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Keywords:

  • long waves;
  • morphodynamics;
  • sediment waves

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. One-Dimensional Perturbations
  5. 3. Two-Dimensional Waves
  6. 4. Planform Waves
  7. 5. Concluding Remarks
  8. Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
  9. Acknowledgments
  10. 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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. One-Dimensional Perturbations
  5. 3. Two-Dimensional Waves
  6. 4. Planform Waves
  7. 5. Concluding Remarks
  8. Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
  9. Acknowledgments
  10. 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]).

image

Figure 1. Sketch of a typical impulse response: (a) absolute instability and (b, c, and d) convective instability. The parameter ωi denotes the temporal growth rate of the perturbation.

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[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

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

[6] Let us start considering a straight rectangular channel, with cohesionless bottom of uniform grain size d*s, width 2B* 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.

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Figure 2. Sketch of the channel and notations.

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[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:

  • equation image
  • equation image
  • equation image

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:

  • equation image

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:

  • equation image
  • equation image

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

  • equation image
  • equation image
  • equation image

The following parameters emerge:

  • equation image

where F0 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

  • equation image

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

  • equation image

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

  • equation image

where equation image 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:

  • equation image

where

  • equation image

[12] Note that the relationship (14) defines, in general, three eigenmodes ωj (j = 1, 2, 3) depending on the wave number λ and on the values attained by the relevant physical parameters F0 and γ. The imaginary part of ωj determines the growth rate of the perturbation, (ωj)i. The perturbation is unstable if (ωj)i > 0, otherwise it decays. Moreover, the perturbation wave speed aj is given by the ratio (ωj)r/λ, with (ωj)r the real part of ωj.

[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 F0 to be finite, it seems reasonable to expand ω in powers of γ as follows:

  • equation image

[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 O0), we find a classical result: one of the eigenvalue (ω03) vanishes and the remaining two eigenvalues reduce to those found in the fixed bed case, namely,

  • equation image

[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.

image

Figure 3. Dependence of the temporal growth rate (ωj)i and of the phase speed aj on the wave number λ for three different values of the basic Froude number. (a and b) Behavior of the hydrodynamic modes (j = 1, 2) obtained at the leading order in the decoupled theory. Note that one of the modes is unstable and describes the formation of roll waves in channels with a fixed bed. (c and d) Behavior of the morphodynamic mode obtained at order O(γ) in the decoupled theory. Waves migrate downstream (upstream) under subcritical (supercritical) conditions.

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

  • equation image
  • equation image

[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 [RIGHTWARDS ARROW] ∞, describes perturbations whose length scale is much smaller than C2D0 (= B*C2D*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 [RIGHTWARDS ARROW] 0, it is easy to find

  • equation image
  • equation image
  • equation image

[21] It then turns out that both the hydrodynamic modes, ω1 and ω2, migrate downstream with constant wave speed equal to 1.5 U0 and 0.5 U0, respectively. Moreover, the first hydrodynamic mode is damped (amplified) if F0 < 2 (F0 > 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 ω3 migrates downstream (upstream) under subcritical (supercritical) conditions with very small wave speed (of order γλ2) which vanishes identically at critical conditions, and is invariably damped, though at a quite slow rate, of order γλ2.

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

  • equation image
  • equation image
  • equation image

[23] The first hydrodynamic mode ω1 migrates invariably downstream with constant wave speed equal to (1 + 1/F0): at criticality the wave speed tends to 2. This mode is damped (amplified) if F0 < 2 (F0 > 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 ω2 migrates downstream (upstream) under supercritical (subcritical) conditions with a constant wave speed equal to (1 − 1/F0): at criticality the wave speed vanishes. The second mode is invariably damped at the constant rate (1 + F0/2).

[25] The morphodynamic mode ω3 migrates downstream (upstream) under subcritical (supercritical) conditions, with constant wave speed γλ/(1 − F02), which becomes unbounded as F0 [RIGHTWARDS ARROW] 1; moreover, the morphodynamic mode is invariably damped at a constant rate which becomes unbounded as F0 [RIGHTWARDS ARROW] 1.

