26.3.2 Additional kinetic energy change due to boundary effects

After cancelling the cosine, the third term in Equation (26.38) is written as

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...{\rho_{i.k.j}}^\lambda \delx(zt_{i,k,j})}^\phi \; \dxui \; \dyuj \; dhu_{i,k,j}$$ (26.42)

where zt_i,k,j is the depth from the ocean surface to the point within cell T_i,k,j. Using the latitudinal version of Equation (21.14) to re-arrange terms in the summation on ``jrow'' gives

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...ot \dyujm}^\phi \cdot \overline{\rho_{i.k.j}}^\lambda \delx(zt_{i,k,j})\; \dxui$$ (26.43)

and substitution from Equation (26.41) yields

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...\cdot \overline{\rho_{i,k,j}}^\lambda \cdot \delx(zt_{i,k,j}) \; \dxui \; \dytj$$ (26.44)

Again, using Equation (21.15) to re-arrange terms in the longitudinal summation on``i'' yields

 

$$\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{... ..._{i-1,k,j}\cdot \overline{\rho_{i-1.k.j}}^\lambda)\; zt_{i,k,j}\; \dxti\; \dytj$$ (26.45)

The fourth term in Equation (26.38) is

 

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...}}^\phi \; \dely(zt_{i,k,j})}^\lambda \; \dxui \; \csuj \; \dyuj \; dhu_{i,k,j}$$ (26.46)

A similar manipulation as given for the third term above reduces this fourth term to

 

$$\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{... ...y(adv\_vnt_{i,k,j-1}\overline{\rho_{i.k.j-1}}^\phi)\; zt_{i,k,j} \dxti \; \dytj$$ (26.47)

Combining Equations (26.43), (26.47) and (26.49) gives the total change in kinetic energy due to pressure terms as

 

$$\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{... ...t_{i,k,j-1}\overline{\rho_{i.k.j-1}}^\phi) \Bigr) \; zt_{i,k,j}\; \dxti\; \dytj$$

$$- adv\_vbt_{i,k-1,j}\; \overline{\rho_{i,k-1,j}}^z \; \dxti\; \cstj \; \dytj \; dhw_{i,k-1,j}$$ (26.48)