In estuaries, the part of a river where the flow is predominantly driven by the tide, the water motion is important to understand. For instance, by understanding the flow, one can better predict how pollutants are transported, or when flooding is likely to occur. However, much is still unknown about certain aspects of the water motion, such as how the hydrodynamics are influenced by the steepness of the river bed.

In this thesis, the effect of steep lateral inclines on the river bed is systematically investigated. Because these steep inclines come with significant mathematical challenges, a new model for the water motion in estuaries is developed.

To study the effects of steepness, the model is applied to short estuaries with increasingly deep channels with increasingly steep inclines in the middle of the river, and shallower waters closer to the banks. The results indicate that during flood, i.e. the moment when flow into the estuary is strongest, water flows in through the deep channel and then from the channel towards the banks. During ebb, this pattern is reversed. For estuaries with steeper beds, currents are stronger, and the previously mentioned pattern is more pronounced.

Through a process known as momentum advection, so-called residual currents appear that are present regardless of the phase of the tidal cycle. Because these residual currents produce net transport by the movement of water, they are crucial in many transport processes involving sand, mud, salt, pollutants or other materials. The structure of the current is as follows: in the deeper channels, the residual current flows from the sea into the channel, and close to the banks,water flows out of the estuary. Steeper beds cause the residual flow to become stronger and (indirectly) because of the Coriolis effect, notably asymmetric.

Differential equations are commonly solved by, first, discretising the domain and then using an iterative method on the resulting system of equations. Refining the mesh gives a more accurate solution. However, the iterative method does not necessarily converge quickly to a good approximate solution anymore. To deal with this issue, we can add a preconditioner to the system. A good preconditioner enhances the speed of convergence.

Two ways of finding a good preconditioner are known. The first uses the properties of the matrix of the system, that is an algebraic preconditioner. The second uses the properties of the differential equation that give rise to the system, which is an operator preconditioner. This thesis makes a connection between both types of preconditioning.

First, we discretise a differential equation and perform numerical test on the system to see if the
algebraic preconditioning works. Then, the domains on which the matrix and preconditioner act are defined in terms of the differential equations.

We conclude that the matrix containing the differential operators acts on H(div, Ω) × H1(Ω) and the preconditioner acts on [L2(Ω)]^2× H2(Ω). So we need to constrain the domains in such a way that the operators both act on the same domain: that is, H(div, Ω) × H2(Ω). If the function is in this domain,then we know that the algebraic preconditioner is an effective preconditioner.

The sea and the shoreline form a complex ecosystem driven by tides. So far, studies often ignore the moving boundary caused by these tides. The focus of this thesis to incorporate this boundary by using a coordinate transformation and a time-explicit numerical method. To achieve this, first the one-dimensional shallow water equations are derived from the 3D Navier Stokes equations. Then these 1D equations are non-dimensionalized and the coordinate transformation is done. This results in a system of non-linear equations. The seabed is modelled as a straight line. At the seaward side there is a periodic forced wave and at the landward side the water depth is 0. The time-explicit numerical method of Lax-Friedrichs is used. This method is stable under a more restricted Courant-Friedrichs-Lewy condition and is convergent for refined grids. For the Ameland inlet system the water depth, velocity and length of the basin results are calculated and compared to a simplified model and complemented by a Fourier analysis. The results are realistic (constant in the beginning of the basin with visible non-linearities at the landward side). An analysis is done to understand how the model behaves for dierent physical parameters, such as: the amplitude of the periodically forced wave, the undisturbed water depth, the length of the basin and the resistance. The model remains stable and the results are realistic.

