26.2.2.1 Example where density varies linearly with depth

If density varies linearly with depth, then

$$\delta_z(\rho_{i,k,j}) = constant$$ (26.24)

and density $\rho_{i,k,j}$ can be expanded in terms of density in the upper level $\rho_{i,k-1,j}$ as

$$\rho_{i,k,j} = \rho_{i,k-1,j} + \delta_z{\rho_{i,k-1,j}}\cdot dhw_{i,k-1,j} \;.$$ (26.25)

Using the above expansion to eliminate $\rho_{i,k,j}$ in Equation (26.24) results in the following expansions:

$$\delx(\rho_{i,k,j}) \delta_z (\rho_{i,k-1,j})\cdot \delx (zt_{i,k,j})$$ = (26.26)

$$\overline{\rho_{i,k,j}}^\lambda \rho_{i,k-1,j} + \delta_z (\rho_{i,k-1,j}) \cdot \overline{dhw_{i,k-1,j}}^\lambda \;.$$ = (26.27)

Substituting the above expansions into Equation (26.24) results in

$${\cal P}_x(k) = 0$$ (26.28)