The topic of this thesis project is to use the particle-tracking framework to study salt intrusion.
This project was proposed by Deltares and is a small part of the SALTISolutions project, that attempts to build a digital twin model of the Rhine-Meuse Delta. From a mathematical perspective, the particle-tracking framework is inspired by the theoretical connection between partial differential equations that describes substance transport and stochastic differential equations (SDEs). The idea is to use replace the continuous substance distribution in the fluid by many particles. After tracking the particles, the consequent substance distribution can be approximated by the particles' distribution at certain time instant. There are two major achievement of this project. Firstly, different implementation of non-penetrating and flux boundary conditions for the particle-tracking framework are tested and compared. The recursion algorithm is preferred for the previous and the ``Neumann-wise'' is recommended for the latter. Secondly, the convergence order of SDE numerical schemes and their influences on the simulation of the vertical transport physics are examined by studying the one-dimensional pure diffusion test case. We see that most simple Euler scheme contaminates the stratification completely in a theoretical setting even using a small timestep, indicating that it might be worthwhile to switch to high-order schemes whenever possible. However, even if one can implement other higher order numerical scheme correctly and efficiently, which is not easy, it might be equivalently important to have enough particles and good interpolation of velocity and diffusion field in the computation domain, because they may introduces error comparable to the diffusion error. Since salt transport is already resolved by the Delft3D-FM model, it is of nonsense to use the particle-tracking framework to study transport. The one-dimensional pure diffusion test case is mainly used for verification of the method. Actually, the particle-tracking framework should be used in combination of the diagnostics timescales and tracer methods. This often requires calculation of the statistics of the particle ensemble. It will be interesting to conduct such analysis on a realistic three-dimensional test case, and this could be the field of future research.