In dit verslag is een eolisch sedimenttransport model geformuleerd en het model van De Vries geïmplementeerd en geanalyseerd. Na de afleiding van de eendimensionale advectie-diffusievergelijking vanuit de driedimensionale continuïteitsvergelijking is eerst het model van De Vries zonder diffusie onderzocht. In het model kan er onderscheid gemaakt worden tussen 3 verschillende situaties: Geen (toevoer van) sediment, Sediment-gelimiteerd en Erosie-gelimiteerd. In elke situatie zijn voor lange tijd\-schalen de toestanden van het model afgeleid vanuit de advectievergelijking. In de resultaten kwam naar boven dat de theoretische toestanden sterk overeenkwamen met de toestanden in de numerieke simulatie. Verder is het effect van numerieke diffusie van het model van De Vries geconstateerd en de grootte van dit effect afgeleid. Om het effect van numerieke diffusie te verminderen is diffusie in het model geïntroduceerd samen met het Lax-Wendroff numerieke schema. Dit schema gebruikt de gewogen upwind methode samen met de gewogen centrale Euler methode om de nauwkeurigheid van het model te verbeteren. In de resultaten is aangetoond dat numerieke diffusie kan worden gecompenseerd door het juist kiezen van dit gewicht. Uit een stabiliteitsanalyse blijkt dat de Lax-Wendorff methode een betere stabiliteitcriterium heeft dan de upwind methode. Hierdoor kan een grotere tijdstap gekozen in een numerieke simulatie. De maximale tijdstap is getest door het model te runnen met verschillende tijdstappen rondom het theoretische maximum. Uit de resultaten bleek dat het model op tijdstap die een fractie kleiner was dan de theoretische maximale tijdstap, stabiele oplossingen simuleerde. Zodra de tijdstap de maximale waarde behaalde werden de oplossingen instabiel. Door het toevoegen van diffusie aan het model is een extra voorgeschreven randvoorwaarde nodig. Door implementatie van verschillende randvoorwaarden aan het einde van het domein is gebleken dat de randvoorwaarde 'extrapoleren tweede orde' het best is. De dimensie-analyse van de advectie-diffusievergelijking wijst uit de eolischsedimenttransport gedomineerd wordt door advectief transport en het effect van diffusie verwaarloosd mag worden.

Tidal flats are important coastal ecosystems that support a diverse range of plants and animals. Accurate prediction of water motion over tidal flats is crucial for the design of coastal infrastructure, hazard assessment, environmental management and oceanography research. Defina [1] derived a new set of shallow water equations that allow for water motion prediction in partially dry domains. Existing methods for predicting water motion are limited by numerical stability and accuracy, and require high-quality input data. Using the periodic behaviour of tides, this thesis introduces a new method in the frequency domain. The method is tested and compared to a fully numerical solution. The results of the linearised classical shallow water equations encourage future research required for the statistical shallow water equations that allow partially dry domains.

Due to their ecological significance and their role in providing grounds for a variety of human activities, understanding how tidal inlet systems evolve is of interest to scientists, engineers, and managers. The seabed profile in such systems evolves due to the erosion and deposition of sediment particles from and onto the seabed. Equivalently, this can be described by the tidally averaged convergences and divergences of sediment transport, which takes place due to diffusive and advective processes. The aim of this thesis is to systematically investigate the importance of the different sediment transport mechanisms in double-inlet systems, in which the back-barrier basin is connected to the open sea by two inlets. The geometry under consideration is that of a shallow tidal inlet with a prescribed length and arbitrary width. Starting from an idealised process-based model, in which all sediment transport mechanisms are included, the model is reduced to a formulation, in which only the dominant transport mechanisms are retained. The results obtained suggest that the transport mechanisms, which must be retained in order to obtain equilibrium bed profiles, comparable to those obtained with the original process-based model, are advection, erosion, and deposition. Modelling the transport of sediment using these terms results in reproducing the equilibrium bed profiles found with the original model in the case when the inlet system is modelled as a rectangular channel. When the width of the channel in non-constant, the equilibrium bed profiles obtained with the two approaches have similar shapes, with slight variations in their exact values.

