0}), we find a classical result: one of the eigenvalue (ω_{03}) vanishes and the remaining two eigenvalues reduce to those found in the fixed bed case, namely,

[17] As shown in Figures 3a and 3b, one of the eigenvalue is associated with perturbations that are invariably stable and migrate downstream. The second eigenvalue is stable for values of the Froude number smaller than two and perturbations may migrate either upstream or downstream.

[18] At next order *O*(γ) small 'morphodynamic' corrections for the two hydrodynamic modes and a third morphodynamic nontrivial mode arise. We find

[19] The dependence of the perturbation growth rate and of the wave speed on wave number for the morphodynamic mode is plotted in Figures 3c and 3d. The morphodynamic mode appears to be invariably stable and migrates upstream (downstream) under supercritical (subcritical) conditions. In particular, it is useful to consider two limiting cases as λ tends, respectively, to 0 and ∞. Note that because of the adopted scaling (6), the short wave limit, λ ∞, describes perturbations whose length scale is much smaller than *C*^{2}*D*_{0} (= *B***C*^{2}*D**_{0}/*B**, with *B** half channel width). Therefore, considering the typical values attained by *C* and *B**/*D**_{0}, it turns out that the above limit can properly describe the behavior of relatively long sediment waves, say of the order of channel width, for which the shallow water approximation is perfectly suitable.

[20] In the long wave limit, λ 0, it is easy to find

[21] It then turns out that both the hydrodynamic modes, ω_{1} and ω_{2}, migrate downstream with constant wave speed equal to 1.5 *U*_{0} and 0.5 *U*_{0}, respectively. Moreover, the first hydrodynamic mode is damped (amplified) if *F*_{0} < 2 (*F*_{0} > 2) and describes the formation of the so-called roll waves. However, in the present 1-D context the growth rate of roll waves increases monotonically as λ increases. The second hydrodynamic mode is invariably damped. The morphodynamic mode ω_{3} migrates downstream (upstream) under subcritical (supercritical) conditions with very small wave speed (of order γλ^{2}) which vanishes identically at critical conditions, and is invariably damped, though at a quite slow rate, of order γλ^{2}.

[22] In the short wave limit λ ∞ (inertial waves) it turns out that

[23] The first hydrodynamic mode ω_{1} migrates invariably downstream with constant wave speed equal to (1 + 1/*F*_{0}): at criticality the wave speed tends to 2. This mode is damped (amplified) if *F*_{0} < 2 (*F*_{0} > 2): the unstable mode corresponds to the well known roll wave instability, whose growth rate keeps constant in this limit.

[24] The second hydrodynamic mode ω_{2} migrates downstream (upstream) under supercritical (subcritical) conditions with a constant wave speed equal to (1 − 1/*F*_{0}): at criticality the wave speed vanishes. The second mode is invariably damped at the constant rate (1 + *F*_{0}/2).

[25] The morphodynamic mode ω_{3} migrates downstream (upstream) under subcritical (supercritical) conditions, with constant wave speed γλ/(1 − *F*_{0}^{2}), which becomes unbounded as *F*_{0} 1; moreover, the morphodynamic mode is invariably damped at a constant rate which becomes unbounded as *F*_{0} 1.

[26] The above picture suggests that the decoupled approach becomes singular as λ ∞ with *F*_{0} 1: in this limit, decoupling is no longer an appropriate procedure. We then show how the singularity can be removed by allowing some coupling between hydrodynamics and morphodynamics.

##### 2.1.2. Coupled Theory: A Perturbation Solution

[27] We may achieve coupling while still seeking solutions in the form of perturbation expansions, as long as we establish a relationship between the relative sizes of the small parameters λ^{−1} and γ. It turns out that the most interesting picture arises if we set

with Γ and *f O*(1) parameters, and expand the eigenvalue ω in powers of λ^{−1} as follows:

[28] Substituting from (26) and (27) into the dispersion relationship (14) and equating likewise powers of λ, at the leading order of approximation, *O*(λ), we find

[29] Hence two nontrivial modes arise such that

[30] It is instructive to consider the particular case obtained by setting ourselves at criticality, i.e., by choosing *f* = 0. Then

[31] We can consider two different cases, depending on the value attained by the parameter Γ: if 0 < Γ < 9/8, then both modes are stable and nonmigrating at the leading order of approximation; if Γ > 9/8, then ω_{01} is stable and migrates downstream while ω_{02} is stable and migrates upstream.

