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Met behulp van een wiskundig model het bepalen van de invloed van de bodemwrijving op de residuele stroming in een open kanaal.

In een estuarium is het belangrijk te weten wat de kenmerken zijn van de stroming van het water om de ecologie te bestuderen. Hiervoor wordt een tweedimensionaal model met behulp van de breedte-gemiddelde ondiep-water-vergelijkingen (shallow water equations) afgeleid. De aandacht wordt daarna vervolgens gefocust op de leidende orde vergelijkingen. Bij het oplossen van deze vergelijkingen wordt de nadruk wordt gelegd op de modellering van de viscositeit in de diepterichting. Er wordt daarbij vooral gekeken naar de invloed van verschillende viscositeitsprofielen op de waterbeweging.

Barrier island coasts are a common feature in many parts of the world. An example is the Wadden coast of The Netherlands, Germany and Denmark. These coasts consist of barrier islands separated by tidal inlets with at the landward side tidal basins.Characteristic for the Wadden Sea is that the tidal basins are not completely separated,but are connected via topographic highs allowing exchange of water between the basins. As a result the tidal inlets that connect the basins to the North Sea will interact. The focus in this thesis is on the effect of this interaction on the cross-sectional equilibrium and stability of tidal inlets that are part of a double inlet system. The knowledge gained in this study will help to develop rational management plans for this kind of system. In determining the equilibrium values and stability of cross-sectional areas of the inlets use is made of flow diagrams. A flow diagram consists of the equilibrium flow curves of each inlet and a flow field showing the adaptation of the inlet cross-sections after the system has been removed from equilibrium. Each intersection of the equilibrium flow curves represents a stable or unstable equilibrium. The equilibrium flow curve for each inlet is the locus of the values of the cross-sectional areas for which the velocity amplitude in the inlet equals the equilibrium velocity i.e. approximately 1 m/s according to ESCOFFIER [1940]. As a start the double inlet system is schematized as a basin connected to the ocean by two channels. The water surface area of the basin is assumed constant and water levels are assumed to fluctuate uniformly. On the seaward side a simple harmonic, semidiurnal tide is used to force the system. Analyzing the double inlet system under these conditions by means of the flow diagrams leads to the conclusion that a stable equilibrium of the two inlets does not exist. Ultimately only one inlet remains open and the other will close. This confirms the earlier conclusions of VAN DE KREEKE [1990] and BORSJE [2003] concerning the cross-sectional stability of multiple inlet systems.

The Dutch Western Wadden Sea, comprising the basins of Texel, Eierlandse Gat and Vlie, suffers a shortage of sediments. This implies that sediment is imported into the basins. The influence of the main forcing agents on the sediment transport is investigated. These forcing agents are the tide, waves and wind. In case of the wind forcing, both the large-scale and local wind forcing are taken into account. The tide induces an import of sediments through the Texel inlet, and export through the inlets of Eierlandse Gat and Vlie. Waves have most influence on the ebb-tidal delta region of the inlets. Due to the sheltering effect of these deltas and the barrier islands they lose most of their energy when they enter the inlets. As a result the influence of the waves on the sediment transports inside the basins is low. Wind forcing generates a residual flow that is of the same order of magnitude as the residual flow generated by the tide. It induces residual sediment transports through the inlets that are of the same order of magnitude as the tide induced sediment transports. A wave and wind climate is constructed to determine the annual residual sediment transports. With the bathymetry of the year 1998, in total a volume of 6306*103m3/yr is imported into the basins. With the help of the semi-empirical equilibrium relations, the required volume of sediments for the basins to be in equilibrium is determined. In total a volume of 13.78*108m3 is added, 88% of this is placed in the Texel basin. As a result the transport through the Texel inlet approaches the equilibrium (little to no residual transport), but through the other two inlets there is a strong residual export. The shift of the tidal divides is one reason for these results. These divides shift towards the Texel inlet, adjusting the tidal propagation in the basins. In the basins of the Eierlandse Gat and Vlie this results in an increase of the exports. This effect was not taken into account with the determination of the required volume of sediments. Another reason for the strong export through the inlets of the Eierlandse Gat and Vlie is the placement of the added sediments. In the basins of these inlets the additional sediments are predominantly placed in the ebb-channels. This increases the residual export of sediment through these channels. This implies that the determined equilibrium bathymetry is not the exact equilibrium. Yet it gives a fair indication of the amount of sediments needed for the basins to be in equilibrium.