In this thesis, the foundations are explored for establishing two-dimensional depth-integrated morphodynamical equilibria for a long embayment with two tidal inlets connected to the open sea. Governing water motion, suspended sediment transport and bed evolution equations had to be rederived from the underlying balances, where the tidal inlets are prescribed by M2 forcing, and the bedload transport is primarily due to the slope gradient. The equilibrium solutions are determined by an asymptotic expansion in a small parameter and applying a temporal Fourier series up to the leading order term. From this, a solvable system of six partial differential equations is derived. Using finite-difference discretization, this system of partial differential equations is turned into a discrete lexicographically ordered system of nonlinear constraints. The Newton-Rapsons method is used in a newly developed code to be able to produce morphodynamical equilibrium solutions. When the cross-sectional discretization contains only one interior point, the solutions are in accordance with the cross-sectional model. In particular, the more complex bifurcation structure arises after variation of the tidal phase, amplitude and inlet depth. This strengthens the recent findings of complex bifurcation structures. These width-averaged equilibria seem to stay accurate equilibria in the cross-sectional expanded model even for the more complex behaviour. The question of these equilibria are stable cannot be answered as firmly as some unexplained behaviour arose during the analysis of the eigenvalues. However, these results hint that the width-averaged equilibria remain stable until the width of the channel is half its length. Further research is needed to prove this.

The change in sea beds in double inlet systems is caused by a complex combination of factors such as sea level dynamics, water movement and suspended sediment in the water. The unchanged water depth, the equilibrium seabed, in inlet systems is important to know for shipping near harbours and wildlife in nature reserves.
An idealized exploratory version of this complex system of morphodynamics to find the equilibrium solutions is usually modelled in a depth- and width-averaged way, which removes the vertical structure. Hence I will investigate if it is possible to model the system in a width-averaged two-dimensional way with the vertical structure intact and what are the consequences of retaining the vertical structure?
To investigate this the construction of a two-dimensional model is described and the solution methods posed. Following this the resulting equilibrium sea beds are compared to the equilibrium sea beds found in depth-averaged one-dimensional models to ascertain the importance of the vertical structure.
It was found that adding the vertical structure makes the resulting governing equations of the model more complex but can be solved with perturbation techniques. Furthermore, for specific sets of parameters describing the system it is possible to have multiple equilibrium sea beds, which suggests the existence of bifurcations. Finally, it is possible to find a relation between the 1-dimensional and 2-dimensional governing equations that results in both models producing the same results.
Concluding, two-dimensional models compared to one-dimensional models add an extra layer of complexity which allows for bifurcations but makes the model computatively more expensive. Instead, using the relation found in this paper in a one-dimensional model is preferred as in theory it results in the same equilibrium sea beds found.

The change in sea beds in double inlet systems is caused by a complex combination of factors such as sea level dynamics, water movement and suspended sediment in the water. The unchanged water depth, the equilibrium seabed, in inlet systems is important to know for shipping near harbours and wildlife in nature reserves.
An idealized exploratory version of this complex system of morphodynamics to find the equilibrium solutions is usually modelled in a depth- and width-averaged way, which removes the vertical structure. Hence I will investigate if it is possible to model the system in a width-averaged two-dimensional way with the vertical structure intact and what are the consequences of retaining the vertical structure?
To investigate this the construction of a two-dimensional model is described and the solution methods posed. Following this the resulting equilibrium sea beds are compared to the equilibrium sea beds found in depth-averaged one-dimensional models to ascertain the importance of the vertical structure. \\ \\
It was found that adding the vertical structure makes the resulting governing equations of the model more complex but can be solved with perturbation techniques. Furthermore, for specific sets of parameters describing the system it is possible to have multiple equilibrium sea beds, which suggests the existence of bifurcations. Finally, it is possible to find a relation between the 1-dimensional and 2-dimensional governing equations that results in both models producing the same results.
Concluding, two-dimensional models compared to one-dimensional models add an extra layer of complexity which allows for bifurcations but makes the model computatively more expensive. Instead, using the relation found in this paper in a one-dimensional model is preferred as in theory it results in the same equilibrium sea beds found.