[32] Proceeding to the next order of approximation, *O*(λ^{0}), the first two modes are slightly corrected and a third morphodynamic nontrivial mode arises. It reads

[33] Hence, for negative values of *f*, such a mode is stable and migrates downstream, while, for positive values of *f*, this mode is unstable and migrates downstream. This is a very weak instability which, as shown in the next section, does not persist in the strongly nonlinear regime.

#### 2.2. Nonlinear One-Dimensional Waves

[34] We now examine to what extent the above linear predictions are confirmed in the nonlinear regime. To this aim, we refer to some results of Siviglia et al. (paper to be presented, 2006), who have developed a numerical code able to cope with the fully nonlinear form of the governing equations. We refer the reader to the latter paper for details of the numerical procedure employed. It suffices here to state that the one-dimensional governing equations were written in a quasi-conservative form [*Toro*, 2005] and solved by a quasi-conservative version of the MacCormack method.

[35] Let us first consider the behavior of short bottom perturbations subject to a supercritical flow. Under these conditions, the uncoupled linear theory (equation (19)) predicts upstream propagation and damping, while the coupled linear theory (equation (31)) predicts growth close to criticality as well as downstream propagation. The nonlinear response to an initial perturbation in the form of a small and short hump is depicted in Figure 4 and displays various interesting features: the perturbation damps, following a short initial stage of weak growth; the hump migrates upstream; a secondary weaker hump is generated, migrates downstream and is also weakly damped; and nonlinearity gives rise to the formation of sharp fronts. Hence the nonlinear response deviates from the linear one in many respects: in particular, the linear instability predicted by the coupled theory does not persist in the nonlinear regime. Moreover, the morphodynamic influence is felt both upstream and downstream through the formation of the secondary hump. Note that the latter feature emerged also in the numerical solutions of *Lyn and Altinakar* [2002].

[36] Let us next consider the behavior of short bottom perturbations subject to a subcritical flow. Under these conditions, the linear uncoupled theory (equation (19)) predicts downstream propagation and damping; similar predictions arise from the coupled linear theory (equation (31)). The nonlinear response to an initial perturbation in the form of a small and short hump is depicted in Figure 5. It shows that the perturbation is damped, following a short initial stage of weak growth; the hump migrates downstream; a secondary weaker hump is generated, migrates upstream, and is also weakly damped; and nonlinearity gives rise to the formation of sharp fronts. Hence the nonlinear response near criticality deviates from the linear one in many respects: in particular, while linear perturbations as predicted by the coupled theory decay, nonlinear effects display an initial growth which, however, does not persist in the strongly nonlinear regime. Moreover, the morphodynamic influence is again felt both upstream and downstream through the formation of the secondary hump.

[37] Let us next consider the behavior of long bottom perturbations subject to a subcritical flow. Under these conditions, the linear theory (equation (22)) predicts that the morphodynamic mode migrates downstream (upstream) under subcritical (supercritical) conditions with very small wave speed (of order γλ^{2}) which vanishes identically at critical conditions. Moreover, the morphodynamic mode is invariably damped, though at a quite slow rate, of order γλ^{2}. The nonlinear response to an initial perturbation in the form of a small and long hump is depicted in Figure 6. It confirms the results obtained at the linear level. Note that no secondary hump is generated in this case and nonlinearity is unable to give rise to the formation of sharp fronts. The latter picture displays essentially the diffusive behavior described as dispersive by *Lisle et al.* [2001].

