22.3.2 Advective velocities for U cells

Another set of advective velocities are defined in a direction normal to U-cell faces. Refer to Figure 22.3b which illustrates a U-cell with all six advective velocities and their indices. Note that a U grid point is within the U-cell and there are four surrounding T points; one at each corner of the U-cell. A horizontal slice through the plane containing the grid points is depicted in Figure 22.3d showing the relation between indices in the central U-cell and four surrounding T-cells. To calculate advective velocities on faces of U cells, the averaging approach indicated by Dukowicz and Smith (1994) and subsequently described by Webb (1995) is used. Their formulae are:

$$adv\_veu_{i,k,j} \overline{adv\_vet_{i,k,j}}^{\lambda\phi}$$ = (22.18)

$$adv\_vnu_{i,k,j} \overline{adv\_vnt_{i,k,j}}^{\lambda\phi}$$ = (22.19)

$$adv\_vbu_{i,k,j} \overline{adv\_vbt_{i,k,j}}^{\lambda\phi}$$ = (22.20)

The above averaging works when the grid resolution is uniform. For MOM, a more general form is used to conserve volume within each U cell when resolution is non-uniform. Volume conservation is insured by taking a weighted average of T cell advective velocities within the plane of the cell face and a linear interpolation in a direction normal to the cell face. Refer to Figure 22.3c. Using the indicated grid distances, the resulting four point averaging operators are given by:

 

$$adv\_vnu_{i,k,j} \frac{1}{dyt_{jrow+1}\cdot dxu_i} \times$$ =

$$\biggl[ \biggl. (adv\_vnt_{i,k,j}\cdot duw_i + adv\_vnt_{i+1,k,j}\cdot due_i)\cdot dus_{jrow+1}$$

$$(adv\_vnt_{i,k,j+1}\cdot duw_i + adv\_vnt_{i+1,k,j+1}\cdot due_i)\cdot dun_{jrow} \biggl. \biggr]$$ + (22.21)

 

$$adv\_veu_{i,k,j} \frac{1}{\dyuj\cdot dxt_{i+1}}\cdot \times$$ =

$$\biggl[ \biggr. (adv\_vet_{i,k,j}\cdot dus_{jrow} + adv\_vet_{i,k,j+1}\cdot dun_{jrow})\cdot duw_{i+1}$$

$$(adv\_vet_{i+1,k,j}\cdot dus_{jrow} + adv\_vet_{i+1,k,j+1}\cdot dun_{jrow})\cdot due_i. \biggr. \biggr]$$ + (22.22)

The vertical velocity at the bottom face of U cells can be calculated by directly integrating the continuity equation

 

$$adv\_vbu_{i,k,j} \frac{1}{\csuj}\cdot\sum_{m=1}^{k} \biggl( \delx(adv\_veu_{i-1,m,j}) + \dely(adv\_vnu_{i,m,j-1})\biggr)\cdot dzt_m$$ = (22.23)

Equivalently, the vertical velocity at the bottom face of U cells can by calculated as an area preserving weighted average of the vertical velocities at the bottom face of four surrounding T cells.

 

$$adv\_vbu_{i,k,j} \frac{1}{\dyuj\cdot dxu_i\cdot \csuj} \times$$ =

$$\biggl[ \biggr. adv\_vbt_{i,k,j}\cdot dus_{jrow}\cdot duw_i\cdot \cstj$$

$$adv\_vbt_{i+1,k,j}\cdot dus_{jrow}\cdot due_i\cdot \cstj$$ +

$$adv\_vbt_{i,k,j+1}\cdot dun_{jrow}\cdot duw_i\cdot \cos\phi^T_{jrow+1}$$ +

$$adv\_vbt_{i+1,k,j+1}\cdot dun_{jrow}\cdot due_i \cdot \cstjp\biggr]$$ + (22.24)

This volume conserving averaging operation eliminates the problem of numerical de-coupling between advective velocities on the bottom of T cells and U cells in the presence of flow over topographic gradients. As discussed by Webb (1995), this de-coupling can introduce a large mis-match between the vertical velocity on T-cells and U-cells, with the U-cell velocity generally very noisy near topography. Implementations of vertical velocity in MOM 1 and previous versions of the GFDL ocean model suffered from such de-coupling and noise. It should be noted that the weighting factors dus, dun, due, and duw are always equal if resolution is uniform. In this case, the averaging used here reduces to that of Webb (1995).

Refer to Sections 22.9.5 and 22.8.7 for details on how advective velocities are used to compute the advective fluxes of momentum and tracers.