In this thesis the water motion and sediment dynamics are investigated in a periodically closed and opened estuary. The water motion in an estuary is mainly driven by the semi-diurnal tide with an period of 12h25m and river discharge. An example of such an estuary is the Ems-Dollard estuary. Recent observations show an increase in tidal range (height difference between high and low tide), suspended sediment concentration and the depletion of oxygen levels (consequently harming the ecosystem). A possible solution, periodically closing and opening the estuary, is investigated. The water motion in a periodically opened and closed estuary is described by the linearised cross-sectionally averaged equations which give the sea-surface elevations and tidal velocity when solved with the eigenfunction expansion method. It was found that the sea-surface elevations and tidal velocity for a periodically opened and closed estuary are again 12h25m periodic. For the sediment transport, when no overtide is considered the residual sediment transport is seaward directed if the estuary is closed at low water and landward directed if the estuary is closed at high water. The barrier location determines the magnitude of the residual sediment. When overtide is included in the forcing of the system no relation is found
between the direction of the residual sediment transport and the closing height and closing position. The location of the barrier and closing height both determine the magnitude of the residual sediment transport and direction. By introducing a barrier that periodically closes and opens we intended to achieve a seaward directed residual sediment transport in the Ems-Dollard estuary. The results suggest that this is not possible. Further research is needed with more
extensive models to confirm this. For future Research I recommend to extend the model to a two-dimensional model with the eigenfunction expansion method. Other possibilities may be to consider a spatial dependent erodible bed.

In this thesis the hydrodynamic behaviour of an idealised relatively short tidal estuary is studied using a cross-sectionally averaged model, i.e. a one dimensional model. The geometry of the estuary is simplified and only the hydrodynamic processes are considered. This model is used to gain insight in the behaviour of currents and water levels influenced by a movable barrier situated in the estuary. It is investigated whether shifting the motion of the barrier in time influences the sedimentation in the embayment.

Estuaries support a unique hydrodynamic system due to the fresh water flowing into a salt water sea. This leads to stratification which can significantly alter internal tidal dynamics in the estuary. A model has been developed by Winant (2010) to describe the water motions of a two-layer system in tidal basin. The aim of this thesis is provide an extension to this model by parameterizing bottom stress using a partial slip parameter and viscosity. Also level variations of the bed are taken into account in this thesis. A width-averaged model of an idealized stratified tidal basin is developed. With increased bottom stress, the velocity near the bed is significantly slowed down, whereas increasing the viscosity slows down the velocity further away from the bed. The interface fluctuates more when the viscosity is low, the wavelength of these fluctuations is decreased with increasing viscosity. Resonance characteristics in the hydrodynamics are dampened severely by increased viscosity, and less by bottom stress. Increasing the slope of the bed results in a lag of tidal dynamics at the end of the embayment compared to at the embayment entrance. A sloped bed also dampens the resonance characteristics.