Tidal basins are observed all over the world's coastline, for example along the north coast of The Netherlands. These basins are important both from an economic and ecological point of view. Complex channel and shoal patterns can be found in these inlets, as they develop due to the interaction of the currents generated by tides, wind and density differences with the erodible bed. External conditions, such as human interferences, can also strongly influence the morphologic behaviour. The aim of this thesis is to elucidate the physical mechanisms resulting in the formation of channel and shoal patterns in short tidal embayments.

In this paper, a random Cellular Automaton model is developed to characterise heterogeneity of geological formations. The CA-model is multilateral and can be easily applied in both two and three dimensions. We demonstrate that conditioning on well data is possible and the method is numerically efficient. To construct the model, the subsurface is subdivided into N cells, with an initial lithology assigned to each cell. Rules to update the current cell states are chosen from a set of rules, independently for each cell. The model converges typically in less than 50 iterations to a steady state or periodic solution. Within one period the realisations exhibit similar statistical properties. The final fraction of the various lithologies can be tuned by choosing the proper initial fractions. In this way, geological knowledge of those fractions can be satisfied.

This study investigates the influence of basin geometry on the cross-sectional stability of double inlet systems. The inlet is in equilibrium when the amplitude of the inlet velocities equals the equilibrium velocity (~1 m s-1). This equilibrium is stable when after a perturbation the cross-sections of both inlets return to their original equilibrium value. The necessary amplitudes of the inlet velocities are obtained using an idealized 2DH hydrodynamic that calculates tidal elevation and flow in a geometry consisting of several adjacent rectangular compartments. Model results suggest that regardless of the inclusion or exclusion of bottom friction in the basin, stable equilibrium states exist. Qualitatively, the influence of basin geometry does not change the presence of stable equilibrium. Quantitatively, however, taking a basin surface area of 1200 km2, equilibrium values can differ up to a factor 2 depending on the geometry of the basin.

The wind-driven ocean circulation at midlatitudes is susceptible to several types of instabilities. One of the simplest models of these flows is the quasigeostrophic barotropic potential vorticity equation in an idealized ocean basin. In this model, the route to complex spatio/temporal flows is through successive bifurcations. The aim of this study is to describe the physics of the destabilization process of a periodic wind-driven flow associated with a secondary bifurcation. Although bifurcation theory has proven to be a valuable tool to determine the physical mechanisms of destabilization of fluid flows, the analysis of the stability of time-dependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degrees-of-freedom involved. The approach followed here is to construct a low-order model using numerical Galerkin projection of the full model equations onto the dynamically active eigenmodes. The resulting reduced model is shown to capture the local dynamics of the full model. The physical mechanism of the destabilization of the periodic wind-driven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to time-dependent increase of the background horizontal shear in the flow.

Tidal straining, which can mathematically be described as the covariance between eddy viscosity and vertical shear of the along-channel velocity component, has been acknowledged as one of the major drivers for estuarine circulation in channelized tidally energetic estuaries. In this paper, the authors investigate the role of lateral circulation for generating this covariance. Five numerical experiments are carried out, starting with a reference scenario including the full physics and four scenarios in which specific key physical processes are neglected. These processes are longitudinal internal pressure gradient forcing, lateral internal pressure gradient forcing, lateral advection, and the neglect of temporal variation of eddy viscosity. The results for the viscosity–shear covariance are correlated across different experiments to quantify the change due to neglect of these key processes. It is found that the lateral advection of vertical shear of the along-channel velocity component and its interaction with the tidally asymmetric eddy viscosity (which is also modified by the lateral circulation) is the major driving force for estuarine circulation in well-mixed tidal estuaries.

