3 A-grid vs C-grid

The most complete description of the approach taken in the DieCAST family of models is given by Sheng et al. (1998). This paper gives a systematic and thorough derivation of the basic numerics and the finite difference formulation on a C-grid. For convenience, we have reproduced the basic model formulation from Sheng et al. (1998) in Appendix A, which includes a discussion of both the A-grid and C-grid formulations. Below, we give a brief discussion of the differences between the C-grid and A-grid versions of the model. Fig. 1a shows the A-grid and C-grid configurations, and Fig. 1b shows the combined A-grid and C-grid used for the CANDIE A-grid model version described by this manual.

Both the C-grid and the A-grid versions of DieCAST are formulated in terms of a rectangular coordinate system in which the model domain is divided into regularly stacked boxes in the vertical. The ocean bottom is thus represented with a staircase approximation. For both model formulations, the temperature, salinity, and pressure are given as the average values over the volume of each box or model cell, and a control volume approach is used in which the average values of quantities within each cell are updated in accordance with the fluxes across the six cell faces. Differences between the two formulations occur in the representations of the horizontal velocity field. This is shown in Fig. 2 which depicts a three-dimensional CANDIE grid cell with the A-grid and C-grid points.

In the case of the C-grid, the velocity components are specified as average values on the faces of the cells: U is determined for the cell faces normal to the X-direction, V for cell faces normal to the Y-direction and W for cell faces normal to the Z-direction. For the A-grid, the vertical component of velocity is still specified on the top and bottom faces of the cell as in the C-grid, but the two horizontal velocity components are specified at the same locations as T, S, and P. (Often it is assumed that this position is at the cell center, but really the A-grid variables represent averages over the cell volume.)

Each of the above choices has its limitations. For the C-grid, there is a problem associated with the treatment of the Coriolis force because updating the X-component of velocity (U) requires knowledge of the Y-component (V) at the U point, and vice versa. Thus, one is forced to interpolate horizontal velocity components in order to deal with the Coriolis force on a C-grid. On the A-grid, no interpolations are required to determine the Coriolis force, but the conservation equations for heat, salt, momentum and tracers each require knowledge of the velocity components on the cell faces, i.e., on the C-grid. Thus interpolations of the horizontal velocity components are again required. This is the only real difference between the C-grid and A-grid formulations of DieCAST. The remainder of this communication deals only with the A-grid formulation.

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