Marco Tubino

Finite-amplitude alternate bars

Following ideas developed in the field of hydrodynamic stability of laminar flows (Stuart 1971) a predictive theory is proposed to determine the development of finite-amplitude alternate bars in straight channels with erodible bottoms. It is shown that an ‘equilibrium amplitude’ of bedforms is reached as t → ∞ within a wide range of values of the parameter (β − β c )/β c , where t is the time, β is the width ratio of the channel and β c is its ‘critical’ value below which bars would not form. The theory leads to relationships for the maximum height and the maximum scour of bars which compare satisfactorily with the experimental data of various authors. Moreover the experimentally detected tendency of the bed perturbation to form diagonal fronts is qualitatively reproduced.

Channel bifurcation in braided rivers: Equilibrium configurations and stability

[1] We investigate the equilibrium configurations and the stability of river bifurcations in gravel braided networks. Within the context of a one-dimensional approach, the nodal point conditions play a crucial rule, as pointed out by Wang et al. [1995] who propose an empirical relationship relating water and sediment flow rates into the downstream branches. In the present paper, an alternative formulation of nodal point conditions is proposed based on a quasi two-dimensional approach. The results show that, if the Shields parameter of the upstream channel is large enough, the system only admits of one solution with both branches open, which is invariably stable. As the Shields parameter of the upstream channel decreases, two further stable solutions appear characterized by a different partition of water discharge into the downstream branches: in this case, the previous solution becomes unstable. Theoretical findings are confirmed by the numerical solution of the nonlinear one-dimensional equations.

FREE-FORCED INTERACTIONS IN DEVELOPING MEANDERS AND SUPPRESSION OF FREE BARS

The coexistence of migrating alternate (free) bars, spontaneously developing in erodible channels as a result of an instability process, with steady point bars, forced by curvature in meandering reaches of rivers is investigated theoretically. A perturbation expansion is set up in terms of two dimensionless small parameters, describing free and forced perturbations. The effect of mixed interactions is found to be responsible for the damping and slowing down of free bars as channel curvature increases. The theory allows the determination of the threshold value of channel curvature above which free bars are suppressed as a function of meander wavenumber for given flow and sediment characteristics. The minimum channel sinuosity for free bar suppression is found to be associated with the resonant wavenumber range of Blondeaux & Seminara (1985). Theoretical predictions compare satisfactorily with experimental observations by Kinoshita & Miwa (1974). The theory also suggests that free bars may appear again in a more advanced stage of meander development in accordance with field observations by Kinoshita (1961).

Free bars in rivers

In the paper we review some recent work on the mechanics of formation and development of river bars. The emphasis is placed on the instability process which leads to the spontaneous development of bars in almost straight reaches of alluvial rivers. A three dimensional formulation of the problem is presented along with a discussion on the relevant closure relationships. Results of linear and non linear theories for free bars under bedload dominated conditions are summarised. Furthermore, account is given on the effect on bar instability induced by suspended load, grain sorting and width variations. Some as yet unpublished results are also presented.

Planimetric instability of channels with variable width

We study the steady three-dimensional flow field and bed topography in a channel with sinusoidally varying width, under the assumptions of small-amplitude width variations and sufficiently wide channel to neglect nonlinear effects and sidewall effects. The aim of the work is to investigate the role of width variations in producing channel bifurcation in braided rivers. We infer incipient bifurcation in cases where the growth of a central bar leads to planimetric instability of the channel, i.e. when the given infinitesimal width perturbation is enhanced. Results of the three-dimensional model suggest that the equilibrium bottom profile mainly consists of a purely longitudinal component, uniformly distributed over the cross-section, which induces deposition at the wide section and scour at the constriction, and of a transverse component in the form of a central bar (wide sections) and scour (constrictions), with longitudinal wavelength equal to that of width variations. A comparison between the results of the three-dimensional model and those obtained by means of a two-dimensional depth-averaged approach shows that the transverse component is mainly related to three-dimensional effects. Theoretical findings display a satisfactory agreement with results of flume experiments. Transverse variations are responsible for the planimetric instability of the channel; we find that in the range of values of Shields stress typical of braided rivers, the incipient bifurcation is enhanced as the width ratio of the channel increases.

