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

CHAPTER 2. THEORY 23
2.4.1 Modelling advection
The movement of a Lagrangian tracer due to advection is described by the
ordinary differential equation
d�xT
dt = �vT (�xT (t)) (2.4.1)
where �xT (t) = (xT (t), yT (t), zT (t)) is the particle position and �vT (�xT (t)) =
(uT (�xT (t)), vT (�xT (t)), wT (�xT (t)) is the corresponding velocity vector. To calculate
the movement of particles one can either apply a standard numerical
integration scheme to the equation of motion (2.4.1) (e.g. the Runge-Kutta
method) or find its analytical solution. The analytical solution is not difficult
to obtain and has the advantage that the results based on it are very
accurate.
2.4.1.1 Interpolation scheme
An interpolation scheme is necessary to transform the velocity components
calculated by GETM from the Eulerian grid to the position �xT of a tracer.
Maier-Reimer [1973] used bilinear interpolation on an Arakawa C-grid considering
four points for each velocity component. As two of these four points
are always located outside the grid box containing the tracer particle, they
are inappropriate to reflect the flow within this grid cell. Furthermore in
GETM, the discretised equation of continuity only takes into account the
velocity points defined on the boundaries of each grid box to guarantee
conservation of mass. Thus, applying bilinear interpolation would clearly
violate the equation of continuity. The linear interpolation applied by Duwe
[1988] only uses the six velocity points defined on the boundaries of each
grid box and thus conserves mass. As a consequence the velocity components
within a grid box are solely functions of their corresponding direction
(i.e. ∂u/∂y = ∂u/∂z = 0). This has two advantages:
1. the interpolation scheme and hence the Lagrangian tracer model is
consistent with GETM (mass conserving) which computes the velocity
and
2. the acceleration of a particle on its path is easily calculated.