H.E. de Swart

Approaches to long-term modelling of coastal morphology: A review

Aspects of long-term mathematical modelling of coastal morphology are inventoried and discussed. They concern reduction techniques for input data, process descriptions and output data, as well as model concepts ranging from statistical extrapolation of the past coastal behaviour, via semi-empirical behaviour models, to formally integrated descriptions of the constituent small-scale processes. All approaches have in common that they reduce the need for detailed descriptions in space and time of the underlying physical processes. They lead not only to more transparent and robust models which require less computational effort, but also to a better insight into which aspects of coastal behaviour are relevant from a long-term point of view and which are not (“signal” vs. “noise”).

On the nonlinear dynamics of free bars in straight channels

A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a GinzburgLandau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.

Multiple morphodynamic equilibria in tidal embayments

The possible morphodynamic equilibria of tidal embayments are investigated within the framework of a one-dimensional model The equilibria are defined by a steady profile of the erodible bottom. The extension with respect to earlier studies is that the embayments have arbitrary lengths L with respect to the tidal wavelength. This implies a much richer dynamics due to the possibility of tidal resonance and new sediment transport contributions that are caused by internally generated overtides and residual currents. If the system is only forced by an externally prescribed M2 tide at the seaward boundary, a unique morphodynamic equilibrium is obtained for all embayment lengths smaller than the frictional length scale of the tide. Bottom friction causes tidal resonance to occur for a shorter length than a quarter of the frictionless tidal wavelength. This shift is smaller than would occur in the case of a fixed bed profile since the equilibrium condition induces larger water depths. If an externally prescribed overtide is added to the forcing, more than one type of morphodynamic equilibria can be found. For L values smaller than the M4 resonance length scale the bottom profiles are strongly concave, with locally large water depths, and the water motion resembles a standing tidal wave. For longer embayments another type of equilibria, characterized by a weakly concave bottom profile and a traveling tidal wave, appears. For sufficiently strong amplitudes of the externally prescribed M4 tide, multiple morphodynamic equilibria are found. The maximum L, beyond which morphodynamic equilibria cease to exist, decreases with increasing influence of external overtide and bottom friction. These model results show an overall good agreement with field observations.

Lateral entrapment of sediment in tidal estuaries: An idealized model study

Two physical mechanisms leading to lateral accumulation of sediment in tidally dominated estuaries are investigated, involving Coriolis forcing and lateral density gradients. An idealized model is used that consists of the three?dimensional shallow water equations and sediment mass balance. Conditions are assumed to be uniform in the along?estuary direction. A semidiurnal tidal discharge and tidally averaged density gradients are prescribed. The erosional sediment flux at the bed depends both on the bed shear stress and on the amount of sediment available in mud reaches for resuspension. The distribution of mud reaches over the bed is selected such that sediment transport is in morphodynamic equilibrium, that is, tidally averaged erosion and deposition of sediment at the bed balance. Analytical solutions are obtained by using perturbation analysis. Results suggest that in most estuaries lateral density gradients induce more sediment transport than Coriolis forcing. When frictional forces are small (Ekman number E 0.02), the lateral density gradient mechanism dominates and entraps sediment in areas with fresher water. Results also show that the lateral sediment transport induced by the semidiurnal tidal flow is significant when frictional forces are small (E ? 0.02). Model predictions are in good agreement with observations from the James River estuary.

Modelling the formation of shoreface-connected sand ridges on storm-dominated inner shelves

A morphodynamic model is developed and analysed to gain fundamental understanding of the basic physical mechanisms responsible for the characteristics of shorefaceconnected sand ridges observed in some coastal seas. These alongshore rhythmic bed forms have a horizontal lengthscale of order 5 km and are related to the mean current along the coast: the seaward ends of their crests are shifted upstream with respect to where they are attached to the shoreface. The model is based on the two-dimensional shallow water equations and assumes that the sediment transport only takes place during storms. The flux consists of a suspended-load part and a bed-load part and accounts for the e ects of spatially non-uniform wave stirring as well as for the preferred downslope movement of sediment. The basic state of this model represents a steady longshore current, driven by wind and a pressure gradient. The dynamics of small perturbations to this state are controlled by a physical mechanism which is related to the transverse bottom slope. This causes a seaward deflection of the current over the ridges and the loss of sediment carrying capacity of the flow into deeper water. The orientation, spacing and shape of the modelled ridges agree well with eld observations. Suspended-load transport and spatially non-uniform wave stirring are necessary in order to obtain correct e-folding timescales and migration speeds. The ridge growth is only due to suspended-load transport whereas the migration is controlled by bed-load transport.