Weakly nonlinear theory of regular meanders

Flow and bed topography in a regular sequence of meanders is shown to be strongly influenced by nonlinear effects within a fairly wide range of aspect ratios of the channel and meander wavenumbers. This finding is associated with the behaviour of meanders as nonlinear resonators in a neighbourhood of the resonance conditions discovered by Blondeaux & Seminara (1985). A weakly nonlinear approach valid for relatively small measures of channel curvature and within a neighbourhood of the resonant conditions displays all the typical features of nonlinear resonators, including non-uniqueness of the channel response

Grain sorting and bar instability

A two-dimensional model of flow and bed topography is proposed to investigate the effect of sediment heterogeneity on the development of alternate bars. Within the context of a linear stability theory the flow field, the bed topography and the grain size distribution function are perturbed leading to an integro-differential linear eigenvalue problem. It is shown that the selective transport of different grain size fractions and the resulting spatial pattern of sorting may appreciably affect the balance between stabilizing and destabilizing actions which govern bar instability. Theoretical results suggest that sediment heterogeneity leads to a damping of both growth rate and migration speed of bars, while bar wavelength is shortened with respect to the case of uniform sediment. The above findings conform, at least qualitatively, to the experimentally detected reduction of bar height, length and celerity. The observed tendency of coarser particles to pile up towards bar crests is also reproduced by theoretical results.

Semianalytical analysis of hyporheic flow induced by alternate bars

[1] We investigate the effects of alternate bar morphology on the hyporheic flow in gravel bed rivers. Our goal is to investigate the relations between residence time distribution of a conservative tracer and the parameters controlling bed form morphology. We assume homogeneous, isotropic or anisotropic hydraulic properties of the streambed sediment and constant flow regime in equilibrium with the bed form, which is considered fixed because its formation timescale is much longer than that of the subsurface flow. Under these assumptions, we solve the in‐stream and hyporheic flow fields analytically in a three‐dimensional domain. We approximate the former with the shallow water equations and model the latter as a Darcian flow. The two systems are linked through the hydraulic head distribution, which is predicted at the streambed by the surface model and applied as a boundary condition to the hyporheic flow model. We solve the solute transport equation in the hyporheic zone for a conservative tracer by means of particle tracking. Our model predicts that the mean value and variance of the hyporheic residence time depend on the alternate bar amplitude at equilibrium. This result is found to be applicable also to discharges that are lower (70% in our simulations) than the formative and submerge the bars entirely. Moreover, our analysis shows that 95% of the hyporheic flow is confined in a near‐bed layer, whose depth is about the width of the channel and shallows from low to steep gradient streams. This causes the hyporheic mean residence time to reach a threshold when the alluvial depth is deeper than the channel width. Our results also show that as the stream slope increases, the streamlines compact near to the streambed, thereby reducing the mean residence time and its variance. Finally, we observe that the hyporheic residence time of pulse injections of passive solutes is lognormally distributed, with the mean value depending in a simple manner on the amplitude of the alternate bars.

Sand bars in tidal channels. Part 1. Free bars

We investigate the basic mechanism whereby bars form in tidal channels or estuaries. The channel is assumed to be long enough to allow neglect of the effects of end conditions on the process of bar formation. In this respect, the object of the present analysis differs from that of Schuttelaars & de Swart (1999) who considered bars of length scaling with the finite length of the tidal channel. The channel bottom is assumed to be cohesionless and consisting of uniform sediments. Bars are shown to arise from a mechanism of instability of the erodible bed subject to the propagation of a tidal wave. Sediment is assumed to be transported both as bedload and as suspended load. A fully three-dimensional model is employed both for the hydrodynamics and for sediment transport. At the leading order of approximation considered, the effects of channel convergence, local inertia and Coriolis forces on bar instability are shown to be negligible. Unlike fluvial free bars, in the absence of mean currents tidal free bars are found to be non-migrating features (in the mean). Instability arises for large enough values of the mean width to depth ratio of the channel, for given mean values of the Shields parameter and of the relative channel roughness. The role of suspended load is such as to stabilize bars in the large-wavenumber range and destabilize them for small wavenumbers. Hence, for large values of the mean Shields stress, it turns out that the first critical mode (the alternate bar mode) is characterized by a very small value of the critical width to depth ratio. Furthermore, the order- m mode being characterized by a critical value of the width to depth ratio equal to m times the critical value for the first mode, it follows that for large values of the mean Shields stress several unstable modes are simultaneously excited for relatively low values of the aspect ratio. This suggests that the actual bar pattern observed in nature may arise from an interesting nonlinear competition among different unstable modes.

Growth of alternate bars in unsteady flow

A theoretical model is formulated to investigate the development of the amplitude of alternate bars in unsteady flows. The problem is tackled by means of a weakly nonlinear analysis developed in a neighborhood of the threshold conditions for bar formation. Bar response to unsteady flow is found to depend on a parameter that is a measure of the ratio between the time scale of the basic flow and the time scale of bar growth. The present theory shows that if is O(1), as often occurs in nature, flow unsteadiness affects the instantaneous growth rate and phase of bar perturbations and controls the final amplitude reached by the bed configuration. A procedure for determining the final amplitude for a given flood event is proposed. Flume experiments were performed to test the main theoretical results. The bed response to unsteady flow was measured for different values of the period of the flood. The observed temporal behavior of the bar amplitude proves to be strongly affected by the unsteady character of the flow for of O(1), as predicted by the theory.