41.2.5 Work done by Buoyancy

In Section A.1.3, it was shown in the continuous equations that the net change in kinetic energy due to pressure forces equals buoyancy effects in the absence of sources and sinks. In what follows, the discrete form of the net change in potential energy is arrived at by summing the horizontal and vertical advection of tracers multiplied by $grav\cdot zt_{k}/\rho_\circ$ over the entire ocean volume. If it is assumed that density $\rho$ is a linear function of tracers (temperature and salinity) then the net change in potential energy is given by

 

$$-\frac{grav}{\rho_\circ} \sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum... ...ta_\phi(adv\_vnt_{i,k,j-1}\cdot \overline{\rho_{i,k,j-1}}^\phi) \right] \biggr.$$

$$\biggl. \delta_z (adv\_vbt_{i,k-1,j} \cdot \overline{\rho_{i,k-1,j}}^z) \biggr) \; \dxti \; \cstj \; \dytj \; dzt_k$$ + (41.98)

Noting that $adv\_vet_{i,k,j}$ and $adv\_vnt_{i,k,j}$ are zero on land boundaries and summing in the horizontal over all cells for any level k eliminates the horizontal advection terms reducing the kernal to

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...{i,k-1,j}\cdot \overline{\rho_{i,k-1,j}}^z) \; \dxti \; \cstj \; \dytj \; dzt_k$$ (41.99)

Using the vertical equivalent of Equation (21.15) to re-arrange the summation on ``k'' in the vertical (for a rigid lid only) yields

 

$$\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{... ...,j}\cdot \overline{\rho_{i,k-1,j}}^z \cdot dzw_{k-1} \; \dxti \; \cstj \; \dytj$$ (41.100)

which should be compared to Equation (A.99) to show that the change in potential energy is exactly compensated for by the change in kinetic energy from horizontal pressure terms.