Random-walk models are a versatile tool for modelling dispersion of both passive and active tracers in turbulent ow. The physical and mathematical foundations of stochastic Lagrangian models of turbulent di usion have become more and more solid over the years. An important aspect of these types of models that has not received much attention is the behaviour of the particles near boundaries. Often, a simple stochastic, numerical scheme is used. Because turbulent mixing in the vertical direction is much more complicated than in the two horizontal directions, it is in the vertical direction that a simple numerical scheme, such as the Euler scheme, may cause problems.
In this paper our main goal is the development of an e cient 3D particle transport model that can be used in stratiΓΏed ow. For this type of situation the vertical direction is of special interest. First, a closer look is taken at some considerations that should be regarded when choosing a numerical scheme. SpeciΓΏcally schemes are investigated that can be used in the vertical direction, where the di usion coe cient is varying in that direction. Experiments are performed regarding the accuracy of di erent numerical schemes in various situations. The behaviour of the particles near an impermeable layer interface is investigated. The stochastic Heun and Runge-Kutta schemes turn out to be very attractive for this type of model.
For the simulation of the transport of various physical quantities, such as salinity, heat, silt, oxygen, or bacteria, di erent types of models are available. In this case we will take a closer look at the modelling of the transport of pollutants from point sources (either instantaneous or continuous transport). For this purpose a 3D particle transport model has been developed that is especially suited for stratiΓΏed situations such as can be found in estuaries. The main idea is to use a simple numerical scheme for the horizontal directions and a higher-order method for the vertical direction. The results play an important role in making speciΓΏc choices for this type of particle transport model.