DieCAST and CANDIE both use a finite difference scheme based on
Cartesian coordinates with unevenly-spaced Z-levels in the vertical.
The early version of CANDIE uses the Arakawa C grid for the spatial
discretization with state variables ** u,v,w** and

The C grid is widely used in the community [e.g. MICOM of Bleck
* et al*., 1992 and the MIT model of Marshall * et al*., 1997] because
of the ease with which the control volume approach can be implemented,
and because the horizontal pressure gradients in the momentum
equations and the flux divergences in both the momentum and continuity
equations can be calculated to the same numerical accuracy as the
other terms in the equations. The main weakness of the C grid, on the
other hand, is that the horizontal staggering of the ** u** and

In DieCAST, the Coriolis force is treated implicitly . The
approach suggested by Dietrich *et al*. [1987, 1990] uses a blend of A
and C grids, thus avoiding the computational burden mentioned above.
The key to this approach is to interpolate the trial velocity
components and to the ** p** points, and update
the advanced trial velocity components at the

Let , calculated using
(23)-(26)), represent the trial velocity components
at the ** p**-points. They can be calculated, for example, using
two-point averaging of the trial velocity components on the C grid
and by:

The indices ** i**,

These velocity components are then interpolated back to the ** u** and

Results show that the smoothing involved in interpolating to the ** p**
points and back again introduces numerical dissipation. For example,
if and

Two approaches have been suggested to reduce this numerical
dissipation. The standard version of DieCAST uses a fourth order
interpolation scheme to reduce the dissipation associated with each
pair of interpolations. Dietrich [1993] shows test cases for a doubly
periodic domain which demonstrate the utility of this approach for
regions which are removed from horizontal boundaries. The effective
filter weights for the fourth order interpolation scheme are **
(1,-18,63,164,63,-18,1)/256)**, corresponding to reduced but still
significant numerical dissipation. Figure 2 shows that the amplitude
reduction using the fourth order interpolation is reduced to about
20\% for waves with wavelengths of . For
comparison, we also plot the amplitude reduction corresponding
to biharmonic lateral dissipation with the non-dimensionalized diffusion coefficient (see Figure 2). Note. however, that
near horizontal boundaries the original version of DieCAST
uses a second order scheme, and the associated dissipation remains
substantial (see the discussion of the canyon test problem in the next
section). Also, this approach still results in dissipation which
increases as .

A second modification, which further reduces the dissipation, is to
interpolate only the * changes* in velocity at ** p** points back to
the staggered

represent the changes in the trial velocity components at ** p** points
due to the Coriolis and vertical diffusion terms. Interpolating
back to the

If this approach is used, then the dissipation associated with the
interpolations s reduced to zero for the special case and considered above. Implementation of this
approach in the standard DieCAST model can reduce numerical
dissipation significantly, particularly when small time steps are
required, but some dissipation remains. Further work is required to
improve the accuracy adjacent to boundaries. This issue warrants
further investigation as it may be critical for problems which are
strongly influenced by the boundary conditions [e.g., Haidvogel et
al., 1992; Jiang * et al*., 1995]. An example from the DieCAST model
which includes both of the above modifications will be discussed in
Section 6.

One further point should be mentioned regarding the treatment of the
Coriolis term in DieCAST. Although and
are updated using (42) and
(43) with , the surface pressure is computed
using (36) and the associated velocity corrections are
computed using (33) and (34) with ,
respectively, that is, as if the Coriolis term were being treated
explicitly. As noted by Dietrich * et al*.[1987], this leads to an error
of order . Sheng * et al*. (in preparation) give an
example where this error has significant effect even for ** F**
substantially less than 1.

In CANDIE we choose to treat the Coriolis force explicitly **(**
**)**, and use the standard four-point averaging of ** u (v)**
[e.g. Heaps, 1972] to determine appropriate estimates at the