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Journal ArticleDOI

The effect of geometry and bottom friction on local bed forms in a tidal embayment

01 Jul 2002-Continental Shelf Research (CONTINENTAL SHELF RESEARCH)-Vol. 22, Iss: 11, pp 1821-1833
TL;DR: In this paper, a 2DH idealized local morphodynamic model for a tidal channel was used to demonstrate that estuarine bars with typical length scales on the order of the tidal excursion length can develop as the result of a positive feedback between water motion, sediment transport and the sandy bottom.
About: This article is published in Continental Shelf Research.The article was published on 2002-07-01 and is currently open access. It has received 75 citations till now. The article focuses on the topics: Length scale & Suspended load.

Summary (3 min read)

1. Introduction

  • The geomorphology of semi-enclosed tidal embayments with a sandy bed often consists of a complex network of channels and shoals.
  • On the other hand, deeper embayments, e.g. those located in the Dutch and German Wadden Sea, are characterized by a fractal pattern of channels (cf. Cleveringa and Oost, 1999; Ehlers, 1988) which appear to scale with the length of the embayment.
  • Their model results apply to narrow, frictionally dominated tidal channels.
  • The results from the two approaches should qualitatively agree in the appropriate limits.
  • Here, the intermediate model will be compared with the results by Seminara and Tubino (1998).

2. Model description

  • The features studied in this paper have length scales which are small compared with the tidal wavelength, the embayment length and the length scale on which variations of the channel width occur.
  • Within the local model, tidal velocities are of the order of 1 m s 1: Since the amplitude of the sea surface elevations is assumed to be much smaller than the undisturbed water depth, the socalled rigid lid approximation can be adopted.
  • Using a procedure first proposed by Lorentz (1922), see also Zimmerman (1992), the non-linear bottom friction can be linearized in such a way that averaged over one tidal cycle the same amount of energy is dissipated in both formulations.
  • The first term on the right-hand side of Eq. (4) models the sediment pick-up function, and the second term the tendency of sediment to settle due to gravity effects.
  • This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates.

3. Linear stability analysis

  • For realistic values of the parameters the 2D system of equations, as described in Section 2, allows for a morphodynamic equilibrium solution Weq ¼ ðu; v;rz;C; hÞeq; which is spatially uniform, i.e. they are independent of both the x- and the ycoordinate.
  • Here, ueq ¼ ðueq; 0Þ and u0 ¼ ðu0; v0Þ are the equilibrium and perturbed velocity vector.
  • The ratio of the tidal period and the morphologic timescale is typically of the order of 10 2–10 4: Since Eqs. (8a)–(8d) evolve on the tidal timescale, the bed perturbation h0 in these equations can be considered fixed.
  • The first two terms on the right-hand side of Eq. (13) give the contribution of the divergence of the advective sediment flux Fadv; while the last two terms model the divergences of fluxes due to diffusive processes (Fdiff ) and bedslope effects (Fbed), respectively.
  • The real part of the eigenvalue RðoÞ denotes the growth rate of the perturbation and IðoÞ=k its migration speed.

4. Results

  • In this section results from the local 2D channel model will be described.
  • Default values which are characteristic for the Western Scheldt will be used, see Table 1.
  • In the remainder of this paper, the authors will only consider advective modes, i.e. horizontal dispersion terms in the momentum and concentration equations are neglected ðm ¼.
  • This is justified since the ratio of dispersive to advective fluxes is of the order 10 1–10 3 for the bed form length scales that are considered in this paper.
  • This is also done in the model adopted by Seminara and Tubino (1998).

4.1. Advective instabilities for linear bottom friction

  • 0Þ: Fig. 3 shows the dimensionless growth rate as a function of the dimensionless longitudinal wavenumber k for various values of the lateral number n:.
  • This result is reminiscent from river morphodynamics where this mode is also found to be the most unstable one if bed slope effects are neglected (Callander, 1969).
  • For long waves, sediment transport is mainly driven by the residual velocity perturbation /u0S:.
  • These features will now be explained in more detail.
  • Expression (17) shows that the growth of long-wave perturbations is primarily governed by the residual perturbed velocity /u0S: Fig. 4 shows a typical example of the behaviour of this quantity as the longitudinal wavenumber k varies.

4.2. Bedslope effects

  • The most unstable mode now occurs for finite n: Eigenfunctions with high modenumber n (i.e. fast spatial oscillations in the lateral direction) are damped.
  • For friction values above the neutral curve, bedforms have positive growth rates.
  • The minimum of the neutral curve is referred to as the critical mode for the specified lateral modenumber n and is characterized by the critical wavenumber kcr and friction parameter values rcr:.
  • Since this destabilizing effect has its maximum value for a finite value of k (see Fig. 3), it is to be expected that both rcr and kcr have finite (non-zero) values.

