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

Abstract: Using a 2DH idealized local morphodynamic model for a tidal channel, it is demonstrated 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. The water motion is modelled by the depth-averaged shallow water equations and driven by an externally prescribed M2 tide. Sediment is mainly transported as suspended load due to advective processes. Convergences and divergences of the tidally averaged sediment fluxes result in the evolution of the bed. It is shown that the combined effect of bottom friction and advective processes can trigger instabilities that lead to the formation of bottom patterns. Bed slope effects are required in order to prevent infinite braiding of these features. With bed slope effects, bars with longitudinal length scales of the order of the tidal excursion length are most likely to become unstable. This result is found to be independent of the ratio of the width to the tidal excursion length as well as the adopted formulation of the bed shear stress. 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. Qualitatively, the results are recovered when bottom friction is linearized. Quantitatively, only small modifications occur: the critical friction parameter is decreased and the longitudinal length scale of the most unstable bed form increases. r 2002 Elsevier Science Ltd. All rights reserved.

## 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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...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....

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...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....

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