[26] The above picture suggests that the decoupled approach becomes singular as λ [RIGHTWARDS ARROW] ∞ with F0 [RIGHTWARDS ARROW] 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

  • equation image

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

  • equation image

[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

  • equation image

[29] Hence two nontrivial modes arise such that

  • equation image

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

  • equation image

[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 ω01 is stable and migrates downstream while ω02 is stable and migrates upstream.

[32] Proceeding to the next order of approximation, O0), the first two modes are slightly corrected and a third morphodynamic nontrivial mode arises. It reads

  • equation image

[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].

image

Figure 4. Example of the nonlinear morphodynamic response to an initial perturbation in the form of a small and short hump (λ = 292) under slightly supercritical conditions (F0 = 1.02, γ = 1.95 10−3, C = 15.3). The bed elevation displays nonlinear damping, following a short initial stage of weak growth, upstream migration of the bump, and the generation of a secondary weaker bump which migrates downstream and is also weakly damped. Also note that nonlinearity gives rise to the formation of sharp fronts.

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[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.

image

Figure 5. Example of the nonlinear morphodynamic response to an initial perturbation in the form of a small and short hump (λ = 303) under slightly subcritical conditions (F0 = 0.98, γ = 1.68 10−3, C = 15.4). The bed elevation displays nonlinear damping following a short initial stage of weak growth, downstream migration of the hump, and the generation of a secondary weaker hump which migrates upstream and is also weakly damped.

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[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 γλ2) which vanishes identically at critical conditions. Moreover, the morphodynamic mode is invariably damped, though at a quite slow rate, of order γλ2. 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].

image

Figure 6. Nonlinear morphodynamic response to an initial perturbation in the form of a small and long hump, which displays a diffusive behavior (very weak migration and weak damping) described as dispersive by Lisle et al. [2001] (F0 = 0.51, γ = 0.001).

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[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.

image

Figure 7. Nonlinear morphodynamic response to the propagation of a hydraulic jump, which shows a very weak effect of sediment transport in (a) the initial phase of fast propagation of the jump followed by (b) a strong interaction in the later stage when the jump slows down. Also note the overshooting of the jump (F0 = 1.27, γ = 0.004, C = 18.3).

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3. Two-Dimensional Waves

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

[39] The information concerning large-scale bed perturbations with a lateral structure is propagated through sediment waves, called bars, characterized by heights scaling with flow depth and wavelengths of the order of several channel widths: these waves are commonly observed in sandy streams as well as in gravel bed rivers. The subject of river bars has been widely investigated in the literature. The reader interested in an overview of the subject is referred to the survey paper of Tubino et al. [1999]. Our goal, here, is to review some more recent results related to our present concern, namely morphodynamic influence. To this aim, we will examine distinctly the cases of migrating versus stationary bars, a classification that will appear to be more appropriate than the classical distinction between free and forced bars introduced by Seminara and Tubino [1989].

3.1. Migrating Bars

[40] It has been known for over two decades that a uniform turbulent open channel flow over an erodible bottom is able to support the formation and migration of large-scale perturbations in the form of alternate sequences of pools and riffles separated by diagonal fronts, arranged either in single rows (alternate bars, observed in sufficiently narrow channels), or in multiple rows (multiple row bars, developing in wide channels). The mathematical treatment of this problem is made somewhat simpler by the large spatial scales involved in the process, which allow the use of the steady shallow water equations for the flow field, coupled to the two-dimensional version of the Exner equation and an appropriate closure for bed load transport over sloping beds [e.g., Seminara, 2006]. The choice to consider a quasi-steady approach, whereby the flow field is assumed to adapt instantaneously to changes in bottom configuration, is motivated by the sharply different timescales governing hydrodynamics and morphodynamics. When a significant portion of sediment is transported in suspension, an advection-diffusion equation for the concentration field is required to evaluate the suspended flux, though the simpler relationship obtained for suspended load in slowly varying flows by Bolla Pittaluga and Seminara [2003] may also be used under appropriate conditions, as recently shown by Federici and Seminara [2006].

