26.3.3 Changes in Potential energy due to partial bottom cells

When vertical thickness of bottom cells at a given depth is a function of $\lambda$ and $\phi$ then extra terms are needed to construct the change in potential energy due to advection. As a consequence of absorbing a vertical grid cell thickness factor into the advective transports of Equations (26.41) and (26.40) , the advection operator for T-cells takes the form

 

$${\cal L}^T(\alpha_{i,k,j}) \frac{ \delta_\lambda (adv\_vet_{i-1,k,j} \cdot \overline{\alpha_... ..._vnt_{i,k,j-1} \cdot \overline{\alpha_{i,k,j-1}}^\phi) } {dht_{i,k,j} \; \cstj}$$ =

$$\delta_z (adv\_vbt_{i,k-1,j}\cdot \overline{\alpha_{i-1,k,j}}^\lambda)$$ + (26.49)

and Equation (A.100) is re-written as

 

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2}^{imt-1} zt_{i,k,j} \; (\dxti \; \cstj \; \dytj \; dht_{i,k,j}) \times$$

$$\biggl( \frac{ \delta_\lambda(adv\_vet_{i-1,k,j}\cdot \overline{\... ...vnt_{i,k,j-1}\cdot \overline{\rho_{i,k,j-1}}^\phi) } {dht_{i,k,j}\cstj} \biggr.$$

$$\biggl. \delta_z (adv\_vbt_{i,k-1,j} \cdot \overline{\rho_{i,k-1,j}}^z) \biggr)$$ + (26.50)

where zt_k has been replaced by zt_i,k,j and dzt_khas been replaced by dht_i,k,j. Note the extra factor of dht_i,k,j in the denominator. Because of these changes, the summation of the first two terms over a horizontal surface will not vanish. After some cancellations, the work done by horizontal advection of density is

 

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2}^{imt-1} zt_{i,k,j} \; \dxti \; \dytj \times$$

$$\biggl( \delta_\lambda(adv\_vet_{i-1,k,j}\cdot \overline{\rho_{i-... ... + \delta_\phi(adv\_vnt_{i,k,j-1}\cdot \overline{\rho_{i,k,j-1}}^\phi) \biggr).$$ (26.51)

which exactly compensates for the loss of kinetic energy due to the first two terms in Equation (26.50). The third term in Equation (26.52) may be re-arranged in the vertical summation to yield

 

$$\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{... ...cdot \overline{\rho_{i,k-1,j}}^z \cdot dhw_{i,k-1,j} \; \dxti \; \cstj \; \dytj$$ (26.52)

which is the work due to buoyancy and exactly compensates for the remaining change in kinetic energy in Equation (26.50).

  

Figure 26.1: Comparing bottom topography for a 1x horizontal resolution a) Using full cells. b) Using partial bottom cells

  

Figure 26.2: Comparing bottom topography for a 2x horizontal resolution a) Using full cells. b) Using partial bottom cells

  

Figure 26.3: Two vertical columns of T-cells with one partial bottom cell

  

Figure 26.4: a) Horizontal arrangement of T-cells and U-cells. b) Two partial bottom T-cells with a partial bottom U-cell. In general, the partial bottom U-cell thickness is determined by the minimum thickness of four surrounding partial bottom T-cells.