39.7.1 Locally referenced potential density equation

Some of the discussion in this section can be found in McDougall (1991) and Appendix B in Griffies et al., (1998). What is of interest is the time tendency of locally referenced potential density

 

$$\partial_{t}\rho = \rho_{\theta} \; \partial_{t}\theta + \rho_{s} \; \partial_{t}s,$$ (39.53)

where

$$\rho_{\theta} \frac{\partial \rho}{\partial \theta} = -\alpha \rho$$ =

$$\rho_{s} \frac{\partial \rho}{\partial s} = \beta \rho$$ = (39.54)

are the partial derivatives of density with respect to the active tracers potential temperature $\theta$ and salinity s. These derivatives are evaluated at the local temperature, salinity, and pressure. The reason there is no pressure time tendency term in equation (39.56) is due to the local referencing used for locally referenced potential density. In other words, locally referenced potential density is a local water mass variable in the sense that it changes only when water mass properties (temperature and salinity) change. Jackett and McDougall (1997) discuss an approximate global water mass variable called neutral density.

Now split the right hand side of equation (39.56) into various processes using the prognostic equations for temperature and salinity

$$\partial_{t}\rho \rho_{\theta} \; \partial_{t}\theta + \rho_{s} \; \partial_{t}s$$ =

$$- \vec{u} \cdot (\rho_{\theta} \; \nabla \theta + \rho_{s} \nabla... ...\theta} \; \nabla \cdot \vec{F}(\theta) + \rho_{s} \; \nabla \cdot \vec{F}(s) )$$ =

$$- [ \rho_{\theta} \; \nabla (\vec{u} \; \theta) + \rho_{s} \nabla... ...{\theta} \; \nabla \cdot \vec{F}(\theta) + \rho_{s} \; \nabla \cdot \vec{F}(s)]$$ = (39.55)

where $\vec{u}$ is the divergence-free current vector. The non-advective tracer flux takes the form

$$\vec{F} = \vec{F}_{I} + \vec{F}_{skew-lap} + \vec{F}_{skew-bih} + \vec{F}_{V} + \vec{F}_{H} ,$$ (39.56)

where

• $\vec{F}_{I}$ is the diffusive flux of tracer along the neutral directions.
• $\vec{F}_{skew-lap}$ is the skew-diffusive flux arising from Laplacian skew-diffusion (e.g., option gent_mcwilliams discussed in Section34.1.6).
• $\vec{F}_{skew-bih}$ is the skew-diffusive flux arising from biharmonic skew-diffusion (e.g., option biharmonic_rm discussed in Section 34.1.8).
• $\vec{F}_{V}$ is the vertical diffusive flux.
• $\vec{F}_{H}$ is the Laplacian horizontal diffusive flux.

Note again that convection is absent in this analysis, as its effects on density are readily diagnosed using the option save_convection. Additionally, the effects from a biharmonic horizontal diffusive flux currently has not been implemented in this diagnostic.