[41] At a linear level, a classical normal mode analysis is readily performed assuming a Fourier decomposition of the perturbations in both horizontal directions: hence perturbations of the longitudinal component of flow velocity are expressed in the form:

  • equation image

with s and n dimensionless longitudinal and lateral coordinate (both scaled with the half channel width B*), with origin at the channel axis, and λ perturbation wave number (scaled by 1/B*). Similar perturbations are assumed for the lateral velocity, the flow depth and bottom elevation. Substituting from the perturbed configuration into the governing differential problem a dispersion relationship is obtained. The latter may be expressed in the general form [Federici and Seminara, 2003]

  • equation image

where N, nj (j = 0, 3) and di (i = 0, 2) are functions of the relevant parameters of the problem, namely the aspect ratio of the channel β(= B*/D*0), the Shields stress of the basic uniform flow τ* and the relative roughness ds, defined as the ratio between the (uniform) grain size and the undisturbed flow depth D*0. Note that (33) holds for both real and complex values of ω and λ and allows one to define two classes of modes, namely temporal and spatial modes.

[42] Temporal modes are characterized by complex values of the frequency ω and real values of the bar wave number λ: for any such mode (given m) and given values of the remaining parameters, (33) establishes a real relationship between the temporal growth rate of perturbations (i.e., the imaginary part ωi of the angular frequency) and the bar wave number λ; a second real relationship is established between the wave speed ωr/λ (with ωr real part of the angular frequency) and the bar wave number λ.

[43] Conversely, spatial modes are characterized by complex values of the bar wave number λ and real values of the frequency ω: for any such mode (given m) and given values of the remaining parameters, (33) establishes a real relationship between the spatial growth rate of perturbations (i.e., the imaginary part λi of the complex wave number) and the angular frequency ω; a second real relationship is established between the perturbation wave speed ω/λr (with λr real part of the bar wave number) and the angular frequency ω.

[44] Temporal modes have been investigated by Colombini et al. [1987]. In that paper, it was recognized that the aspect ratio β plays the role of the control parameter for bar instability, a framework later generally employed: hence, for given values of τ* and ds, the linear theory predicts a threshold value of the aspect ratio (βcm) above which bars of mode m are expected to grow and defines a most unstable wave number (λcm). The latter work, based on the assumption of bed load dominated conditions, has later been extended to the case when a significant fraction of sediment is transported in suspension [see Tubino et al., 1999]. The picture emerging from the above efforts is in general accordance with the field observations of alternate bars in straight channels and multiple row bars in braided rivers. More recently, Federici and Seminara [2003] have tackled the problem of interest for the present overview, namely the problem of morphodynamic influence. Using Briggs' [1964] criterion to distinguish between convective and absolute instabilities, they proved analytically that the nature of bar instability is invariably convective, at least at the linear level. Moreover, the wave group of perturbations was found to be positive, hence migrating bars propagate their information downstream. Finally, the convective nature of bar instability implies that the development of bars commonly observed in rivers as well as in laboratory flumes requires the existence of a persistent forcing of some perturbation. In this sense, migrating bars cannot be thought of as actually free perturbations: rather, understanding the process whereby they develop in space and time requires an investigation of the forced growth of spatial modes.

[45] This has been recently done through the numerical solution of the fully nonlinear equations governing the morphodynamic problem [Federici and Seminara, 2003; Defina, 2003]. The picture emerging from these works confirms the convective nature of bar instability in the nonlinear regime (see Figure 8). More precisely, starting from a nonpersistent (either randomly distributed or localized) perturbation of bottom topography, bars are found to grow and migrate downstream leaving the source area undisturbed. In the presence of a persistent perturbation located at the initial cross section, the nonlinear development leads to the formation of a periodic pattern characterized by an equilibrium amplitude independent of the amplitude and harmonic content of the initial forcing; bars slow down and lengthen throughout the growth process in accordance with the classical experimental observations of Fujita and Muramoto [1985]. The distance from the initial cross section at which equilibrium is achieved depends on the amplitude of the initial perturbation: a result which suggests the need for a careful examination of laboratory observations reported in the literature in order to ascertain that equilibrium had indeed been reached in the reported experiments.