4.3. Non-linear friction and the influence of channel width

  • The authors will extend their model by including non-linear bottom friction, which means that sb in Eq. (2) reads sb ¼ r#rjjujju; ð20Þ where #r ¼ 3pr=ð8UÞ follows from the Lorentz linearization procedure that was mentioned below Eq. (2).
  • The effect of non-linearity on the growth of bedforms can be inferred from Fig. 7 which shows the neutral curves for both linear and nonlinear friction.
  • Also, the most unstable wavenumber shifts towards a higher value, i.e. the critical mode occurs on a shorter longitudinal length scale.
  • The explicit dependence of the non-linear friction parameter #rjjujj on velocity thus yields a decrease of bottom friction above shallow (deep) parts of the channel.
  • So far, the authors have considered a socalled wide channel for which width and tidal excursion length are of the same order of magnitude.

5. Discussion and conclusions

  • The formation of bottom patterns that scale with the tidal excursion length has been studied within a 2D idealized model.
  • In the case that sediment diffusion can be neglected, this instability is mediated by advective processes, in particular through residual flows that arise from tide-topography interactions.
  • Bed slope effects act as a means to prevent the emergence of both longitudinal and lateral smallscale features.
  • All these discrepancies may in principle yield qualitatively different outcome.

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Journal ArticleDOI
TL;DR: In this paper, the authors discuss the morphodynamics of different units that characterize a tidal inlet system: the overall system, the ebb-tidal delta, the tidal channels, channel networks, tidal bars and meanders, and finally the intertidal zone of tidal flats and salt marshes.
Abstract: In this review we discuss the morphodynamics of tidal inlet systems that are typical of barrier coasts formed during a period of continuous sea-level rise during the Holocene. The morphodynamics concerns feedbacks between three major components: the hydrodynamics of tidal currents and wind waves; the erosion, deposition, and transport of sediment under the action of the former hydrodynamic agencies; and the morphology proper, which results from the divergence of the sediment transport. We discuss the morphodynamics of the different units that characterize a tidal inlet system: the overall system, the ebb-tidal delta, the tidal channels, channel networks, tidal bars and meanders, and finally the intertidal zone of tidal flats and salt marshes. In most of these units, stability analysis is a major guide to the establishment of equilibrium structures.

285 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the long-term evolution of estuarine morphodynamics with special emphasis on the impact of pattern formation using a two-dimensional (2-D), numerical, process-based model.
Abstract: [1] The research objective is to investigate long-term evolution of estuarine morphodynamics with special emphasis on the impact of pattern formation. Use is made of a two-dimensional (2-D), numerical, process-based model. The standard model configuration is a rectangular 80 km long and 2.5 km wide basin. Equilibrium conditions of the longitudinal profile are analyzed using the model in 1-D mode after 8000 years. Two-dimensional model results show two distinct timescales. The first timescale is related to pattern formation taking place within the first decades and followed by minor adaptation according to the second timescale of continuous deepening of the longitudinal profile during 1600 years. The resulting longitudinal profiles of the 1-D and 2-D runs are similar apart from small deviations near the mouth. The 2-D results correspond well to empirically derived relationships between the tidal prism and the channel cross section and between the tidal prism and the channel volume. Also, comparison between the current model results and data from the Western Scheldt estuary (in terms of bar length, hypsometry, percentage of intertidal area and values for the ratio of shoal volume and channel volume against the ratio of tidal amplitude and water depth) shows satisfying agreement. On the basis of the model results a relationship for a characteristic morphological wavelength was derived on the basis of the tidal excursion and the basin width and an exponentially varying function was suggested for describing a dimensionless hypsometric curve for the basin. Furthermore, special attention is given to an analysis of the numerical morphodynamic update scheme applied.

248 citations


Cites background or result from "The effect of geometry and bottom f..."

  • ...Seminara and Tubino [2001], Schuttelaars and De Swart [1999], Schramkowski et al. [2002] and Dronkers [2005] extensively describe the prevailing processes in bar development for the linear domain where the bars themselves do not yet significantly influence the velocity field....

    [...]

  • ...The reason why a rectangular configuration was chosen is twofold.First,it makes comparison possible with earlier research carried out by Schuttelaars and De Swart [1996, 1999, 2000], Seminara and Tubino [2001], Schramkowski et al. [2002, 2004] and Hibma et al. [2003b] who all used rectangular basins or assumedthatnosignificantchangesofthebasinwidthoccurred over a typical length scale of the developing bars....