A three-dimensional numerical model with a prognostic salinity field is used to investigate the effect of a partial slip bottom boundary condition on lateral flow and sediment distribution in a transect of a tidally dominated channel. The transect has a symmetrical Gaussian cross-channel bottom profile. For a deep, well-mixed, tidally dominated channel, partial slip decreases the relative importance of Coriolis deflection on the generation of cross-channel flow patterns. This has profound implications for the lateral distribution of residual salinity that drives the cross-channel residual circulation pattern. Transverse sediment transport, however, is always found to be governed by a balance between advection of residual sediment concentration by residual lateral flow on the one hand and cross-channel diffusion on the other hand. Hence, the changes in the cross-channel distribution of residual salinity modify the lateral sediment distribution. For no slip, a single turbidity maximum occurs. In contrast, partial slip gives a gradual transition to a symmetrical density distribution with a turbidity maximum near each bank. For a more shallow, partially mixed tidal channel that represents the James River, a single turbidity maximum at the left bank is found irrespective of the near-bed slip condition. In this case, semi-diurnal contributions to sediment distribution and lateral flow play an important role in cross-channel sediment transport. As vertical viscosity and diffusivity are increased, a second maximum at the right bank again exists for partial slip.

An idealized model for tide propagation and amplification in semi-enclosed rectangular basins is presented, accounting for depth differences by a combination of longitudinal and lateral topographic steps. The basin geometry is formed by several adjacent compartments of identical width, each having either a uniform depth or two depths separated by a transverse topographic step. The problem is forced by an incoming Kelvin wave at the open end, while allowing waves to radiate outward. The solution in each compartment is written as the superposition of (semi)-analytical wave solutions in an infinite channel, individually satisfying the depth-averaged linear shallow water equations on the f plane, including bottom friction. A collocation technique is employed to satisfy continuity of elevation and flux across the longitudinal topographic steps between the compartments. The model results show that the tidal wave in shallow parts displays slower propagation, enhanced dissipation and amplified amplitudes. This reveals a resonance mechanism, occurring when the length of the shallow end is roughly an odd multiple of the quarter Kelvin wavelength. Alternatively, for sufficiently wide basins, also Poincaré waves may become resonant. A transverse step implies different wavelengths of the incoming and reflected Kelvin wave, leading to increased amplitudes in shallow regions and a shift of amphidromic points in the direction of the deeper part. Including the shallow parts near the basin’s closed end (thus capturing the Kelvin resonance mechanism) is essential to reproduce semi-diurnal and diurnal tide observations in the Gulf of California, the Adriatic Sea and the Persian Gulf.

The initial growth of bed perturbations on planar sloping beaches under the forcing of obliquely incident, breaking waves is investigated using a state‐of‐the‐art numerical model. This allows for a systematic investigation of the sensitivity of the spatial structures of the bed perturbations and their growth and migration rates to different model formulations and parameterizations. If the sediment is only transported in the direction of the net current velocity and sediment stirring is taken proportional to the wave height squared, growing up‐current oriented crescentic bars are found with a preferred spacing of 800 m and a down‐current migration rate of 10 m d−1. Varying the angle of wave incidence, drag coefficient and bed slope results in qualitatively similar growing bed forms. Using an Engelund and Hansen transport formula, very oblique down‐current oriented bars are obtained that grow in time. No preferred wavelength, however, is found. Using the Bailard transport formula results in growing, up‐current oriented bars with a preferred spacing smaller than 300 m for wave angles smaller than 7°. When using either the Engelund and Hansen or Bailard sediment transport formulation, it is essential to take the transport in the direction of the wave orbital velocity into account in order to have growing bed perturbations.