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Figure 8. A sample of the temporal evolution of the bed topography generated by Federici and Seminara [2003] starting from an initial, small and randomly distributed bottom perturbation (β = 12; τ* = 0.09; ds = 0.04). A bar train forms and progressively migrates downstream, thus leaving the bed unperturbed.

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[46] In a recent paper, Federici and Seminara [2006] have extended the above analysis to the case of sandy rivers, in which a nonnegligible amount of sediment is transported in suspension. The overall picture of the process does not change qualitatively, i.e., the instability keeps invariably convective with wave groups migrating always downstream.

3.2. Stationary Bars

[47] A uniform turbulent open channel flow over an erodible bottom is also able to support the formation of stationary large-scale perturbations of bed topography. In the linear case, they are readily found as particular cases of the class of perturbations investigated in section 3.1 by simply setting ω to vanish in the dispersion relationship (33): hence these perturbations do not amplify/decay in time (but may amplify/decay in space) and do not migrate. They must then satisfy the dispersion relationship:

  • equation image

[48] For each lateral mode m, four solutions for the complex wave number λ are defined by (34). They were first investigated by Olesen [1983] and Struiksma et al. [1985] and were later reconsidered by Seminara and Tubino [1992] in different contexts. Typically, as shown in Figure 9, (iλm1) and (iλm2) are purely real, O(1) quantities, with opposite signs (iλm1 > 0, iλm2 < 0): they describe nonoscillatory spatial perturbations which decay fairly fast either downstream or upstream. The other two solutions iλm3, iλm4 are complex conjugate. Since their real (λm3r = −λm4r) and imaginary (λm4i = λm3i) parts attain values ranging about 0.1–0.3, they describe oscillatory spatial perturbations which decay fairly slowly, spreading their influence over a considerable channel length. Moreover, −λm3i is found to be negative or positive depending on the aspect ratio of the channel being respectively smaller or larger than a threshold value, βrm. At the threshold conditions these perturbations are purely oscillatory and, for the lower mode (β = βr1 = βr), they describe a periodic sequence of nonmigrating and nonamplifying alternate bars, quite similar to the pattern of point bars observed in meandering channels. Such modes play an important role in the mechanism of meander formation as they can be resonantly excited in meanders with a periodic distribution of channel curvature [Blondeaux and Seminara, 1985]. For this reason, we refer to channels characterized by aspect ratios β < βr as subresonant, while channels with aspect ratios β > βr will be described as superresonant.

image

Figure 9. Typical behavior of the the solutions of the dispersion relationship for steady perturbations of the alternate bar type, m = 1 (τ* = 3, ds = 0.005, dune covered bed).

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[49] Recently, Federici and Seminara [2006] have extended the analysis of stationary waves to the case of sandy streams where a significant fraction of sediments is transported in suspension: the picture emerging in this case confirms the main findings described above, though various features are altered both qualitatively and quantitatively: in particular, the resonant values of the aspect ratio are significantly lowered by the presence of suspended load.

[50] Essentially, stationary waves of the above type represent spatially growing or decaying bed undulations which may arise in response to different forcing effects, e.g., nonuniform but steady initial conditions in a straight channel, channel curvature, variations in channel geometry such as abrupt or gradual widening/narrowing.

3.3. Upstream-Downstream Influence Driven by Channel Curvature

[51] Let us next consider an erodible channel with slowly varying, bur otherwise arbitrary, distribution of channel curvature. A number of models for flow and bed topography in such channels have been proposed in the literature of the 1980s: they have been reviewed and compared by Parker and Johanneson [1989]. In the following, we concentrate on the model of Zolezzi and Seminara [2001, hereinafter referred to as ZS], an extension of the original model of Blondeaux and Seminara [1985]. It is a linear model, i.e., assumes that perturbations of flow and bed topography driven by deviations of channel alignment from the straight configuration are small, an assumption which has been shown to be valid [Seminara and Solari, 1998] provided some scour parameter δ attains values not larger than about 10. The latter parameter reads

  • equation image

where ν0 is the curvature ratio B*/r*0, with r*0 typical radius of curvature at the bend apex (see Figure 10), τ* is a characteristic average Shields stress, Cf0 is a characteristic friction coefficient and r is an empirical parameter ranging about 0.5.

image

Figure 10. Sketch of a meandering channel and notations.