[38] Let us finally consider the response of an erodible bed to the propagation of hydrodynamic fronts. We focus here on the effect of bed load transport on the propagation of hydraulic jumps as well as on the issue of existence of stationary hydraulic jumps in erodible channels. Results of this analysis will also provide reasonable suggestions for the somewhat related problem of roll waves. Recently, *Bellal et al.* [2003] have observed experimentally that the propagation of a hydraulic jump, originated by imposing an abrupt increase of the water level at the downstream channel end, undergoes two stages: a first stage of fairly fast propagation, dominated by the hydrodynamics and practically nonaffected by sediment transport, followed by a stage where the jump has slowed down sufficiently for its hydrodynamic timescale to be comparable with the morphodynamic response time. At this stage, the sharp reduction of the sediment transport capacity of the stream through the jump has a chance to produce its effects: a sediment front is generated and propagates downstream producing the progressive disappearance of the hydraulic jump and the tendency of the stream to reach its final uniform equilibrium state. This picture is perfectly reproduced by the numerical simulations reported in Figure 7. The latter plots also show that in the initial stage of the process, the migration of the hydraulic jump overshoots the location where a steady jump would exist on a fixed bed. We have not yet performed simulations concerning the propagation of roll waves on an erodible bed. However, it is quite reasonable to expect that roll waves migrate too fast to “feel” the presence of a cohesionless bed with any significant intensity.

### 4. Planform Waves

- Top of page
- Abstract
- 1. Introduction
- 2. One-Dimensional Perturbations
- 3. Two-Dimensional Waves
- 4. Planform Waves
- 5. Concluding Remarks
- Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
- Acknowledgments
- References

[55] Planform waves in erodible channels develop in response to the effects of bank erosion: in straight channels this may lead to width variations coupled with aggradation-degradation propagated throughout the channel reach; in meandering rivers the propagated information is a variation of channel alignment. While the former problem has not been thoroughly investigated yet (but see *Cantelli et al.* [2004]), the latter is a fairly mature subject, stemming from the seminal contribution of *Ikeda et al.* [1981], who first showed that river meandering can be explained as the outcome of a planimetric instability driven by erosion of outer banks and deposition at the inner bends, a mechanism described as bend instability. It is not our aim, here, to review the wide literature on the mechanics of river meandering. The reader interested in an overview of the subject is referred to the recent survey paper of *Seminara* [2006]. Our goal is to attempt clarifying some aspects of the problem, related to the main issue considered in this paper, namely morphodynamic influence.

[56] In order to set the scene for this discussion let us briefly summarize the main ingredients of the formulation of the problem. The framework employed to predict the planform evolution of erodible channels with constant width is based on the scheme originally proposed by *Ikeda et al.* [1981]: the channel is identified through its centreline and its motion is described by stipulating that each point moves in the normal direction with a lateral migration speed ζ* driven by bank erosion. On the basis of purely geometrical arguments, the latter framework leads to a nonlinear partial integrodifferential equation governing planform evolution. *Seminara et al.* [1994, 2001] have derived a compact intrinsic form of such equation, which reads

where θ(*s*, *t*) is the angle that the local tangent to the channel axis forms with a Cartesian axis *x** (see Figure 10), *s* is the longitudinal coordinate, scaled by half the channel width *B**, ζ is the lateral migration speed of the channel, scaled by some reference speed *U**_{0}, while *t* is time, scaled by *B**/*U**_{0}.

[57] The latter equation must be coupled with a model able to provide an integrated description of bank erosion: in other words, the model must replace the actual intermittent process by an effective, spatially and temporally continuous, process. This may be achieved through the rule proposed by *Ikeda et al.* [1981]: the shift of channel centreline is driven by the differential excess of flow speed at the outer and inner banks:

where the depth averaged longitudinal velocity U is scaled by *U**_{0} and *E* is a dimensionless long-term erosion coefficient.

[58] The problem is closed once a suitable model for flow and bed topography in sinuous channels with erodible bed is employed to evaluate the quantity ζ. Below, along with the ZS model, we will consider also the pioneer model of *Ikeda et al.* [1981, hereinafter referred to as IPS], which has enjoyed a great popularity in the American communities of geomorphologists and physicists [e.g., *Howard and Knutson*, 1984; *Sun et al.*, 1996; *Stolum*, 1996; *Edwards and Smith*, 2002]. The latter model has been later modified by *Johannesson and Parker* [1989, hereinafter referred to as JP] to account for the coupling between hydrodynamics and morphodynamics.

