Bottom topography is discretized to model levels as described in Chapter 18. Fig 26.1a is an example of a typical section of bottom topography compared with the resulting discretized bottom. The solid line represents the ``true'' bottom which the discretization is trying to capture. Ocean T-cells are indicated with grid points and land T-cells are shaded. The same bottom discretized with partial bottom cells is indicated in Fig 26.1b where the bottom T-cell in each column is allowed to be partially filled with land. Note that the topography is significantly more accurately approximated when partial cells are used. Most of the change in regions of steep slopes is captured by changes in the number of levels. Even so, partial cells give a better estimation of the true slope in these regions. In general, T-cell (and U-cell) thickness becomes a function of latitude, longitude and depth when partial cells are used. Note how the grid points follow the topography in Fig 26.1b. In effect, a one level ``sigma coordinate'' has been added by the partial cells.
Refer to Fig 26.2 which indicates what happens when horizontal resolution in Fig 26.1 is doubled but vertical resolution remains unchanged. Comparing Fig 26.1a with Fig 26.2a indicates that discretized bottom topography using full cells does not significantly improve when horizontal resolution is increased. In fact, it remains about the same. However, comparing Fig 26.1b with Fig 26.2b, the improvement using partial cells is significant.
In addition to ocean volume and f/H being more accurately represented by partial-cells, topographic wave speeds are also more accurate and therefore so is the dispersion relation for topographic waves as discusssed in Pacanowski and Gnanadesikan (1997). The disadvantage of partial-cells is that the integration time is 10 to 15% longer than for full-cells for the same number of vertical levels. However, to achieve the same accuracy, partial-cells are significantly more computationally efficient than using full-cells and increasing the number of vertical levels26.1. There is also a pressure gradient error with partial-cells but this is tiny as discussed in Pacanowski and Gnanadesikan (1997). In the interior of the ocean, equations are second order accurate (Treguier et al., 1996) but reduce to first order at bottom boundaries because the thickness of partial-cells breaks the analytical stretching of grid cell thickness in the vertical. Although equations drop from second order in the interior to first order for non full-cells at the bottom, a leading order error in the position of the topography has been corrected. Globally, the solution remains second order accurate.