Initial formation of channels and shoals in a short tidal embayment

It is demonstrated, by using a simple model, that bedforms in a short tidal embayment can develop due to a positive feedback between tidal currents, sediment transport and bedforms. The water motion is modelled by the depth integrated shallow water equations. The system is forced by a prescribed free-surface elevation at the entrance of the embayment. For the sediment dynamics a diffusively dominated suspended load transport model is considered. Tidal averaging is used to obtain the bottom profiles at the long morphological time scale. The stability of a constantly sloping equilibrium bottom profile is studied for various combinations of the model parameters. It turns out that without a mechanism that generates vorticity this equilibrium profile is stable. In that case small-scale perturbations can at most become marginally stable if no bedload term in the bottom evolution equation is incorporated. If vorticity is generated, in our model by bottom friction torques, the basic state is unstable. The spatial patterns of the unstable modes and their growth rates depend, among other things, on the strength of the bottom friction, the width of the embayment and the grain size: if the sediment under consideration consists of large particles, the equilibrium will be more stable than when smaller particles are considered. Without a diffusive term in the bed evolution equation, small-scale perturbations become unstable. To avoid this physically unrealistic behaviour bedload terms are included in the sediment transport. Furthermore, it is shown that using an asymptotic expansion for the concentration as given in earlier literature is only valid for small or moderate mode numbers and the technique is extended to large mode numbers. A physical interpretation of the results is also given.

Holocene tidal conditions and tide-induced sand transport in the southern North Sea

A numerical model of Holocene tides and sand transport in the North Sea was used to establish potential tidal sand transport patterns in the Southern Bight and sand budgets for the Dutch and Belgian coast. The sand transport formulation yields magnitude and direction in addition to erosion/deposition values, in contrast to previous studies. All palaeobathymetries were obtained using the present bathymetry and sea level change results from an isostatic rebound model. The sea first invaded the Southern Bight through the Strait of Dover, and the microtidal range increased rapidly to mesotidal. The amphidromic system developed when the land bridge between Britain and the continent flooded and the northern tidal influence increased. Simultaneously, the current ellipses rotated from shore normal to shore parallel, while the flow velocities increased. The net tidal sand transport direction changed from toward the Dutch and Belgian shore to northward alongshore in the north and southward in the south. Until 7000 B.P. erosion dominated at sea, and deposition dominated near shore. Subsequently, erosion occurred in the south, and sedimentation occurred in the north. To the north of the Southern Bight, transport rates were low. Since 7500 B.P., sand has been transported mainly in suspension. Dominant transport by tidal asymmetry changed to transport by tidal residual currents and back in large areas. Comparison with previous studies illustrates that isostatic rebound is important for constructing palaeobathymetries. Most sediment was supplied to the Dutch and Belgian coast before 6000 BP. Substantial tidally induced mean sea level changes occurred in the early Holocene.

THE EFFECT OF GEOMETRY AND BOTTOM FRICTION ON LOCAL BED FORMS IN A TIDAL EMBAYMENT

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.

Morphodynamics of ebb-tidal deltas: a model approach

Abstract The results of 2DH numerical models of the Frisian Inlet (located in the Dutch Wadden Sea) are discussed to gain further knowledge about the physical mechanisms causing the presence of both ebb-tidal deltas and of channels and shoals in tide-dominated inlet systems. A hydrodynamic model, extended with sediment transport formulations, was used to verify earlier conceptual models that deal with ebb-tidal delta characteristics. The model does not confirm their hypothesis concerning the observed spatial asymmetry of ebb-tidal deltas and suggests that long-term morphological simulations are needed to understand this aspect. Furthermore, the model indicates that the initial formation of the ebb-tidal delta is mainly due to convergence of the tidally averaged sediment flux related to residual currents, whilst the net sediment transport in the basin is mainly caused by tidal asymmetry. A second model (accounting for feedbacks between tidal motion and the erodible bottom) was used to simulate the long-term bathymetric evolution of the Frisian Inlet under fair weather conditions. This model reproduces the gross characteristics of the observed morphology: the presence of a double-inlet system with two distinct ebb-tidal deltas having different sizes and the presence of channels and shoals. The role of the ‘Engelsmanplaat’, a consolidated shoal in the middle of the Frisian Inlet, was not found to be crucial for the morphodynamic stability of this inlet system.

Effect of wave-topography interactions on the formation of sand ridges on the shelf

[1] The role of wave-topography interactions in the formation of sand ridges on microtidal inner shelves is investigated with an idealized morphodynamic model. The latter uses the two-dimensional shallow water equations to describe a storm-driven flow on an inner shelf with an erodible bottom and a transverse slope. Both bed load and suspended load sediment transport are included. New are the incorporation of a wave module based on physical principles and a critical shear-stress for erosion. A linear stability analysis is used to study the initial growth of bed forms, by analyzing the initial growth of small perturbations evolving on an alongshore uniform basic state, which describes a storm-driven flow on a microtidal inner shelf. Model simulations show that wave-topography interactions cause the ridges to become more trapped to the coast. Both growth and migration of the ridges are controlled by suspended load transport. The physical mechanism responsible for ridge growth is related to transport by the stormdriven current of sediment that is entrained due to wave orbital motions induced by bed forms. This new mechanism even acts in absence of a transverse bottom slope. The orientation, spacing and shape of the modeled ridges agree well with field observations from different shelves.