Estuaries have always been important for mankind and therefore it is essential to have a good understanding of the flow and sediment dynamics there. The goal of this thesis is twofold. One objective is to use an existing model to gain a more thorough understanding of the flow dynamics in an estuary. The second objective is to extend the model such that it can also compute the suspended sediment concentration in a cross-section. The equations governing the flow dynamics are the shallow water equations. The advection-diffusion equation governs the sediment dynamics. To compute the flow and suspended sediment concentration in a cross-section, conditions are assumed to be uniform in the along-channel direction. To solve the equations a coordinate transformation is applied first. After the transformation, the cross-section of the channel is represented in the computational domain by a rectangle. In the vertical direction, an eigenfunction expansion is used with eigenfunctions derived from a special case of the Sturm-Liouville eigenvalue problem. In the horizontal direction, derivatives are approximated with a central finite difference scheme. In the frequency domain, variables are expressed as the sum of tidal components. The Galerkin method is applied in both the vertical direction and the frequency domain to optimise the weight functions for every location along the transect. The system obtained with the Galerkin method is solved using Newton-Raphson iterations and an LU-decomposition. To find the distribution of the erosion coefficient corresponding to a morphodynamic equilibrium, a time integration method is used. The effect of several parameters on the advective contribution to the cross-channel flow is systematically investigated. The results show that the steepness of the bottom slope affects the magnitude of the advective contribution to the residual lateral flow. For a steep bottom slope the contribution is large and for a gradual bottom slope the contribution is small. The curvature of the channel strongly affects the total cross-channel flow, depending on the magnitude of the radius of curvature, but hardly affects the flow caused by advection. The lateral density gradient can largely affect the \mbox{cross-channel} flow. Especially, the amplitude of the M$_2$ tidal component of the density gradient affects the advective contribution to the flow. Both the magnitude and characteristics of the advective contribution change when the amplitude of the M$_2$ tidal component of the density gradient is varied. The phase of the M$_2$ tidal component of the density gradient hardly affects the cross-channel flow and advective contribution of the flow. Measurement data of a cross-section of the Ems is compared with a simulation of this situation. The magnitude of the lateral flow is similar for the measurements and model results but there is a difference in the direction of the flow in the upper part of the water column. This deviation could be caused by the description of the free surface. In the measurements there is a time-varying thickness of the water column whereas the rigid lid assumption is applied in the model. However, other differences between the simulation and the actual situation could have contributed to a deviation between the measurements and model results as well. The results for the sediment module show that the model works as expected for a prescribed erosion coefficient and for computing the erosion coefficient in morphodynamic equilibrium for situations with only diffusive transport. For simple situations the analytical solution is approximated and for more complicated situations the results agree with the physical intuition. The main recommendation for further research is to investigate how the model can be extended such that it is also possible to compute the erosion coefficient in morphodynamic equilibrium for situations with both advective and diffusive sediment transport.

Tidal inlet systems are often highly valuable and sometimes even unique ecosystems. However, field measurements show that tidal inlet systems are sensitive to changing exogenous conditions, such as rising sea levels. This thesis aims to investigate to what extent the adjacent sea influences the stability and equilibrium state of the tidal inlet. A one-dimensional idealised model is used to model the interaction between the sea and the inlet. The water motion is forced by the tide and the inlet is assumed to be narrow and short. At equilibrium, an increasingly sloping bottom is found in the sea and a constantly sloping bottom in the inlet. This equilibrium bottom profile seems to be in reasonable agreement with observations. The sea-inlet bottom profile is less stable than the inlet bottom profile, nevertheless, the sea-inlet bottom profile is still asymptotically linear stable. Moreover, the results in this thesis suggest that for one-dimensional idealised models consisting solely of a tidal inlet, the correct seaward boundary condition is a properly chosen fixed entrance depth. For a two-dimensional semi-infinite sea, it is shown that a Perfectly Matched Layer is a convenient method to incorporate the Sommerfeld radiation condition and that the narrow tidal inlet cannot be modelled as a point source forcing in the two-dimensional sea domain.

Using an idealized width-averaged model, the influence of spring-neap cycles on the transport and trapping of suspended fine sediment in tide-dominated estuaries is investigated. To this end we introduce a multiple time scale expansion. This provides a mathematically sound argument for treating the fast ebb-flood cycle and the slow spring-neap variations as independent time scales. With this expansion, semi-analytical approximations of the water motion and suspended sediment concentrations can be found as functions of the spring-neap time scale. The advantage of this is that the transport of sediment is then investigated in a tidally-averaged sense using analytically obtained temporal dependencies and model simulations based on the conditions in 2005 of the Ems-Dollard estuary. We found that sediment import is strongly enhanced during spring tide and that a combination of vanishing import processes and sediment export due to river discharge resulted in a net export of sediment during neap tide. Furthermore the spring-neap varying deposition of fine sediment in a bottom pool on the river bed is investigated. We found that the time needed for the system to adjust to new equilibrium conditions can not be neglected within a spring-neap cycle as temporal lag effects are clearly visible in the dynamic behaviour of the bottom pool and suspended sediment concentrations. These temporal lag effects are sensitive to the choice of the parameter governing erosion. Assuming different values of the erosion parameter we have shown that the long-term characteristics of the bottom pool are sensitive to these temporal lag effects. Choosing a low value of the erosion parameter results in the presence of a bottom pool throughout the spring-neap cycle. With a high value of the erosion no bottom pool is formed at any time in the spring-neap cycle.

