39.7.2.1 Equation of state considerations

As described in Section 15.1.2, the density is evaluated as a cubic approximation (Bryan and Cox, 1972) to the UNESCO equation of state

$$\rho(\tilde{t},\tilde{s},k) (c_{k,1} + (c_{k,4} + c_{k,7}*\tilde{s})*\tilde{s} +$$ =

$$(c_{k,3} + c_{k,8}*\tilde{s} + c_{k,6}*\tilde{t})*\tilde{t})*\tilde{t} +$$

$$(c_{k,2} + (c_{k,5} + c_{k,9}*\tilde{s})*\tilde{s})*\tilde{s}$$

$$c_{k,1} \; \tilde{t} + c_{k,2} \; \tilde{s} + c_{k,3} \; \tilde{t}^{2} + c_{k,4} \; \tilde{s} \; \tilde{t} + c_{k,5} \; \tilde{s}^{2}$$ =

$$c_{k,6} \; \tilde{t}^{3} + c_{k,7} \; \tilde{s}^{2} \; \tilde{t} + c_{k,8} \; \tilde{s} \; \tilde{t}^{2} + c_{k,9} \; \tilde{s}^{3}$$ + (39.86)

where

$$\tilde{t}_{i,k,j} t_{i,k,j,1,\tau} - T^{ref}_k$$ = (39.87)

$$\tilde{s}_{i,k,j} t_{i,k,j,2,\tau} - S^{ref}_k$$ = (39.88)

are tracer anomalies with respect to some pre-specified reference values. The choice of computing density anomalies is based on the increased numerical accuracy inherent in this approach (see Section 15.1.2 for further discussion). The nine polynomial coefficients c_k,1-9 in this equation depend on the depth level k. They are defined at the tracer points, which means that the density is defined there as well. The first partial derivative of the density with respect to the active tracers are given in MOM by the quadratic expressions

$$(\rho_{\theta})_{i,k,j} = \partial_{\tilde{t}} \; \rho (\tilde{t},\tilde{s},k) c_{k,1} + 2 \; c_{k,3} \; \tilde{t} + c_{k,4} \; \tilde{s} + 3 \;... ...ilde{t}^{2} + c_{k,7} \; \tilde{s}^{2} + 2 \; c_{k,8} \; \tilde{s} \; \tilde{t}$$ = (39.89)

$$(\rho_{s})_{i,k,j} = \partial_{\tilde{s}} \; \rho(\tilde{t},\tilde{s},k) c_{k,2} + c_{k,4} \; \tilde{t} + 2 \; c_{k,5} \; \tilde{s} + 2 \;... ...ilde{t} \; \tilde{s} + c_{k,8} \; \tilde{t}^{2} + 3 \; c_{k,9} \; \tilde{s}^{2}$$ = (39.90)

The second partial derivatives of density with respect to the active tracers are given by the linear expressions

$$(\rho_{\theta \theta})_{i,k,j} = \partial_{\tilde{t} \; \tilde{t}} \; \rho(\tilde{t},\tilde{s},k) 2 \; c_{k,3} + 6 \; c_{k,6} \; \tilde{t} + 2 \; c_{k,8} \; \tilde{s}$$ = (39.91)

$$(\rho_{\theta s})_{i,k,j} = \partial_{\tilde{t} \; \tilde{s}} \; \rho(\tilde{t},\tilde{s},k) c_{k,4} + 2 \; c_{k,7} \; \tilde{s} + 2 \; c_{k,8} \; \tilde{t}$$ = (39.92)

$$(\rho_{ss})_{i,k,j} = \partial_{\tilde{s} \; \tilde{s}} \; \rho(\tilde{t},\tilde{s},k) 2 \; c_{k,5} + 2 \; c_{k,7} \; \tilde{t} + 6 \; c_{k,9} \; \tilde{s}$$ = (39.93)

These expressions are tabulated in the routine dens.h. The second partial derivatives of the density with respect to an active tracer and pressure are given by

$$\rho_{\theta \; p} z_{p} \; \frac{\partial \rho_{\theta}}{\partial z}$$ =

$$-\left(\frac{1}{\rho \; g} \right) \partial_{z} \rho_{\theta}$$ = (39.94)

$$\rho_{s \; p} z_{p} \; \frac{\partial \rho_{s}}{\partial z}$$ =

$$-\left(\frac{1}{\rho \; g} \right) \partial_{z} \rho_{s},$$ = (39.95)

where the hydrostatic approximation has been used. Consistent with the Boussinesq approximation used in MOM, the $\rho \, g$ term will be evaluated as $\rho_{o} \, g$, where $\rho_{o} = 1.035 g/cm^{3}$ and g=980.6 cm/sec². The vertical derivatives are discretized in a centered fashion:

$$(\rho_{\theta p})_{i,k,j} = (\partial_{\tilde{t} \; p} \; \rho)_{i,k,j} -\frac{1}{ \rho_{o} \; g} \left( \frac { (\rho_{\theta})_{i,k-1,j} - (\rho_{\theta})_{i,k+1,j}} {dhwt_{i.k-1,j} + dhwt_{i,k,j}} \right)$$ = (39.96)

$$(\rho_{s p})_{i,k,j} = (\partial_{\tilde{s} \; p } \; \rho)_{i,k,j} -\frac{1}{\rho_{o} \; g} \; \left( \frac{ (\rho_{s})_{i,k-1,j} - (\rho_{s})_{i,k+1,j} } {dhwt_{i.k-1,j} + dhwt_{i,k,j}} \right).$$ = (39.97)

Next to surface and bottom boundaries, the values of $(\rho_{\theta p})_{i,k,j}$ and $(\rho_{s p})_{i,k,j}$ are equated to the value one level away. Note that these vertical derivatives compute differences in in situ density; i.e., the two densities are not referenced to the same temperature, salinity, and pressure. The reason is that one is interested in gradients between different potential density surfaces when computing thermobaricity and halobaricity.