    [...]

  • ...The relation of a typical morphological length to the tidal excursion is also found by Hibma et al. [2003b, 2003c] and Schramkowski et al. [2002] ....

    [...]

  • ...Swart [1999], Seminara and Tubino [2001], Schramkowski et al. [2002] and Van Leeuwen and De Swart [2004] describe initial channel/shoal formation in a highly schematized tidal environment....

    [...]

  • ...…is that they can be related to the (short) basin length, the embayment width and the relative importance of diffusive and advective transports [Van Leeuwen and De Swart, 2004], or the tidal excursion length implicitly taking into account the impact of friction and depth [Schramkowski et al., 2002]....

    [...]

Journal ArticleDOI
TL;DR: In this article, a 2DV model is developed to investigate the changes in relative roughness (friction) and mixing, resulting in fundamental changes in the char- acteristics of the horizontal (velocity) and vertical tides (sea surface elevation).
Abstract: Over decades and centuries, the mean depth of estuaries changes due to sea-level rise, land subsi- dence, infilling, and dredging projects. These processes produce changes in relative roughness (friction) and mixing, resulting in fundamental changes in the char- acteristics of the horizontal (velocity) and vertical tides (sea surface elevation) and the dynamics of sediment trapping. To investigate such changes, a 2DV model is developed. The model equations consist of the width- averaged shallow water equations and a sediment balance equation. Together with the condition of mor- phodynamic equilibrium, these equations are solved analytically by making a regular expansion of the vari- ous physical variables in a small parameter. Using these analytic solutions, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying various forcings separately. As a case study, we consider the Ems es- tuary. Between 1980 and 2005, successive deepening of the Ems estuary has significantly altered the tidal and sediment dynamics. The tidal range and the surface sediment concentration has increased and the position of the turbidity zone has shifted into the freshwater zone. The model is used to determine the causes of these historical changes. It is found that the increase of the tidal amplitude toward the end of the embayment is the combined effect of the deepening of the estuary and a 37% and 50% reduction in the vertical eddy viscosity and stress parameter, respectively. The phys- ical mechanism resulting in the trapping of sediment, the number of trapping regions, and their sensitivity to grain size are explained by careful analysis of the var- ious contributions of the residual sediment transport. It is found that sediment is trapped in the estuary by a delicate balance between the M2 transport and the residual transport for fine sediment (ws = 0.2 mm s −1 ) and the residual, M2 and M4 transports for coarser sediment (ws = 2 mm s −1 ). The upstream movement of the estuarine turbidity maximum into the freshwa- ter zone in 2005 is mainly the result of changes in tidal asymmetry. Moreover, the difference between the sediment distribution for different grain sizes in the same year can be attributed to changes in the temporal settling lag.

164 citations


Cites background from "The effect of geometry and bottom f..."

  • ...Hence, the partial slip condition can be rewritten as Avuz = su (Schramkowski et al. 2002)....

    [...]

  • ...Following Friedrichs and Hamrick (1996) and Schramkowski et al. (2002), this dependency is taken to be linear in the local water depth, i.e., s = s0 H(x)H0 ....

    [...]

Journal ArticleDOI
TL;DR: In this article, the formation of channel and shoal patterns in a schematic estuary was investigated using a 2D depth-averaged numerical model based on a description of elementary flow and sediment transport processes.
Abstract: The formation of channel and shoal patterns in a schematic estuary is investigated using a 2-D depth-averaged numerical model based on a description of elementary flow and sediment transport processes. The schematisations apply to elongated inland estuaries, sandy, well-mixed and tide-dominated. The model results show how, due to non-linear interactions, a simple and regular pattern of initially grown perturbations merges to complex larger-scale channel/shoal patterns. The emerging patterns are validated with field observations. The overall pattern agrees qualitatively with patterns observed in the Westerschelde, The Netherlands, and in the Patuxent River estuary, Virginia. Quantitative comparison of the number of channels and meander length scales with observations and with an analytical model gives reasonable accordance. Complementary to other research approaches, this model provides a tool to study the morphodynamic behaviour of channels and shoals in estuaries.