The linear and nonlinear morphological behavior of double-barred coastal systems under the forcing of obliquely incident waves is studied using a nonlinear numerical model. The linearly most unstable bed forms consist of crescentic patterns (rip channels), whose spacing depends on the magnitude of the longshore current velocity. Using the nonlinear model, six morphodynamic experiments have been performed with various initial bed perturbations in order to assess, among others, the influence of the initial bed perturbation on the morphodynamic evolution. The nonlinear experiments have been pursued well into the nonlinear regime, showing that after a phase of initial exponential growth, a highly dynamic behavior is observed and no equilibrium state is reached. The spacings predicted with the linear stability analysis are observed during the exponential phase of the nonlinear experiments. In the dynamic phase, however, four to seven modes significantly contribute to the resulting bed features. In this final stage, the apparent wavelength of 1000 m of the resulting bed forms on the inner bar is quite insensitive to the initial bed perturbation. On the outer bar it seems that the longer the wavelength of the initial bed perturbation, the longer the wavelength of the final bed forms in the dynamic phase and the larger the migration celerity. In general, the bed forms can be characterized as crescentic or undulating bed patterns. Good correspondence between simulated and observed spacings, shapes and migration celerities are found.

In many tidal embayments, complex patterns of channels and shoals are observed. To gain a better understanding of these features, an idealized model, that describes the interaction of water motion, sediment transport and bed evolution in a semi-enclosed, rectangular basin, is developed and analysed. To explain the initial formation of channels and shoals, two-dimensional perturbations superposed on a laterally uniform equilibrium bottom are studied. These perturbations evolve due to convergences of various residual suspended sediment fluxes: a diffusive flux, a flux related to the bed topography, an advective flux resulting from internally generated overtides and an advective flux due to externally prescribed overtides. For most combinations of these fluxes, perturbations start to grow if the bottom friction is strong enough. Their growth is mainly a result of convergences of diffusive and topographically induced sediment fluxes. Advective contributions due to internally generated overtides enhance this growth. If only diffusive sediment fluxes are considered, the underlying equilibrium is always unstable. This can be traced back to the depth dependence of the deposition parameter. Contrary to the results of previous idealized models, the channels and shoals always initiate in the shallow, landward areas. This is explained by the enhanced generation (compared to that in previous models) of frictional torques in shallow regions. The resulting initial channel–shoal formation compares well with results found in complex numerical model studies. The instability mechanism and the location of the initial formation of bottom patterns do not change qualitatively when varying parameters. Changes are mainly related to differences in the underlying equilibrium profile due to parameter variations.

For a proper understanding of flow patterns in curved tidal channels, quantification of contributions from individual physical mechanisms is essential. We study quantitatively how such contributions are affected by crosschannel bathymetry and three alternative eddy viscosity parameterisations. Two models are presented for this purpose, both describing flow in curved but otherwise prismatic channels with an (almost) arbitrary transverse bathymetry. One is a numerical model based on the full threedimensional shallow water equations. Special feature of this diagnostic model is that assumptions regarding the relative importance of particular physical mechanisms can be incorporated in the computations by switching corresponding terms in the model equations on or off. We also present an idealized model that provides semi-analytical approximate solutions of the shallow water equations for all three considered alternative eddy viscosity parameterisations. It forms an aid in explaining and theorising about results obtained with the numerical model. Observations regarding Chesapeake Bay serve as a reference case for the present study. We find that the relative importance of both along channel advective forcing and transverse diffusive forcing depends on local characteristics of the cross-sectional bottom profile rather than global ones. In our reference case, tide-residual along-channel flow induced by these forcings is not small compared to the total tidal residual. Building on this observation, we present an indicative test to judge whether advective processes should be included in leading order in modelling tide-dominated estuarine flow. Furthermore, depending on the applied eddy viscosity parameterisation (uniformly or parabolically distributed over the vertical), we find qualitatively different spatial patterns for the along-channel advective forcing.

