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5 3D Level Modelling
Fig. 5.1 The three-dimensional Arakawa C-grid
5.2.2 Treatment of the Advection Terms
Using the product rule of differentiation, the advection terms for a property B can
be written as:
u
∂ B
∂ x
+ v
∂ B
∂ y
+ w
∂ B
∂z
=
∂(u B)
∂ x
+
∂(v B)
∂ y
+
∂(w B)
∂z
− B
∂u
∂ x
+
∂v
∂ y
+
∂w
∂z
(5.7)
The last term involves the continuity equation (Eq. 5.4) and therefore should
vanish in theory. Nevertheless, this term is retained in the numerical advection
scheme for elimination of small round-off errors that otherwise could accumulate
during a simulation. A control-volume approach is used to calculate advection of
scalars and the nonlinear terms in the momentum equations (Fig. 5.2). With this
approach, fluxes of a property into this control volume are calculated from fluxes
through its east/west, north/south and top/bottom faces, respectively. Again, individual fluxes are computed by means of a Total-Variation-Diminishing (TVD) advection scheme using the Superbee limiter (see Sect. 3.6). Implementation for the newly
introduced y-direction should be straight forward.