26.2.2 Pressure gradient

When vertical thickness of cells is a function of $\lambda$, $\phi$, and depth, extra terms are needed to construct the pressure gradient. Refer to Pacanowski and Gnanadesikan for a discussion of the pressure gradient. Another way to look at it is the following: With the aid of Leibnitz's Rule (Equation (4.65), the zonal and meridional pressure gradients are written as

  

$$-\frac{grav}{\rho_\circ a\cdot \cos\phi} \int_{z}^{0} \frac{\partial \rho}{\partial \lambda} dz^\prime -\frac{grav}{\rho_\circ a\cdot \cos\phi} \Bigl( \frac{\partial}{\... ...o \; dz^\prime \bigr) \;\; -\;\; \rho\frac{\partial z}{\partial \lambda} \Bigr)$$ = (26.14)

$$-\frac{grav}{\rho_\circ a} \int_{z}^{0} \frac{\partial \rho}{\partial \phi} dz^\prime -\frac{grav}{\rho_\circ a} \Bigl( \frac{\partial}{\partial \phi} ... ... \rho\; dz^\prime \bigr) \;\; -\;\; \rho\frac{\partial z}{\partial \phi} \Bigr)$$ = (26.15)

where $\rho$ is density and the acceleration due to gravity is grav=980.6 cm/sec². Let ${\cal P}_x(k)$ symbolize the the discrete form for the right hand side of the zonal pressure gradient in Equation (26.14) and ${\cal P}_y(k)$ symbolize the discrete form for the right hand side of the meridional pressure gradient in Equation (26.15). The discrete forms are

  

$${\cal P}_x(k) -\frac{1}{\rho_\circ \csuj} \Bigl( \overline{\delx(p_{i,k,j}) \;\; - grav\cdot \overline{\rho_{i,k,j}}^\lambda \cdot \delx(zt_{i,k,j})}^\phi \Bigr)$$ = (26.16)

$${\cal P}_y(k) -\frac{1}{\rho_\circ} \Bigl( \overline{\dely(p_{i,k,j}) \;\; - grav\cdot \overline{\rho_{i,k,j}}^\phi \cdot \dely(zt_{i,k,j})}^\lambda \Bigr)$$ = (26.17)

where the $\overline{( )}^\phi$ in ${\cal P}_x(k)$ and the $\overline{( )}^\lambda$ in ${\cal P}_y(k)$ are used to move the results onto a U-cells. The pressure p_i,k,j is defined on T-cells and is calculated by vertically integrating the density from the surface to the depth of the T-grid point at level ``k''.

$$p_{i,k,j} = grav\cdot \rho_{i,1,j}\cdot dhw_{i,0,j} \;+ \; gr... ...ot \sum_{m=2}^{k} \overline{\rho_{i,m-1,j}}^z \; dhw_{i,m-1,j}$$ (26.18)

where dhw_i,k,j is the vertical distance between a T-grid point at level ``k'' and its neightbor at ``k+1'' for any coordinate index ``i,j''. In the first term, dhw<sub>i,0,j</sub> is the distance between the ocean surface and the grid point within level k=1. Partial cells are only allowed for levels k>1 and so dhw<sub>i,0,j</sub> is constant for all ``i,j''. Using Equation (26.18), the term $\delx(p_{i,k,j})$ in ${\cal P}_x(k)$ can be written as

$$\delx(p_{i,k,j}) = grav\cdot dhw_{i,0,j}\cdot \delx(\rho_{i,1... ..._{m=2}^{k} \overline{\rho_{i,m-1,j}}^z \; dhw_{i,m-1,j} \bigr)$$ (26.19)

Refer to Fig 26.3 which illustrates two vertical columns of T-cells with one partial bottom cell at level ``k''. Consider the vertically stratified case where density is a function of depth only. This condition implies

  

$$\delx(\rho_{i,k^\prime,j}) \dely(\rho_{i,k^\prime,j}) = 0 \;\;\;\; \;\; for\;\; k^\prime < k$$ = (26.20)

$$\overline{\rho_{i,k^\prime,j}}^\lambda \overline{\rho_{i,k^\prime,j}}^\phi = \rho_{i,k^\prime,j} \;\;\;\; for\;\; k^\prime < k$$ = (26.21)

which reduces Equation (26.19) to

$$\delx(p_{i,k,j}) grav\cdot \delx \bigl( \overline{\rho_{i,k-1,j}}^z \; dhw_{i,k-1,j} \bigr)$$ = (26.22)

Distributing the derivative and reducing further yields

$$\delx(p_{i,k,j}) grav\cdot \Bigl( \overline{\rho_{i,k-1,j}}^{z\lambda}\cdot \delx ... ...erline{dhw_{i,k-1,j}}^{\lambda}\cdot \delx (\overline{\rho_{i,k-1,j}}^z) \Bigr)$$ =

$$\frac{grav}{2}\cdot \Bigl( (\overline{\rho_{i,k-1,j}}^\lambda + \... ...k,j}) \;+\; \overline{dhw_{i,k-1,j}}^{\lambda}\cdot \delx (\rho_{i,k,j}) \Bigr)$$ = (26.23)

Substituting the above form for $\delx(p_{i,k,j})$ into Equation (26.16) yields

 

$${\cal P}_x(k) = -\frac{grav}{2 \; \rho_\circ \csuj} \Bigl(\overli... ...{i,k,j}) - \overline{\rho_{i,k,j}}^\lambda \cdot \delx(zt_{i,k,j})}^\phi \Bigr)$$

In general, density can be divided into two pieces: one that varies linearly with depth and another that varies non-linearly with depth. The non-linear part will lead to non-zero pressure gradients in the vicinity of partial cells. The linear portion of the density variation with depth goes to zero.