1 

Residuele stromingen in een getijdenkanaal ten gevolge van de bodemwrijving
Met behulp van een wiskundig model het bepalen van de invloed van de bodemwrijving op de residuele stroming in een open kanaal.

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2 

Simulation of drying and ﬂooding in a tidal embayment using the level set method

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3 

Waterbeweging in een estuarium: analytische modellering van getijde beweging voor diepteafhankelijke viscositeitsprofielen met behulp van de breedtegemiddelde ondiepwatervergelijkingen.
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 breedtegemiddelde ondiepwatervergelijkingen (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.

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4 

Morphodynamic equilibria for a width varying tidal embayment
In this thesis a morphodynamic model of a tidal embayment, first studied in Schuttelaars & de Swart (1996), will be described and investigated. The difference with the rectangular embayment in Schuttelaars & de Swart (1996) is that the width is described with an exponential function. We focus here on the width averaged solutions. This gives us a crosssectional averaged system of equations similar to an onedimensional system. For the sediment transport a bed load and a suspended load transport model are considered. We also distinguish the situation with and without an externally prescribed overtide. In the situation without overtide the forcing at the seaward side is described with a basic tide. Including the overtide means that we also use higher harmonics in the forcing at the seaward side. The equilibrium bed levels are calculated for each sediment transport model. This shows that for bed load transport the width variations do not have a lot of influence. The resulting equilibrium bed level is exactly the same as the one in Schuttelaars & de Swart (1996). For suspended load transport are three models used: advectively dominated, diffusively dominated and combined sediment transport. The equilibrium bed level for advectively dominated transport is hardly influenced by the varying width because the velocity is compensating the effects of the change in width. For diffusively dominated transport the equilibrium bed level becomes convex for a converging embayment and concave for a diverging embayment. The combined suspended load transport without the externally prescribed overtide shows that the equilibrium bed level is similar to the diffusively dominated transport. For the situation with the externally prescribed overtide the effective length of the embayment decreases for large values of the relative width scale. For these values the model does not apply. Physical interpretations of these results are also given. Finally it is shown that all the equilibrium solutions are stable.

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5 

The Influence of Retention Basins on Tidal Dynamics in Estuaries
The Ems estuary suffers from high concentrations of suspended sediment, which is harmful for the life in the river. Sediment import occurs because of tidal asymmetry, a difference between the velocity of the current during high tide and low tide. The use of retention basins are a possible measure to reduce the tidal asymmetry.
An analytic model, using partial differential equations derived from the Shallow Water equations and NavierStokes equations, was used to investigate the influence of retention basins on the tidal dynamics and sediment transport in estuaries in general and the Ems estuary specifically. For these investigations, this analytic model is useful because system conditions can easily be changed.
Results show that the locations of retention basins are of great importance for their performance. In the Ems estuary the location minimizing the sediment import is at the end of the estuary, near the weir. Enlarging the retention basin is better than placing a second basin at a different location in the estuary.

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6 

Equilibrium and stability of a double inlet system
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 crosssectional 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 crosssectional 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 crosssections 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 crosssectional 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 crosssectional stability of multiple inlet systems.

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7 

Tidal Embayments: Modelling and understanding their morphodynamics
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.

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8 

Resonance characteristics of tides in branching channels

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9 

Sediment transport patterns in the Dutch Western Wadden Sea: influence of forcing and bathymetry
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 largescale 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 ebbtidal 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 semiempirical 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 ebbchannels. 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.

