26.3.1 Changes in Kinetic energy due to partial bottom cells

When vertical thickness of cells is a function of $\lambda$, $\phi$, and depth, extra terms are needed to construct the change in kinetic energy due to pressure forces. Multiplying the zonal pressure gradient given in Equation (26.14) by u and the meridional pressure gradient given in Equation (26.15) by v and integrating over the entire ocean volume yields the change in kinetic energy. The finite difference equivalent of this expression is

 

$$-\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=... ...,j}}^\phi)}{\csuj} + u_{i,k,j,2,\tau}\cdot \dely( \overline{p_{i,k,j}}^\lambda)$$

$$u_{i,k,j,1,\tau} \frac{grav \; \overline{\overline{\rho_{i.k.j}}^... ...verline{\overline{\rho_{i.k.j}}^\phi \dely(zt_{i,k,j})}^\lambda \biggl. \biggr)$$ - (26.36)

where the velocity cell volume element is

$$dvol_{i,k,j} = \dxui \; \csuj \; \dyuj \; dhu_{i,k,j} \; .$$ (26.37)

Note that there are four terms in Equation (26.38). The first two are similar to Equation (A.89) except that dzt_k in the U-cell volume element has been replaced by dhu_i,k,j which is the vertical thickness of a U-cell as given by Equation (26.1).

The reduction of Equation (26.38) is similar to the one given for Equation (A.89) except for differences needed to account for spatial dependence of dhu_i,k,j. In Equation (A.91), dzt_k must be removed and dhu_i,k,j is placed inside both derivatives with subscripts matching those on the velocity. In Equation (A.92), dzt_k is eliminated and dhu_i,k,j is placed inside both averages with subscripts matching those on the velocity. The horizontal advective velocites become advective transports defined as

  

$$adv\_vnt_{i,k,j} \frac{\overline{u_{i-1,k,j,2,\tau}\cdot\dxuim\cdot dhu_{i-1,k,j}}^\lambda}{\dxti}\; \csuj$$ = (26.38)

$$adv\_vet_{i,k,j} \frac{\overline{u_{i,k,j-1,1,\tau}\cdot\dyujm\cdot dhu_{i,k,j-1}}^\phi}{\dytj}$$ = (26.39)

but the vertical advective velocities remain as velocities and the continuity equation becomes

$$\frac{adv\_vet_{i,k,j}-adv\_vet_{i-1,k,j}}{\dxti\cdot\cstj} +... ...}{\dytj\cdot\cstj} + adv\_vbt_{i,k-1,j} - adv\_vbt_{i,k,j} = 0$$ (26.40)

which leads to a change in kinetic energy given by

 

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...j}\cdot \overline{\rho_{i,k-1,j}}^z \; \dxti \; \cstj \; \dytj \; dhw_{i,k-1,j}$$ (26.41)

Note that thickness factor dzw_k-1 in Equation (A.99) has been replaced by its spatially dependent counterpart dhw_i,k-1,j which is the vertical distance between grid points within cells T_i,k,j and T_i,k-1,j. The change in kinetic energy due to the third and fourth terms is given next.