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[52] Moreover, in this model, morphology is fully coupled with hydrodynamics, a crucial feature in many respects. In particular, Zolezzi and Seminara [2001] derived a depth averaged form of the hydrodynamic equations, accounting for the dispersive effect of the centrifugally induced secondary flow which develops in curved channels. Taking advantage of the fact that in natural alluvial channels the effects of curvature are usually weak, ZS were then able to derive the exact general solution of the linear problem of morphodynamics, expanding the flow field in powers of the typically small curvature parameter ν0 as well as in Fourier series in the lateral direction n. Hence ZS write

  • equation image
  • equation image
  • equation image

where M = (2m + 1) π/2, and fd, fh are linear functions of the channel curvature and of its derivatives. Moreover, U, V are the dimensionless longitudinal and transverse components of the depth-averaged velocity, scaled by U*0, D is the dimensionless local flow depth, scaled by D*0, and H is the free surface water elevation above a given reference level, scaled by F02D*0. The scaling quantities U*0, D*0 and F0 represents, respectively, the cross-sectionally averaged speed, the flow depth and the Froude number of a uniform flow characterized by the same flow discharge, channel width and average slope of the meandering reach under investigation. ZS solve the nonhomogeneous fourth-order ordinary differential equation for um obtained substituting from the expansions (36) into the governing equations, and find

  • equation image

where s is the dimensionless longitudinal curvilinear coordinate, scaled by B*, s0 denotes the beginning of the meandering reach, &#55349;&#56478;(s) is the local value of the dimensionless channel curvature, scaled by 1/r*0, Am(= (−1)m 2/M2) is a coefficient quantifying the decaying contribution of higher lateral Fourier modes, gjk (j = 1, 4; k = 0, 1) are coefficients depending on the relevant physical parameters (β, Cf0 and τ*), cmj are integration constants, to be specified on the basis of the boundary conditions at the channel ends, and λmj (m = 0, ∞; j = 1, 4) are four characteristic exponents, solutions of the dispersion relationship (34).

[53] The structure of (39) is of crucial importance for the issue of morphodynamic influence. In fact, the flow and bed topography at a given section are affected by the local value of curvature but also, through the four convolution integrals, by the hydrodynamics and morphodynamics of the reaches located upstream (downstream influence) or downstream (upstream influence). Two different scenarios arise under subresonant and superresonant conditions: their characteristics emerge clearly if the solution for um is rewritten in the form

  • equation image

where

  • equation image

with ϕ a phase function and L the overall intrinsic length of the meandering reach. From (40) and (41) it follows that in the subresonant case (β < βr) the influence of the downstream distribution of channel curvature is rather weak, being associated with the convolution integral involving the characteristic exponent λm1, which decays rapidly downstream (i λm1O(1); iλm1(s − ξ) < 0). On the contrary, the downstream influence of the upstream distribution of curvature is quite strong being determined by the convolution integral involving the complex characteristic exponent λm3 which leads to an oscillatory pattern, weakly decaying upstream (∣λm3i∣ = O(10−1); −λm3i(s − ξ) < 0). A further weak contribution to downstream influence is associated with the convolution integral involving the real characteristic exponent λm2 which, indeed, decays rapidly upstream (∣iλm2∣ = O(1); iλm2(s − ξ) < 0). Finally, the terms involving the constants cmj (j = 1, 4) describe the influence of the upstream and downstream boundary conditions: again, the oscillatory decaying behavior characterizing the terms involving the complex conjugate exponents imply that the influence of the upstream boundary condition spreads over a channel length much larger than the length of the channel reach influenced by the downstream boundary condition. The above scenario is completely reversed when the channel falls in the superresonant regime (β > βr).

