22.3.3.2 Discrete vertical velocity at the ocean bottom

As described in Chapter 18, the discretized ocean bottom is defined by land T-cells beneath the deepest ocean T-cells. The deepest ocean T-cells are given by T_i,kb,jwhere kb=kmt_i,jrow and

$$2 \le kb \le km \; .$$ (22.28)

Refer to Fig 22.4a which illustrates the ocean bottom using land T-cells (the land cells are drawn with solid lines). The bottom face of the deepest ocean T-cells^22.12 is marked with an ``x''. In total, all exposed faces of these land T-cells define the material surface across which no flow can pass (i.e. normal velocity component at the material surface is zero). To keep the figure simple, only one vertical column of T-cells extending from the ocean surface to the ocean bottom is illustrated (the ocean T-cells are drawn with dashed lines). Let the location of this vertical column be given by coordinate indices ``i,j'' where cell T_i,kb,j is the deepest ocean T-cell in the column and the bottom face of cell T_i,kb,j is marked with an ``x''.

Vertical velocity at the bottom face of any ocean T-cell can be found by vertically integrating as in Equaton (22.17) from the resting ocean surface z=0 to the bottom face of the cell in question. Vertical velocity on the bottom face of cell T_i,kb,j(marked by ``x'') is zero because bottom faces are horizontally flat and the normal velocity component on a material surface which is fixed in time is zero. That is, the model's bottom topography is piece-wise flat on the T-cell grid and so the vertical velocity must vanish there. Hence, the vertical component to the advective velocity on the T-cell vanishes at the bottom of the T-cell grid

$$adv\_vbt_{i,kb,j} adv\_vbt_{i,0,j} + \frac{1}{\cstj}\sum_{k=1}^{kb} (\delx (adv\_vet_{i-1,k,j}) + \dely (adv\_vnt_{i,k,j-1}))\cdot dzt_k$$ =

= 0. (22.29)

The vertical advection velocity at the ocean surface $adv\_vbt_{i,0,j}$ vanishes when using a rigid lid option, but is generally nonzero if using a free surface option. Similarly, vertical velocity is zero on all other bottom T-cell faces marked by ``x''.

Now consider the ``black dot'' in Figure 22.4a. This dot marks the bottom face of the deepest ocean U-cell having coordinate indices ``i,j''. The vertical column of U-cells extending from the ocean surface to the deepest ocean cell U_i,kbu,j is indicated in Fig 22.4b (the ocean U-cells are drawn with dashed lines). The depth index ``kbu'' is calculated as the minimum of the four surrounding T-cell depth indices:

$$kbu = kmu_{i,jrow} = min (kmt_{i,jrow},\; kmt_{i,jrow+1},\; kmt_{i+1,jrow},\; kmt_{i+1,jrow+1})$$ (22.30)

Importantly, note that the bottom face of the U-cell U_i,kbu,j is not a material surface. This fact is evident in Figure 22.4c which illustrates partial ocean U-cells existing below the bottom face of cell U_i,kbu,j (the U-cells are drawn with dashed lines). These ocean U-cells are only partially full grid cells since they are truncated by the material surfaces of the surrounding land T-cells.

Since the bottom face of cell U_i,kbu,j is not a material surface, it follows that the vertical velocity at the ``black dot'' in Figure 22.4a is not generally zero

$$adv\_vbu_{i,kbu,j} \ne 0.$$ (22.31)

Instead, using Equation (22.17) to vertically integrate from the surface downwards to the ``black dot'' in Figure 22.4b yields

 

$$adv\_vbu_{i,kbu,j} \overline{adv\_vbt_{i,0,j}}^{\lambda\phi} + \frac{1}{\csuj}\sum_{... ...iggl( \delx(adv\_veu_{i-1,m,j}) + \dely(adv\_vnu_{i,m,j-1}) \biggr) \cdot dzt_m$$ =

The above considerations always started from the ocean surface. Volume conservation ensures that one can equivalently start from the bottom of the U-cells and integrate upwards. For this purpose, it is necessary to define the deepest partially full cell U_i,kmax,j as shown in Figure 22.4c. The depth index ``kmax'' for this cell is calculated as the maximum of the four surrounding T-cell depth indices

$$kmax = max (kmt_{i,jrow},\; kmt_{i,jrow+1},\; kmt_{i+1,jrow},\; kmt_{i+1,jrow+1})$$ (22.32)

The vertical advective velocity $adv\_vbu_{i,kmax,j}$ vanishes since it represents the vertical velocity at the flat material surface which is the bottom of a U-cell column. Integrating Equation (22.17) upwards from the bottom of the U-cell column at kmax to the ``black dot `` at kbu yields

 

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

Volume conservation ensures that this result is identical to equation (22.32). The question now arises as to why Equation (22.34) does not have the appearance of a discretized version of Equation (22.27)? The answer is that Equation (22.27) should be discretized at the location of the ``open circle'' in Figure 22.4 rather than at the ``black dot'' itself. The reason is that location of the ``open circle'' is determined as the maximum distance above the bottom face of cell U<sub>i,kmax,j</sub> for which both the northern and western cell faces are material surfaces. For purposes of completeness, the discrete form of Equation (22.27) can be written for the location of the ``open circle'' through the following considerations. Let ``ko'' be the vertical index of the U-cell whose top face contains the ``open circle''. Also, let the height of the ``open circle'' from the base of cell U_i,kmax,j be given by

H^b = zw(kmax) - zw(ko). (22.33)

Integrating upwards from the bottom face of level ``kmax'' to ``ko'' yields

 

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

Defining the average advective velocity normal to cell faces as

$$\frac{1}{H^b} \sum_{m=ko}^{kmax} adv\_veu_{i,m,j} \; dzt_m$$ U^b_i,j = (22.34)

$$\frac{1}{H^b} \sum_{m=ko}^{kmax} adv\_vnu_{i,m,j} \; dzt_m$$ V^b_i,j = (22.35)

and substituting into Equation (22.36) yields

 

$$adv\_vbu_{i,ko,j} -\frac{1}{\csuj} \biggl(H^b \delx(U^b_{i-1,j}) + H^b\dely(V^b_{i,j-1})\biggr)$$ = (22.36)

Since the north and west face of the cells summed over in Equation (22.36) are material surfaces

U^b_i-1,j = 0 (22.37)

V^b_i,j = 0 (22.38)

which leads to the finite difference counterpart of Equation (22.27).

 

$$adv\_vbu_{i,ko,j} \frac{1}{\csuj} \biggl(-U^b_{i,j} \frac{H^b}{dxu_i} + V^b_{i,j-1}\frac{H^b}{dyu_{jrow}}\biggr)$$ =

$$\frac{1}{\csuj} \biggl( U^b_{i,j}\cdot \delx(H^b) + V^b_{i,j-1} \cdot \dely (H^b) \biggr).$$ = (22.39)