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10 

Turbulence modelling in environmental flows: Improving the numerical accuracy of the kepsilon model by a mathematical transformation
Numerical modelling for environmental flow applications, such as for rivers, lakes, estuaries and coastal flows, faces a tradeoff between the numerical accuracy and the required computation time. This tradeoff results in grids which typically contain 10 to 100 layers in the vertical direction. Such a grid resolution poses severe limitations to the numerical accuracy of the model. The turbulence model determines a significant part of this accuracy. This research therefore investigates an unexplored method of using transformations to improve the numerical accuracy of twoequation turbulence models at a low resolution.
The kepsilon model is used as starting point for this method. The equation for epsilon is transformed to equations for omega and tau. This results in three turbulence models, the kepsilon, komega and ktau models, which are physically equivalent, but possess different numerical properties. This research identifies these different numerical properties in order to explain when and why a certain transformation is beneficial to the numerical accuracy. The three turbulence models are tested in six cases of homogeneous and stratified flows in a onedimensional vertical (1DV) numerical model, which is representative for the implementation in the 3D simulation system Delft 3DFLOW.
It is shown that the ktau model yields more accurate results than the kepsilon and komega models in boundary friction dominated flows, such as those found in rivers, partially stratified estuaries and along the coast. This improved performance is explained from the profile of tau, which is linear near the frictional boundary and therefore accurately approximated on a low resolution grid. The profiles of epsilon and omega are hyperbolic near the frictional boundary and therefore not accurately represented on such a grid.
The boundary condition for tau is wellposed, while no natural boundary conditions for epsilon and omega exist. Dirichlet boundary conditions for epsilon and omega are therefore inaccurate. The Neumann boundary condition is found to be the most accurate alternative boundary condition for epsilon and omega. An adjusted Dirichlet conditions used in Delft 3DFLOW improves on the result of the ordinary Dirchlet condition, but shows bad convergence behaviour, with results being significantly worse at 100 vertical layers than at 10 vertical layers. A new adjusted Dirichlet condition is developed, which has better convergence behaviour, but is still somewhat worse than the Neumann condition.
The ktau and komega models contain a number of diffusive terms, the implementation of which may introduce numerical diffusion in the model. Some of these diffusive terms are essential to the stability of the model. Others are optional. It is argued that the choice whether or not to include such optional diffusive terms should be based on both physical and numerical arguments, because the numerical diffusion associated with the implementation of the terms may have a significant desired or undesired effects on the model results.
It is found in the cases in this research that convergence of the turbulence models with increasing grid resolution is typically found between 100 and 1000 grid cells in the vertical direction. One case of temperature modelling of a lake has been tested in which convergence did not occur up to 2000 grid cells. So converged results are generally beyond the range of generally used vertical resolution in 3D models. Within the feasible range of 10 to 100 layers it is found that the results of the turbulence models do not necessarily become more accurate if higher resolution grids are used. So monotonous convergence of the turbulence models is not guaranteed.

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11 

On the effect of tidal variations of turbulent mixing on flow and salt transport in estuaries
The intrusion of sea water into estuaries creates a complex flow that results from the density difference between fresh river runoff and salty sea water. In combination with other processes, such as tides, these complex flows are of interest because they affect the transport of e.g. salt, sediments and nutrients in water, which determine the future shape and ecology of the estuary. This research focusses on mechanisms that result in exchange flows and transport of salt.
The model study has led to new insights into the way in which tides and baroclinic pressure gradients contribute to the straining circulation, i.e. the exchange flow that is induced by the interaction of temporal variations of turbulence and velocity. The straining circulation is traditionally associated with the tidal flow. To make a distinction between this traditional view and the new findings, we will call this the {\em tidal straining circulation}. It is shown that the tidal straining circulation explains only a small part of the total straining circulation in a parameter space typical for wellmixed and partially stratified estuaries.
This research identifies a new and more important contribution to the straining circulation. This is caused by interactions between the gravitational circulation and temporal variations of turbulent mixing, which we will call the {\em gravitational straining circulation}. The gravitational straining circulation increases nonlinearly with increasing temporal variations of turbulence. Large tidal variations of turbulent mixing are typically found in wellmixed and partially stratified estuaries. Such temporal variations of turbulence can be caused by straininduced periodic stratification (SIPS), asymmetric mixing or symmetric variations of mixing, such as the variation of turbulence with the tide.
The dominant contribution of gravitational straining circulation to the total straining circulation explains why the straining circulation is larger than the gravitational circulation in partially stratified estuaries and why both the gravitational circulation and the straining circulation have the same dependency on the alongchannel salinity gradient. It also explains why the straining circulation is much smaller in strongly stratified estuaries, where the tidal variations of turbulence are not as large as in partially stratified estuaries. The direction and magnitude of the gravitational straining circulation is additionally shown to be independent of the timing of temporal variations of turbulent mixing. The magnitude and direction of the tidal straining circulation depend not only on the timing of temporal variations of mixing, but are shown to also depend on the bed roughness and the rate of mixing. This implies that the tidal straining circulation can act in the opposite direction as is expected from current theory in certain model parametrisations.
Concerning the transport of salt, it is shown that temporal variations of turbulent mixing are able to create a strong salt transport. A potentially large salt transport is caused by temporal correlations of the salinity and the velocity. This salt transport contribution is shown to be highly sensitive to the phase of the salinity, which is strongly dependent on the phase of turbulent mixing. Accurate modelling of turbulence is therefore essential to obtaining accurate results for the salinity.

