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

CHAPTER 2. THEORY 11
for advection
�
∆x
∆tint < min ,
umax
∆y
�
vmax
(2.1.17)
where umax and vmax are the maximum horizontal advective velocities.
Advection in the internal and external mode of the model is discretised using
an explicit numerical scheme for the time differencing while diffusion in the
vertical is discretised with an implicit scheme.
In GETM, a macro time step ∆t is used for the internal mode while the
external mode uses a micro time step ∆tm. The latter is an integer fraction
M of the first one and is limited by means of Eq. (2.1.16). The macro
time step is limited by the maximum current speed in Eq. (2.1.17). The
organisation of the time stepping is shown in Fig. 2.1.1.
2.1.6 Spatial discretisation
The model equations are discretised on an Arakawa-C finite difference grid
(Arakawa and Lamb [1977]) which belongs to a class of staggered grids. In
a C-grid, quantities such as ζ and topographic height H are defined at the
centre of the grid while the zonal velocity component u is transposed half a
grid to the west of the centre and the meridional component v is displaced
half a grid to the south of the centre. The spatial coordinates x and y are
located at the corners of each horizontal grid cell and indexing is carried
out with i-indices in eastern, j-indices in northern and k-indices in upward
direction. Thus, each grid point is defined by a triple (i, j, k). In GETM,
the centre of a grid cell is referred to as the tracer point (T-point) because
here, all tracers such as temperature T , salinity S, the general tracers C i
and the density ρ are computed. The layout of the horizontal grid is shown
in detail in Fig. 2.1.2b. It should be noted that GETM is capable to run on
a curvilinear grid. Since all simulations were carried out using a Cartesian
rectangular grid with horizontal spatial increments ∆x and ∆y, this will not
be discussed here any further.
For vertical discretisation, the water column is divided into N non-intersecting
layers hk (k = 1, · · · , N) ranging from the bottom at z = −H(x, y) to the
surface at z = ζ(x, y, t). This is achieved by introducing N −1 internal levels
zk (k = 1, · · · , N − 1). The vertical location and the depth hk = zk − zk−1
of each surface layer depends on the horizontal position (x, y) and time t.
All physical quantities are computed on a vertically staggered Arakawa Cgrid
which consists of control volumes around the T-points. To resolve the
surface and bottom boundary layers, the vertical equations of the model are