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 − βc)/βc, where t is the time, β is the width ratio of the channel and βc 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.

A two-dimensional model of flow and bed topography in sinuous channels with erodible boundaries is developed and applied in order to investigate the mechanism of meander initiation. By reexamining the problem recently tackled by Ikeda, Parker & Sawai (1981), a previously undiscovered ‘resonance’ phenomenon is detected which occurs when the values of the relevant parameters fall within a neighbourhood of certain critical values. It is suggested that the above resonance controls the bend growth, and it is shown that it is connected in some sense with bar instability. In fact, by performing a linear stability analysis of flow in straight erodible channels, resonant flow in sinuous channels is shown to occur when curvature ‘forces’ a ‘natural’ solution represented by approximately steady perturbations of the alternate bar type. A comparison with experimental observations appears to support the idea that resonance is associated with meander formation.

[1 ] This contr ibution inves tigates the morp hodynam ic equil ibrium of funnel -shape d well-mixe d estua ries and/or tida l channel s. The one-di mensional d e Saint Venant and Exner equations are solve d numerical ly for the ideal case of a fric tionally domi nated estua ry consisti ng of noncoh esive sedim ent and with insignif icant interti dal storage of wat er in tida l flats and salt mars hes. This class of estuaries turns out to be invariabl y flood dominated . The resul ting asym metries in surfa ce elevat ions and tidal curren ts lead to a ne t sediment flux within a tida l cycle which is directed landw ard. As a consequ ence, sedim ents are trapp ed within the estua ry and the bott om profi le evolve s asym ptotical ly toward an equil ibrium confi guration, allo wing a vanishing net sedim ent flux everyw here and, in accordan ce with fiel d observ atio ns, a nearly const ant value of the maxi mum flood /ebb speed. Such an equil ibrium bed profi le is characteri zed by a concavi ty increasing as the estua ry co nvergence incre ases and by a unique ly deter mined value of the depth at the inlet section. The final length of the estua ry is fixed by the longi tudinal extens ion of the very shallow area which tends to form in the landw ard portion of the estuary. Note that sediment advect ion is neglec ted in the analys is, an a ssumption appropriat e to the case of not too fine sediment .I NDEX TERMS: 4235

In the last few decades cooperation among fluid dynamicists and geomorphologists has allowed the construction of a rational framework for the quantitative understanding of several geomorphologic processes involved in the shaping of the Earth's surface. Particular emphasis has been given to the dynamics of sedimentary patterns, features arising from the continuous dynamic interaction between the motion of a sediment-carrying fluid and an erodible boundary. It is this interaction which ultimately gives rise to the variety of natural forms, often displaying a high degree of regularity, observed in rivers, estuaries, coasts, as well as in the deep submarine environment. Theoretical analyses and laboratory experiments have shown that the nature of most of the observed patterns is related to fundamental instability mechanisms whose particular character lies in the fact that it is the mobile interface between the fluid and the erodible boundary, rather than the flow itself, that is unstable. Developments have been general enough to reach the status of a distinct branch of fluid mechanics, geomorphological fluid mechanics. This paper concentrates on the mechanics of fluvial meandering. Our aim is to provide the reader with a systematic overview of the fundamental aspects of the subject, assessing, with the help of recent and novel results, settled as well as unsettled issues

Abstract.We revisit the problem of one-dimensional tide propagation in convergent estuaries considering four limiting cases defined by the relative intensity of dissipation versus local inertia in the momentum equation and by the role of channel convergence in the mass balance. In weakly dissipative estuaries, tide propagation is essentially a weakly nonlinear phenomenon where overtides are generated in a cascade process such that higher harmonics have increasingly smaller amplitudes. Furthermore, nonlinearity gives rise to a seaward directed residual current. As channel convergence increases, the distortion of the tidal wave is enhanced and both tidal wave speed and wave lenght increase. The solution loses its wavy character when the estuary reaches its "critical convergence"; above such convergence the weakly dissipative limit becomes meaningless. Finally, when channel convergence is strong or moderate, weakly dissipative estuaries turn out to be ebb dominated. In strongly dissipative estuaries, tide propagation becomes a strongly nonlinear phenomenon that displays peaking and sharp distortion of the current profile, and that invariably leads to flood dominance. As the role of channel convergence is increasingly counteracted by the diffusive effect of spatial variations of the current velocity on flow continuity, tidal amplitude experiences a progressively decreasing amplification while tidal wave speed increases. We develop a nonlinear parabolic approximation of the full de Saint Venant equations able to describe this behaviour. Finally, strongly convergent and moderately dissipative estuaries enhance wave peaking as the effect of local inertia is increased. The full de Saint Venant equations are the appropriate model to treat this case.