The Rhine Region of Fresh Water Influence (ROFI) is a shallow frictional river plume in front of the Dutch coast. Each tidal cycle a new tidal plume front with fresh-water is released. Recently, internal gravity waves have been observed in this plume. Using a Froude number analysis, support for the internal wave generation mechanism by a tidal plume front is found. It is shown, that on averaged neap tides the density stratification is large. This results in a larger area where the internal waves can be released from the tidal plume fronts compared to spring tides. As the Rhine ROFI is located in shallow water, it is investigated what the effect of bathymetry variation is on internal waves. This is studied by extending the standard KdV model derivation to account for a variable bed. This resulted in a small correction on the propagation speed inside the KdV equation. Observations of internal waves have been used to validate the KdV model. By scaling analysis of the observed waves it is obtained that the relative wave height and relative depth balance for most of the observed events. For these events the wave period and velocity amplitude of the KdV model are well matched with the measurements. This showed that the KdV model may be used for a first estimate of internal waves in the Rhine ROFI. By developing a TGE fitting procedure, it was possible to obtain the pycnocline depth and the direction of wave propagation with only limited data available. Therefore, the parameters have been varied and the error between the TGE solution and the velocity potential obtained from the velocity measurements has been minimized.

Agragrians in the Dutch Wadden sea region struggles with salinizing farmlands. Acacia Water's project Spaarwater enabled agrarians to measure salt concentrations in the ditches of their land using their mobile phone. All these measurements would ideally be used to evaluate the effects of salinization mitigation measures and to predict water quality over time. This gives rise to this thesis for an inquiry into a fast and simple surface water flow model which eventually could even be run on a mobile phone. The water model is a first necessary step for developing a water quality model with salt concentrations. The main focus of this thesis is on developing finite volume methods for the kinematic wave equation. It is a simplification of the full shallow water equations which are the governing equations for surface water flow. Finite volume methods for the diffusion wave equation and the full shallow water equations are also developed to serve as a comparison to the kinematic wave. It is shown that the kinematic wave and the diffusion wave have comparable features. Even though the kinematic wave is not the closest representation of reality, high computational gains could be made by using the kinematic wave equation instead of the full shallow water equations. For a simple straight ditch, the kinematic wave has shown to be 16 times faster than the full shallow water equations. For a network, however, it is not as straightforward. When comparing a kinematic wave network with a network in SOBEK, a software program that uses the full shallow water equations, the latter still has lower computational times.

Over time, the width-averaged depth of estuaries changes due to a complex interaction of hydrodynamics and suspended sediment transport. In many estuaries one specic location with a suspended sediment concentration (SSC) higher than in the sea or in the upstream river, is found, which is called sediment
trapping. The location of the maximum SSC is called the estuary turbidity maximum (ETM). Understanding the dynamics is important to maintain a healthy ecosystem while making anthropogenic changes. To investigate such changes, a two-dimensional model is developed, considering the Ems estuary as a case study. The model equations consist of the width-averaged shallow water equations and a SSC equation. Assuming a morphodynamic equilibrium, these equations are solved mostly analytically by making a regular expansion of each physical variable in a relatively small parameter. Using this method, we are able to gain insight into the fundamental physical processes resulting in sediment trapping in an estuary by studying the influence of various forcings separately. One of the hydrodynamic forces is vertical mixing. This force has been assumed to be constant over time in previous studies [1]. In this thesis vertical mixing as a function that varies on the tidal timescale has been added to the model and is analysed. As a result of the salinity gradient in the estuary the mixing is stronger during flood and weaker during ebb. Using the model it is found that time variations in vertical mixing result in tidally averaged non-zero, and therefore contributing, transports. They cause a narrowing of the location where sediment is trapped. If vertical mixing is exactly maximal when the flood is maximal and minimal when ebb is maximal, the ETM shifts downstream. But if the vertical mixing is lagging the tidal stream, which is more plausible, the ETM will stay or shift upstream.