16.2.3 Non-uniform resolution

Within a region of constant resolution, all T and U grid points are located at the centers of their respective cells. When resolution is non-uniform, this is not the case. Within MOM 2, there are two methods to discretize non-uniform resolution onto T cells and U cells. Based on Equations (16.1) and (16.2) in section 16.1, and the averaging operator given by

$$\overline{(\xi)}^m = \frac{(\xi)_m +(\xi)_{m+1}}{2},$$ (16.4)

the two methods are

1.

$\Delta^T_m = \frac{\Delta^U_\alpha + \Delta^U_\beta}{2} - \frac{\Delta^U_\alpha -\Delta^U_\beta}{2}\cos(\pi\frac{m-0.5}{N})$; $\;\;\;\;\Delta^U_m = \overline{\Delta^T_m}^m$

2.

$\Delta^U_m = \frac{\Delta^U_\alpha + \Delta^U_\beta}{2} - \frac{\Delta^U_\alpha -\Delta^U_\beta}{2}\cos(\pi\frac{m}{N})$; $\;\;\;\;\Delta^T_m=\overline{\Delta^U_m}^m$

where $\Delta^U_\alpha$ and $\Delta^U_\beta$ are resolution of U cells at the bounding surfaces $\alpha$ and $\beta$, N is the number of cells given by Equation (16.1) in section 16.1, and the subscript m refers to longitude index i or latitude index j. These methods can also be applied to cells in the vertical, even though T and U cells are not staggered vertically. Refer to Figure 16.5 and assume phantom Wcells (of thickness dzw_k) staggered vertically such that the Wgrid point within cell W_k lies at the bottom of cell T_k and the T grid point within cell T_k lies at the top of cell W_k. Now replace U by W in the expressions for both methods given above.

The motivation for method 2 is first to notice that on a non-uniform grid, advective velocities are a weighted average of velocities but the denominator is not the sum of the weights as indicated in Section 22.3. This form of the advective velocities is implied by energy conservation arguments as given in Section A.2.4. Secondly, the average of the quantity being advected is not defined coincident with the advecting velocity. Redefining the average operator in Equation (16.4) differently results in second moments not being conserved as indicated in Chapter A. Method 2 remedies both problems by simply redefining the location of grid points within grid cells. All equations remain the same. Both method 1 and 2 conserve second moments.

In method 1, U cell size is the average of adjacent T cell sizes. This means that T points are always centered within T cells, but U points are off center when the grid is non-uniform. This was the method used in model versions prior to MOM 2. In method 2, the construction is the other way around: T cell size is the average of adjacent U cell sizes. Accordingly, U points are always centered within U cells but T points are off center when the resolution is non-uniform. It should be noted that MOM 2 allows both methods. Although the default grid construction is by method 2, enabling option centered_t^16.2 when compiling will result in grid construction by method 1.