41.2.4 Work done by pressure terms

In Section A.1.3, it was shown in the continuous equations that the net change in kinetic energy due to pressure forces equals the net change in energy due to buoyancy. The discrete counterpart of this result is given below using the definition of variables, indices and the relation between jrow and j as described in Section 14.2.1. In this terminology of MOM, the change in kinetic energy due to pressure forces summed over all ocean U cells is given by

 

$$-\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=... ...j,2,\tau}\cdot\delta_\phi(\overline{p_{i,k,j}}^\lambda) \right) \; dvol_{i,k,j}$$

where the U-cell volume element dvol_i,k,j is

$$dvol_{i,k,j} = \dxui\;\csuj\;\dyuj\; dzt_k$$ (41.88)

and pressure p is defined on T cells. Applying Equation (21.15) to the ``i'' summation for the first term in Equation (A.89) and similarly to the ``jrow'' summation for the second term yields

 

$$\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2... ...\lambda(u_{i-1,k,j,1,\tau})\; \overline{p_{i,k,j}}^\phi\; \dxti\; \dyuj\; dzt_k$$

$$\delta_\phi(u_{i,k,j-1,2,\tau}\; \csujm )\; \overline{p_{i,k,j}}^\lambda\; \dxui\; \dytj\; dzt_k$$ + (41.89)

Applying Equation (21.14) to the ``jrow'' summation for the first term in Equation (A.91) and to the ``i'' summation for the second term yields

 

$$\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2... ...{\delta_\lambda(u_{i-1,k,j-1,1,\tau}) \; \dyujm}^\phi \; \dxti \; dzt_k \right.$$

$$\left. \overline{\delta_\phi(u_{i-1,k,j-1,2,\tau}\; \csujm ) \;\dxuim}^\lambda \; \dytj \; dzt_k \right) \; p_{i,k,j}$$ + (41.90)

Defining the advective velocities on the eastern and northern face of a T-cell as

  

$$adv\_vet_{i,k,j} \frac{\overline{u_{i,k,j-1,1,\tau}\cdot\dyujm}^\phi}{\dytj}$$ = (41.91)

$$adv\_vnt_{i,k,j} \frac{\overline{u_{i-1,k,j,2,\tau}\cdot\dxuim}^\lambda}{\dxti}\csuj$$ = (41.92)

and substituting into Equation (A.92) yields

 

$$\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2... ...ft( \delta_\lambda(adv\_vet_{i-1,k,j,1,\tau}) \;\dytj \; \dxti \; dzt_k \right.$$

$$\delta_\phi(adv\_vnt_{i,k,j-1,2,\tau}) \;\dxti \; \dytj \; dzt_k \left. \right) \; p_{i,k,j}$$ + (41.93)

Note that the finite difference counterpart of incompressibility, Equation (4.3), for T-cells uses advective velocities defined on the faces of T cells

$$\frac{1}{\cstj} (\delx(adv\_vet_{i-1,k,j}) + \dely(adv\_vnt_{i,k,j-1})) + \delta_z (adv\_vbt_{i,k-1,j}) = 0$$ (41.94)

Substituting Equation (A.96) into Equation (A.95) yields

 

$$-\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=... ...t-1} p_{i,k,j}\cdot \delta_z (adv\_vbt_{i,k-1,j})\; \dxti\; \cstj\; \dytj dzt_k$$ (41.95)

Once again, using Equation (21.15) to re-arrnage the summation on ``k'' in Equation (A.97) yields

 

$$\frac{1}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_{i=2... ...vbt_{i,k-1,j}\cdot \delta_z(p_{i,k-1,j}) \; \dxti \; \cstj\; \dytj \; dzw_{k-1}$$ (41.96)

Substituting the discrete hydrostatic equation given by Equation (21.42) reduces Equation (A.98) to

 

$$-\frac{grav}{\rho_\circ}\sum_{jrow=2}^{jmt-1}\sum_{k=1}^{km}\sum_... ...,k-1,j}\cdot \overline{\rho_{i,k-1,j}}^z \; \dxti \; \cstj\; \dytj \; dzw_{k-1}$$ (41.97)

Equation (A.99) is the discrete counterpart of a result derived for the continuum in equation (A.18). It represents the change in kinetic energy due to horizontal pressure terms. Comparing with Equation (A.102) indicates that the change in kinetic energy due to horizontal pressure forces is compensated by an equal change in energy due to buoyancy effects.