[54] A prototype configuration on which the predictions of Zolezzi and Seminara [2001] have been tested is the problem of overdeepening, a phenomenon observed by Struiksma et al. [1985] in a laboratory flume with movable bed, consisting of a constant curvature reach connected to two straight reaches located upstream and downstream. Essentially, the fully developed topography in a bend is established through a spatial transient consisting of a damped oscillatory pattern; moreover, the deformed topography established at the bend exit decays in the straight downstream reach through a similar damped oscillatory pattern. However, the analysis of Zolezzi and Seminara [2001] suggests that overdeepening may also occur in the straight reach located upstream of the bend entrance under superresonant conditions. In order to test the latter theoretical prediction, Zolezzi et al. [2005] carried out two series of experiments in a laboratory channel composed by a straight reach connected with a bend of constant curvature, spanning 270°, and a flume consisting of two fairly long straight reaches, connected through a 180° bend of constant curvature. Experimental observations conclusively confirm the theoretical predictions, indicating the existence of two different morphodynamic regimes depending on the value of the width ratio β. As shown in Figure 11, the presence of geometrical discontinuities associated with sharp variations of channel curvature is only felt downstream in sufficiently narrow channels with β < βr (Figure 11a), while it dominantly affects the upstream reach in wider channels such that β > βr (Figure 11b).

image

Figure 11. Steady longitudinal bed profiles measured close to the left (open circles) and right (solid circles) walls of a 180° bend by Zolezzi et al. [2005] under (a) subresonant and (b) superresonant conditions. The phenomenon of the upstream overdeepening is clearly seen to occur only under superresonant conditions: indeed, the presence of the bend is felt in the upstream reach through the formation of stationary alternate bars, slowly decaying upstream.

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4. Planform Waves

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. One-Dimensional Perturbations
  5. 3. Two-Dimensional Waves
  6. 4. Planform Waves
  7. 5. Concluding Remarks
  8. Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams
  9. Acknowledgments
  10. 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

  • equation image

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:

  • equation image

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 λr and βr 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 < βr) migrate downstream, while superresonant meanders (β > βr) 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.

image

Figure 12. Response of the planform of an erodible channel to small random perturbations of an initial straight configuration. All the simulations were carried to incipient neck cutoff, setting E = 1.85 · 10−8 and assuming a dune-covered bed. The following boundary conditions were adopted: (a, c, and f) no constraints at the channel ends and (b and d) periodic boundary conditions. Figures 12a and 12b are characterized by subresonant conditions (β = 8, τ* = 0.3, ds = 0.005) and show a convective instability: wave groups migrate downstream leaving the upstream reach unperturbed. Figures 12c and 12d are characterized by superresonant conditions (β = 15, τ* = 0.3, ds = 0.005): wave groups migrate upstream leaving the downstream reach unperturbed. Figure 12e is characterized by superresonant conditions (β = 25, τ* = 0.7, ds = 0.005), but the instability is absolute: wave groups migrate upstream but spread over the entire domain. Note that each plot shows the planform patterns at the beginning of the simulation, at incipient neck cutoff, and at two intermediate times.

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[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.

5. Concluding Remarks

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

[68] The first issue deserving some final comment is coupling versus decoupling in one-dimensional modeling. Our findings suggest that coupling is necessary when simulating the evolution of fairly short perturbation in a neighborhood of the critical state. On the contrary, very long bottom waves, even if transcritical, display a quasi-diffusional behavior which can be satisfactorily modeled using a decoupled approach. This picture, emerged from our linear analysis, persists in the nonlinear regime. Moreover, the propagation of hydrodynamic fronts is too fast to be affected by sediment transport: however, as the jump slows down, then disequilibrium of sediment transport leads to the slow propagation of a sediment front which interacts strongly with the hydrodynamics, in a fashion which requires a coupled approach. A second comment is deserved by the picture emerged from our overview of the state of the art concerning modeling of bar and planform waves. A warning appears to be in order here. The discussion on the performances of the various models presented in Section 4 clearly suggests that short-term predictions of planform evolution is rather strongly dependent on the choice of the model employed to describe the hydrodynamics and morphodynamics of meandering channels. A number of further developments of the subject await to be pursued. The role of transport in suspension has not been fully explored. Planform waves driven by channel widening and deepening in erosional settings also need further attention: note that they excite a fourth mode whereby information can be propagated through one-dimensional waves. The problem of morphodynamic influence in braided rivers is completely unexplored, though an interesting, further example of upstream influence has been observed by Bertoldi et al. [2005] in laboratory experiments performed on a model of bifurcation, displaying the formation of a stationary bar upstream of the bifurcation. Finally, the issue of long-term meander evolution will deserve further attention, and a systematic validation of the predicted planforms, through the comparison with already available and new field data, is strongly needed.