The exact solution of the problem of river morphodynamics derived in Part 1 is employed to formulate and solve the problem of planimetric evolution of river meanders. A nonlinear integrodifferential evolution equation in intrinsic coordinates is derived. An exact periodic solution of such an equation is then obtained in terms of a modified Fourier series expansion such that the wavenumbers of the various Fourier modes are time dependent. The amplitudes of the Fourier modes and their wavenumbers satisfy a nonlinear system of coupled ordinary differential equations of the Landau type. Solutions of this system display the occurrence of two possible scenarios. In the sub-resonant regime, i.e. when the aspect ratio of the channel is smaller than the resonant value, meandering evolves according to the classical picture: a periodic train of small-amplitude sine-generated meanders migrating downstream evolve into the classical, upstream skewed, train of meanders of Kinoshita type. Evolution displays all the experimentally observed features: the meander growth rate increases up to a maximum and then decreases, while the migration speed decreases monotonically. No equilibrium solutions are found. In the super-resonant regime the picture is essentially reversed: downstream skewing develops while meanders migrate upstream.Numerical solutions of the planimetric evolution equation are obtained for the case when the initial channel pattern exhibits random small perturbations of the straight configuration. Under these conditions, the evolution displays the typical features of solutions of the Ginzburg–Landau equation, in particular, the occurrence of spatial modulations of the meandering pattern which organizes itself in the form of wavegroups. Furthermore, multiple loops develop in the advanced stage of meander growth.

.[1] Observational evidence is presented on the geometry of meandering tidal channels evolved within coastal wetlands characterized by different tidal, hydrodynamic, topographic, vegetational and ecological features. New insight is provided on the geometrical properties of tidal meanders, with possible dynamic implications on their evolution. In particular, it is shown that large spatial gradients of leading flow rates induce important spatial variabilities of meander wavelengths and widths, while their ratio remains remarkably constant in the range of scales of observation. This holds regardless of changes in width and wavelength up to two orders of magnitude. This suggests a locally adapted evolution, involving the morphological adjustment to the chief landforming events driven by local hydrodynamics. The spectral analysis of local curvatures reveals that Kinoshita's model curve does not fit tidal meanders due to the presence of even harmonics, in particular the second mode. Geometric parameters are constructed that are suitable to detect possible geomorphic signatures of the transitions from ebb-to flood-dominated hydrodynamics, here related to the skewness of the tidal meander. Trends in skewness, however, prove elusive to measure and fail to show detectable patterns. We also study comparatively the spatial patterns of evolution of the ratios of channel width to depth, and the ratio of width to local radius of curvature. Interestingly, the latter ratio exhibits consistency despite sharp differences in channel incision. Since the degree of incision, epitomized by the width-to-depth ratio, responds to the relevant erosion and migrations mechanisms and is much sensitive to vegetation and sediment properties, it is noticeable that we observe a great variety of landscape carving modes and yet recurrent planar features like constant width/curvature and wavelength/width ratios.

[1] According to the Bagnold hypothesis for equilibrium bed load transport, a necessary constraint for the maintenance of equilibrium bed load transport is that the fluid shear stress at the bed must be reduced to the critical, or threshold, value associated with incipient motion of grains. It was shown in a companion paper [Seminara et al., 2002], however, that the Bagnold hypothesis breaks down when applied to equilibrium bed load transport on beds with transverse slopes above a relatively modest value that is well below the angle of repose. An investigation of this failure resulted in a demonstration of its lack of validity even for nearly horizontal beds. The constraint is here replaced with an entrainment formulation, according to which a dynamic equilibrium is maintained by a balance between entrainment of bed grains into the bed load layer and deposition of bed load grains onto the bed. The entrainment function is formulated so that the entrainment rate is an increasing function of the excess of the fluid shear stress at the bed over the threshold value. The formulation is implemented with the aid of a unique set of laboratory data that characterizes equilibrium bed load transport at relatively low shear stresses for streamwise angles of bed inclination varying from nearly 0°to 22°. The formulation is shown to provide a description of bed load transport on nearly horizontal beds that fits the data as well as that resulting from the Bagnold constraint. The entrainment formulation has the added advantage of not requiring the unrealistically high dynamic coefficient of Coulomb friction resulting from the Bagnold constraint. Finally, the entrainment formulation provides reasonable and consistent results on finite streamwise and transverse bed slopes, even those at which the Bagnold formulation breaks down completely.