Three-dimensional Lagrangian Tracer Modelling in Wadden Sea ...

CHAPTER 2. THEORY 21
Eulerian grid (TUDELM) to model transport in the North Sea. A major
disadvantage of this method is the progressive deformation of the grid which
in extreme cases leads to numerical instability (e.g. a two-dimensional grid
cell degenerates to a straight line). Another approach by Maier-Reimer
and Sündermann [1982] proposed to interpret the water body not as a continuum
but as finite set of water particles with fixed physical properties
(tracer method). The physical state of an Eulerian variable on the grid
points is defined as the mean value of all particles situated in a grid cell.
This approach can be generalised to include all physical properties such as
velocity, density, temperature and dissolved substances. In the case of a
dissolved substance, it is represented by a discrete number of tracer particles
which is proportional to the concentration C in the grid cell. Instead
of modelling advection and diffusion in terms of concentration, the motions
of the tracer particles due to the mean current field and turbulent velocity
fluctuations and hereby their paths (trajectories) are calculated. Compared
to the established finite techniques, tracer methods have the great profit of
almost completely avoiding unwanted numerical diffusion and thereby assure
more accurate results. Maier-Reimer [1973] pointed out the criteria a
numerical method for the advection-diffusion equation has to satisfy in order
to reproduce the dynamics of the physical property which is transported:
1. Positivity: this criterion ensures that positive properties stay positive.
2. Conservation of local mass: in absence of external sources and sinks,
the total quantity of a property must be constant with time
� � �
∂
C(x, y, z, t) dxdydz = 0. (2.3.1)
∂t
V
3. No numerical diffusion: in the case where turbulent diffusion is absent,
the numerical method must describe transport as it occurs solely due
to advection.
In general, it is difficult to satisfy all three criteria with a numerical scheme in
Eulerian coordinates. See, however, Pietrzak [1998] for a method of incorporating
monotone high-order advection schemes with low numerical diffusion
in three-dimensional models. This method has also been applied by Burchard
and Bolding [2002] for GETM. Nevertheless, particle methods have
benefits which can be summarised (Dimou and Adams [1993]) as follows:
1. Sources are more easily represented in a particle tracking model, whereas
concentration models have difficulty resolving concentration fields whose
spatial extent is small compared to that of discretisation.