126 citations

Journal ArticleDOI
TL;DR: In this paper, the authors investigated the interaction between different scales and spatial scales of the morphodynamic system in alluvial, coastal plain estuaries and the evolution of the estuarine morphodynamic evolution.
Abstract: The morphodynamic system in alluvial, coastal plain estuaries is complex and characterized by various timescales and spatial scales. The current research aims to investigate the interaction between these different scales as well as the estuarine morphodynamic evolution. Use is made of a process-based, numerical model describing 2-D shallow water equations and a straightforward formulation of the sediment transport and the bed level update. This was done for an embayment with a length of 80 km on a timescale of 3200 years, with and without bank erosion effects. Special emphasis is put on analyzing the results in terms of energy dissipation. Model results show that the basins under consideration evolve toward a state of less morphodynamic activity, which is reflected by (among others) relatively stable morphologic patterns and decreasing deepening and widening of the basins. Closer analysis of the tidal wave shows standing wave behavior with resonant characteristics. Under these conditions, results suggest that the basins aim for a balance between the effect of storage and the effect of fluctuating water level on wave celerity with a negligible effect of friction. Evaluating the model results in terms of energy dissipation reflects the major processes and their timescales (pattern formation, widening, and deepening). On the longer term the basin-wide energy dissipation decreases at a decreasingly lower rate and becomes more uniformly distributed along the basin. Analysis by an entropy-based approach suggests that the forced geometry of the configurations prevents the basins from evolving toward a most probable state.

126 citations

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Additional excerpts

  • ...The evolution equation for C reads (Van Rijn, 1993) Ct þ ðuC mCxÞx þ ðvC mCyÞy ¼ S aðu2 þ v2Þ gC; ð4Þ where m denotes the horizontal coefficient for sediment diffusion and S the difference between erosion and sedimentation at the top of the active layer....

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Additional excerpts

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TL;DR: In this paper, the effects of space discretization on wave propagation are investigated and a detailed treatment of boundary conditions is given. But the results are limited to three-dimensional shallow-water flows.
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TL;DR: In this paper, the authors formulated a fluid flow model with a series of approximate but reasonable assumptions for straight channels with non-cohesive sand and silt banks and showed that the model can be applied to a more general treatment of natural rivers, which would include various complicating factors such as meandering, sediment sorting and seepage.
Abstract: Rivers and canals with perimeters composed of non-cohesive sand and silt have self-formed active beds and banks. They thus provide a most interesting fluid flow problem, for which one must determine the container as well as the flow. If bed load alone occurs across the perimeter of a wide channel, gravity will pull particles down the lateral slope of the banks; bank erosion is accomplished and the channel widens. In order to maintain equilibrium, this export of material from the banks must be countered by an import of sediment from the channel centre.The mechanism postulated for this import is lateral diffusion of suspended sediment, which overloads the flow near the banks and causes deposition. The model is formulated analytically with the aid of a series of approximate but reasonable assumptions. Singular perturbation techniques are used to define the channel geometry and obtain rational regime relations for straight channels. A comparison with data lends credence to the model.It is hoped that a first step has been made towards a more general treatment, which would include various complicating factors that are important features of natural rivers but are not essential to the maintenance of channel width. Among these factors are meandering, sediment sorting and seepage.

288 citations


"The effect of geometry and bottom f..." refers result in this paper

  • ...This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates....

    [...]

Journal ArticleDOI
TL;DR: In this article, the effect of a sloping bed on the direction of sediment transport is determined by conducting bed-levelling experiments, and a distinction is made between laboratory conditions and natural rivers.
Abstract: Laboratory experiments have been conducted to provide data for modelling the direction of sediment transport on a transverse sloping alluvial bed. Conditions with prevailing bed-load transport, and conditions in which a significant part of the bed material is transported as suspended-load are studied. The effect of a sloping bed on the direction of sediment transport is determined by conducting bed-levelling experiments. Comparison of the results with data of curved flume experiments and experience gained with numerical computation of the bed topography in natural rivers yields the conclusion that, at least for bed-load transport, a distinction should be made between laboratory conditions and natural rivers. For conditions with suspended sediment transport the transverse slope effect can not be modelled identical as for bed-load transport.

262 citations


"The effect of geometry and bottom f..." refers result in this paper

  • ...This seems to be consistent with literature where it is suggested that the transport due to the bed slope terms cannot be neglected (see Parker, 1978; Talmon et al., 1995), even if suspended-load transport dominates....

    [...]

Frequently Asked Questions (1)
Q1. What are the contributions mentioned in the paper "The effect of geometry and bottom friction on local bed forms in a tidal embayment" ?

In the case that the width is much smaller than the tidal excursion length and non-linear bottom friction is used, there is good qualitative agreement with results from 3D models reported in literature which were applied to the same parameter regime.