We review the importance of the physical mechanisms involved in river meandering by comparing some existing linear models and extensions thereof. Such models are hierarchically derived from a common and general mathematical framework and then analyzed with a detailed discussion of the physical processes and relevant hypotheses that are involved. Experiments and field data are also used to discuss the related morphodynamic processes. The analysis of the models shows the importance of the closure of secondary currents especially in the modeling of eddy viscosity. This aspect confirms the usefulness of using simplified models for some practical applications, provided the secondary currents are modeled in detail. On the other hand, the free response of the sediments, the phase lag of secondary currents, and the momentum redistribution due to the coupling between the main and the transverse flow are shown to be less relevant. Hence the second-order models, which neglect the effect of superelevation induced by the topography-driven lateral flow on the longitudinal flow, can reasonably be considered a good approximation for both predictive analysis and the computation of the resonant conditions. Finally, the analysis of higher harmonics suggests that the multilobed pattern can intrinsically be present in both second- and fourth-order models.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2005RG000185

Hierarchy of models for meandering rivers and related morphodynamic processes

In spite of notable advances in the description of river morphodynamics, the long-term dynamics of meandering rivers is still an open question, in particular, regarding the existence of a possible statistical steady state and its scaling properties induced by the competing action of cutoffs and reach elongation. By means of extensive numerical simulations, using three fluid dynamic models of different complexity and analysis of real data from the Amazon, North America, and Russia, we show that the reach cutoffs, besides providing stability and self-confinement to the meander belt, also act as a dynamical filter on several hydrodynamic mechanisms, selecting only those that really dominate the long-term dynamics. The results show that the long-term equilibrium conditions are essentially governed by only one spatial scale (proportional to the ratio of the river depth and the friction coefficient) and one temporal scale (proportional to the square of the spatial scale divided by the river width, the mean longitudinal velocity, and the erodibility coefficient) that contain the most important fluid dynamic quantities. The ensuing statistical long-term behavior of meandering rivers proves to be universal and largely unaffected by the details of the fluid dynamic processes that govern the short-term river behavior.

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On the long-term behavior of meandering rivers

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

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2008WR007298

A nonlinear model for river meandering

Biomorphodynamic modelling of inner bank advance in migrating meander bends

[1] The mathematical modeling of the long-term evolution of meandering rivers needs an efficient computation of the flow field. Indeed, the estimate of the near bank velocity, needed to determine the rate at which the outer bank migrates, cannot rely on the full numerical solution of the governing equations when considering the river evolution on geological time scales. The aim of the present contribution is twofold: determining the complete linear response of a meandering river to spatially varying channel axis curvature and width, exploiting the ability of the model to describe the morphological tendencies of alluvial rivers; and developing a computationally efficient tool that can be easily incorporated in long-term planform evolution models. The centrifugally induced secondary flow associated with channel axis curvature and longitudinal convection is accounted for by a suitable parametrization based on the structure of the three-dimensional flow field. Cross section width variations are accounted for through a suitable stretching of the transverse coordinate. The relevant momentum and mass conservation equations are then linearized by taking advantage of the fact that alluvial rivers often exhibit mild and long meander bends, as well as evident but relatively small width variations. The input data needed by the analytical solution are the spatial distribution of channel axis curvature and width variations, the mean slope of the investigated river reach, the characteristic grain size of the sediment bed and the flow discharge. The performances of the model, as well as its intrinsic limitations are discussed with reference to the comparison with the bed topography surveyed in a 21 km long reach of the Po River. The results indicate that, in the presence of wide, mildly curved and long bend and weak width variations, the river topography is described with a good accuracy, thus supporting the use of the model to investigate how a river could react to changes in planform geometry or external forcing. Moreover, the analytical character of the model implies a limited computational effort, facilitating a straightforward integration within the models used to simulate the planimetric evolution of alluvial rivers on geological time scales.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/jgrf.20084

R. B. Dean

Papers

Reynolds number dependence of skin friction in two-dimensional rectangular duct flow and a discussion of the law of the wake