[59] What are the common features of the above models? They are all linear. All of them predict bend instability: in other words, all models predict that a small perturbation of the straight alignment grows in time selecting some preferred wavelength characterized by maximum growth rate. All of them are able to reproduce the conditions for incipient neck cutoff, none of them is able to generate chute cutoffs.

[60] On the contrary, because of the approximations involved in the different derivations, various properties of flow and bed topography in meandering channels do not arise in all models. Let us list some of them.

[61] The peak of linear bend instability occurs for values of the dimensionless wave number λ (scaled by 1/*B**) and of the aspect ratio β, close to the values λ_{r} and β_{r} corresponding to the resonant excitation of some free mode consisting of the stationary alternate bars discussed in section 3.2: the occurrence of resonance is predicted by ZS and JP, but cannot emerge in IPS as in the last model bed topography is not solved for.

[62] Bend migration is driven by a phase lag between bank erosion and curvature [*Seminara*, 2006]. Now, as in any linear oscillator, a change in the sign of the phase lag of the response to the forcing is associated with resonance: in other words, subresonant meanders (β < β_{r}) migrate downstream, while superresonant meanders (β > β_{r}) migrate upstream. This behavior cannot emerge in IPS.

[63] The nature of bend instability is consequently altered in IPS. Indeed, S. Lanzoni and G. Seminara (On the nature of meander instability, submitted manuscript, 2006, hereinafter referred to as Lanzoni and Seminara, submitted manuscript, 2006) have shown that in the subresonant regime, bend instability is invariably convective and the group velocity of meander trains is positive, while, under superresonant conditions, bend instability is generally, although not invariably, convective and the group velocity is negative. On the contrary, in IPS instability is invariably convective with positive group velocity.

[64] The nonlinear planform evolution also displays different characteristics: in particular single meanders are typically upstream (downstream) skewed under subresonant (superresonant) conditions, unlike in IPS, where only the subresonant behavior can be reproduced.

[65] Most of these features emerge in the Figures 12a and 12c where it appears that under subresonant (superresonant) conditions and in the absence of a persisting disturbance located at the upstream (downstream) end, perturbations of channel alignment develop into wave groups which migrate downstream (upstream), leaving the upstream (downstream) reach unperturbed, i.e, recovering the initial straight configuration. A different scenario is observed for large enough values of the parameters β and τ_{*}, when, in accordance with the linear findings, a transition from convective to absolute instability is observed (Lanzoni and Seminara, submitted manuscript, 2006): in this case meander trains spread over the entire flow domain, as shown in the Figure 12e.

[66] The nature of bend instability bears a great relevance for numerical simulations of the planform evolution of alluvial rivers. In fact, the choice of the boundary conditions to be applied at the channel ends is not arbitrary: as shown in Figures 12b and 12d, imposing periodic boundary conditions is actually not compatible with the convective nature of bend instability and leads to planform patterns significantly different from those obtained assuming free end conditions.

[67] The simulations reported in Figure 12 have all been stopped before the occurrence of the first neck cutoff, and therefore they give information only on the short-term behavior of meander evolution. On longer timescales, the repeated occurrence of cutoff events imposes further constraints which prevent the infinite growth of meander bends. The values attained by meander sinuosity, i.e., the ratio of intrinsic to cartesian wavelength, are also limited by cutoffs, attaining values lower than 2–2.5: this is well known from several field observations [*Leopold and Wolman*, 1960; *Chitale*, 1970; *Hey and Thorne*, 1986]. Comparing the planform patterns resulting from numerical simulations based on the use of different morphodynamic models, *Camporeale et al.* [2005] have recently suggested that the statistics of fully developed meandering patterns tend to be almost similar, owing to the filtering action exerted by cutoff events. Numerical simulations were performed assuming subresonant conditions, plane bed and forcing the channel ends to be locally straight. To what extent the latter assumptions affect the conclusions of *Camporeale et al.* [2005] is, in our opinion, an open issue which will deserve further attention in the near future.