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12 

Subsurface Characterization Using a Cellular Automaton Approach
In this paper, a random Cellular Automaton model is developed to characterise heterogeneity of geological formations. The CAmodel 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.

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13 

Impact of estuarine convergence on residual circulation in tidally energetic estuaries and inlets
Estuarine convergence (landward reduction of width and/or depth) is known to have the potential to significantly enhance estuarine circulation, a result theoretically derived under the assumption of constant eddy viscosity. Recent studies of longitudinally uniform energetic tidal channels indicate that tidal straining, a process driven by tidally varying eddy viscosity, is a major driver of estuarine circulation. The combined effect of estuarine convergence and tidal straining is investigated, for the first time, in this paper. The present idealized numerical study shows that estuarine convergence is reducing or even reversing tidal straining circulation in such a way that estuarine circulation can be weakened. This is a counterintuitive hydrodynamic effect of estuarine convergence, which may reduce (rather than increase) upestuary particulate matter transport in estuaries and tidal inlets.

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14 

The influence of basin geometry on equilibrium and stability of double inlet systems
This study investigates the influence of basin geometry on the crosssectional stability of double inlet systems. The inlet is in equilibrium when the amplitude of the inlet velocities equals the equilibrium velocity (~1 m s1). This equilibrium is stable when after a perturbation the crosssections 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.

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15 

Instability of timedependent winddriven ocean gyres
The winddriven 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 winddriven 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 timedependent (for example, periodic) flows, using this methodology, is computationally unpractical, due to the large number of degreesoffreedom involved. The approach followed here is to construct a loworder 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 winddriven flow is deduced from the reduced model. While there are several stabilizing processes, notably rectification, the destabilization occurs due to timedependent increase of the background horizontal shear in the flow.

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16 

Observations of barrier island length explained using an exploratory morphodynamic model
Barrier coasts display a chain of islands, separated by tidal inlets that connect a backbarrier basin to a sea or ocean. Observations show that barrier island length generally decreases for increasing tidal range and increasing basin area. However, this has neither been reproduced in model studies nor explained from the underlying physics. This is the aim of our study. Here we simulate barrier coast dynamics by combining a widely used empirical relationship for inlet dynamics with a processbased model of the tidal hydrodynamics. Our model results show stable inlet systems with more than one inlet open that support the observed qualitative relationships and fit in existing barrier coast classifications. To explain this, we identify a competition between a destabilizing mechanism (bottom friction in inlets, tending to reduce the number of open inlets) and a stabilizing one (spatially varying pressure gradients over the inlets, tending to keep the inlets open).

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17 

Analysis of tidal straining as driver for estuarine circulation in wellmixed estuaries
Tidal straining, which can mathematically be described as the covariance between eddy viscosity and vertical shear of the alongchannel 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 alongchannel 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 wellmixed tidal estuaries.

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18 

Seasonal variability in M2 and M4 tidal constituents and its implications for the coastal residual sediment transport
We use an observational data set of tidal gauges in the North Sea to investigate the annual cycle of the M2 and M4 amplitudes and phases. The sea surface elevation amplitude of the M2 can vary by 8–10% and the M4 amplitude by 12–30% over the course of the year, with larger amplitudes in summer. The annual phase variations are in the range of 3–15◦. The reason for these variations is the thermal structure of the North Sea: a welldeveloped thermocline in summer and wellmixed water column during winter. The interaction of the M2 and M4 tides is one of the main drivers of the residual sediment transport. Using an analytical model, the seasonal variability in residual sediment transport is estimated. This transport can vary by 10–50% over the course of the year. These variations are mainly related to the seasonal variability of the M2 and M4 amplitudes.

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19 

Effect of bottom stress formulation on modelled flow and turbidity maxima in crosssections of tidedominated estuaries
A threedimensional 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 crosschannel bottom profile. For a deep, wellmixed, tidally dominated channel, partial slip decreases the relative importance of Coriolis deflection on the generation of crosschannel flow patterns. This has profound implications for the lateral distribution of residual salinity that drives the crosschannel 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 crosschannel diffusion on the other hand. Hence, the changes in the crosschannel 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 nearbed slip condition. In this case, semidiurnal contributions to sediment distribution and lateral flow play an important role in crosschannel sediment transport. As vertical viscosity and diffusivity are increased, a second maximum at the right bank again exists for partial slip.

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20 

Influence of topography on tide propagation and amplification in semienclosed basins
An idealized model for tide propagation and amplification in semienclosed 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 depthaveraged 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 semidiurnal and diurnal
tide observations in the Gulf of California, the Adriatic Sea and the Persian Gulf.

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