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.

The growing amount of plastic pollution in the oceans is worldwide a huge problem. There is an increasing amount of projects that try to clean up this pollution. To remove the pollution efficiently, it is useful to know how plastic litter moves, degrades and sinks. Models are being developed that can describe what happens with plastic in the ocean.

To model the behaviour of plastic in the ocean a lot of parameters need to be taken into account. The transport of plastic particles is largely determined by processes like the wind and the current but their movement also has a random character due to dispersion. Due to this random movement, the future position of a particle can be described by a probability distribution. To find that probability distribution, the Kolmogorov forward equation (or Fokker-Planck equation), in this context the same as the advection-diffusion equation, needs to be solved.

It is also possible that the plastic particles sink and settle on the bottom of the ocean. One goal of this project was to incorporate the settling of particles as an extra term into the advection-diffusion equation.

Forward models are used to predict where particles will go to. It can also be useful to have reverse time models that can describe where particles had their origin. To make reverse time models, it is necessary to solve a reverse time advection-diffusion equation. The main goal of this project was to derive and solve the reverse time advection-diffusion equation that includes the settling of particles. This is done by finding the adjoint of the Kolmogorov forward equation and the settling term.