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 Coriolis mechanism dominates and accumulates sediment on the right bank (looking up‐estuary in the Northern Hemisphere). On the other hand, when frictional forces are moderate to high (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.

The generation of residual circulation in a tidally energetic estuary with constant longitudinal salinity gradient and parabolic cross section is examined by means of a two-dimensional cross-sectional numerical model, neglecting river runoff and Stokes drift. It is shown how the longitudinal and lateral residual circulation can be decomposed into contributions from various processes such as tidal straining circulation, gravitational circulation, advectively driven circulation, and horizontal mixing circulation. The sensitivity of the residual circulation and its components from various processes to changes in forcing is investigated by varying the Simpson number (nondimensional longitudinal buoyancy gradient) and the unsteadiness parameter (nondimensional tidal frequency), as well as the bed roughness and the width of the estuary. For relatively weak salinity gradient forcing, the tidal straining circulation dominates the residual exchange circulation in support of classical estuarine circulation (up-estuary flow near the bed and down-estuary flow near the surface). The strength of the longitudinal estuarine circulation clearly increases with increased salinity gradient forcing. However, when the Simpson number exceeds 0.15, the relative contributions of both gravitational circulation and advectively driven circulation to estuarine circulation increase substantially. Lateral residual circulation is relatively weak for small Simpson numbers and becomes flood oriented (divergent flow near the bed and convergent flow near the surface) for larger Simpson numbers because of increasing contributions from gravitational and advectively driven circulation. Increasing the unsteadiness number leads to decreased longitudinal and lateral residual circulation. Although changes in bed roughness result in relatively small changes in residual circulation, results are sensitive to the width of the estuary, mainly because of changes in residual exchange circulation driven by tidal straining.

In many tidal embayments, bottom patterns, such as the channel-shoal systems of the Wadden Sea, are observed. To gain understanding of the mechanisms that result in these bottom patterns, an idealized model is developed and analyzed for short tidal embayments. In this model, the water motion is described by the depth- and width-averaged shallow water equations and forced by a prescribed sea surface elevation at the entrance of the embayment. The bed evolves due to the divergence and convergence of suspended sediment fluxes. To model this suspended-load sediment transport, the three-dimensional advection–diffusion equation is integrated over depth and averaged over the width. One of the sediment fluxes in the resulting one-dimensional advection–diffusion equation is proportional to the gradient of the local water depth. In most models, this topographically induced flux is not present. Using standard continuation techniques, morphodynamic equilibria are obtained for different parameter values and forcing conditions. The bathymetry of the resulting equilibrium bed profiles and their dependency on parameters, such as the phase difference between the externally prescribed M2 and M4 tide and the sediment fall velocity, are explained physically With this model, it is then shown that for embayments that are dominated by a net import of sediment, morphodynamic equilibria only exist up to a maximum embayment length. Furthermore, the sensitivity of the model to different morphological boundary conditions at the entrance of the embayment is investigated and it is demonstrated how this strongly influences the shape and number of possible equilibrium bottom profiles. This paper ends with a comparison between the developed model and field data for the Wadden Sea’s Ameland and Frisian inlets. When the model is forced with the observed M2 and M4 tidal constituents, morphodynamic equilibria can be found with embayment lengths similar to those observed in these inlets. However, this is only possible when the topographically induced suspended sediment flux is included. Without this flux, the maximum embayment length for which morphodynamic equilibria can be found is approximately a third of the observed length. The sensitivity of the model to the topographically induced sediment flux is discussed in detail.

Over decades and centuries, the mean depth of estuaries changes due to sea-level rise, land subsidence, infilling, and dredging projects. These processes produce changes in relative roughness (friction) and mixing, resulting in fundamental changes in the characteristics 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 widthaveraged shallow water equations and a sediment balance equation. Together with the condition of morphodynamic equilibrium, these equations are solved analytically by making a regular expansion of the various 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 estuary. 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 physical 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 various 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 freshwater 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.