26.2 Discrete Equations

The essential point is to note that part of the northern, southern, eastern, or western face of a grid cell may be partially land and this must be taken into account when fluxing information across cell faces. The discrete equations in Section 21.3 must be altered to account for vertical cell thickness which is a function of $\lambda$, $\phi$, and depth. In general, vertical cell thickness dzt_k is replaced by dht_i,k,j which is the vertical thickness of T-cells and dhu_i,k,j which is the vertical thickness of U-cells. The thickness of T-cells dht_i,k,j is determined when bottom topography is discretized. Although ocean cells may be partial, all land cells are considered as full-cells. Since material surfaces are defined by faces of T-cells, it follows that

$$dhu_{i,k,j} = min \; (dht_{i,k,j},\; dht_{i+1,k,j},\; dht_{i,k,j+1},\; dht_{i+1,k,j+1} )$$ (26.1)

Refer Fig 26.4 depicts the relationship between partial bottom T-cells and U-cells.