Appendix A:: A Formal Derivation of the Mass Conservation Equations for Open Channel Flows on Erodible Streams

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

[69] In the recent literature on the subject of 1-D wave propagation in erodible streams [e.g., Lyn, 1987; Cao et al., 2002; Lyn and Altinakar, 2002; Cao and Carling, 2003; Cui et al., 2005] various contributions have suggested that the governing equations classically employed in the engineering literature might not be appropriate as terms associated with sediment transport would be missing in the hydrodynamic equations. Cui et al. [2005] have already clarified this point. In order to ascertain to what extent the additional terms may play some role in the present analysis which focuses on the coupled approach required under near critical conditions, we will formally derive the governing equations starting from first principles.

A1. Mass Conservation of the Fluid Phase

[70] Let us consider a dilute suspension of sediment particles flowing in an open channel with a cohesionless bottom. Let c the instantaneous (small) volume concentration of sediments and u the fluid velocity, both averaged over turbulence. Mass conservation of the fluid phase contained within any finite control volume V bounded by a surface S with outward normal n requires that the following integral condition be satisfied:

  • equation image

[71] Hence applying the Ostrogradski transformation and following classical arguments on the validity of (A1) for arbitrary choice of the control volume V, we end up with a slightly modified version of the continuity equation for an incompressible fluid, namely,

  • equation image

[72] Integrating (A2) between the bed interface (z = η(x, y, t)) and the free surface (z = h(x, y, t)) we obtain

  • equation image

[73] The boundary conditions associated to (A3) impose

  • equation image
  • equation image

having denoted by unh and unη the normal components of the speed of the free surface and bed interface, respectively. Moreover, cM is the solid concentration of the bed. The conditions (A4) and (A5) state that the relative water flux through the free surface must vanish while the relative water flux through the bed interface is determined by the difference between the volume fractions of the fluid phase at the two sides of the bed interface. Recalling that

  • equation image
  • equation image

with ℱh = zh and ℱη = z − η, the above boundary conditions can be written as

  • equation image
  • equation image

With the help of (A8) and (A9) and using Leibnitz rule, equation (A3) eventually takes the form

  • equation image

having denoted by p( =1 − cM) the bed porosity and by capital letters the depth averaged quantities. The equation (A10) is the 2-D version of the continuity equation for the fluid phase. If the dominant form of transport is bed load, then (A10) reduces to the simpler form

  • equation image

The 1-D version of (A10), (A11), is readily obtained by performing a further integration in the lateral direction y and imposing a vanishing water flux at the nonerodible channel banks. Denoting by Bf the width of the active portion of the bed we then find

  • equation image

where Ω is the cross sectional area, Q is the flow rate and an overbar denotes a quantity averaged over the entire cross section. In the bed load dominated case, the 1-D version of the continuity equation for the fluid phase becomes

  • equation image

The formulation (A12) is essentially identical with the one derived by Cui et al. [2005].

A2. Mass Conservation of the Solid Phase

[74] The derivation of the mass conservation equation for the solid phase can be performed in a quite similar fashion [see Seminara, 1998]. The outcomes of these derivations read

  • equation image
  • equation image

where qs = (qsx, qsy) and Qs are the vector sediment flux per unit width and the total sediment flux in the cross section, respectively. The above formulations simplify in an obvious way in the bed load dominated case.

Acknowledgments

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

[75] This paper is an expanded version of an invited lecture delivered by the senior author (G.S.) at the fourth meeting on River Coastal and Estuarine Morphodynamics, held in Urbana Champaign on 4–7 October 2005. The work has been funded by Cariverona (Progetto MODITE). Partial support has also come from the Ministry of Research and the University of Padua (Progetto di Ateneo Analisi del comportamento morfodinamico di alvei meandriformi in contesti ambientali diversi.)

References

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