39.7.1.2 Summary of the terms forcing locally referenced potential density

In summary, the time tendency of locally referenced potential density can be written

 

$$\partial_{t}\rho \mbox{advection} + \mbox{vertical diffusion} + \mbox{Laplacian horizontal diffusion}$$ =

$$\mbox{Laplacian skew-diffusion} + \mbox{biharmonic skew-diffusion}$$ +

$$\mbox{cabbeling} + \mbox{thermobaricity} + \mbox{halobaricity},$$ + (39.75)

where

   

$$\mbox{advection} - [ \rho_{\theta} \; \nabla \cdot (\vec{u} \; \theta) + \rho_{s} \nabla \cdot (\vec{u} \; s) ]$$ = (39.76)

$$\mbox{vertical diffusion} - [\rho_{\theta} \nabla \cdot \vec{F}_{V}(\theta) + \rho_{s} \nabla \cdot \vec{F}_{V}(s)]$$ = (39.77)

$$\mbox{Laplacian horizontal diffusion} - [\rho_{\theta} \nabla \cdot \vec{F}_{H}(\theta) + \rho_{s} \nabla \cdot \vec{F}_{H}(s)]$$ = (39.78)

$$\mbox{Laplacian skew-diffusion} - [\rho_{\theta} \nabla \cdot \vec{F}_{skew-lap}(\theta) + \rho_{s} \nabla \cdot \vec{F}_{skew-lap}(s)]$$ = (39.79)

$$\mbox{Biharmonic skew-diffusion} - [\rho_{\theta} \nabla \cdot \vec{F}_{skew-bih}(\theta) + \rho_{s} \nabla \cdot \vec{F}_{skew-bih}(s)]$$ = (39.80)

$$\mbox{cabbeling} \nabla \theta \cdot \vec{F}_{I}(\theta) \; \rho_{a b} V^{a} V^{b}$$ = (39.81)

$$\mbox{thermobaricity} \rho_{\theta \, p} \vec{F}_{I}(\theta) \cdot \nabla p$$ = (39.82)

$$\mbox{halobaricity} \rho_{s \, p} \vec{F}_{I}(s) \cdot \nabla p.$$ = (39.83)

Notice that each term has the dimensions of density per time.

A steady state ocean represents a balance between the terms on the right hand side of equation (39.78). For the simplest case, there is only advection by the resolved current $\vec{u}$. In this case, the steady state current is aligned parallel to the neutral directions: $\vec{u} \cdot (\rho_{\theta} \; \nabla \theta + \rho_{s} \nabla s) = \vec{u} \cdot \nabla \rho = 0$. Such a flow field is typically associated with steady state adiabatic fluid flow. When there is time dependence, and/or when cabbeling, thermobaricity, halobaricity, diffusion, or skew-diffusion are present, there will be a nonzero dianeutral component to the velocity field: $\vec{u} \cdot \nabla \rho \ne 0$. Note that since the cabbeling term is sign-definite, the balance between advection and cabbeling,

$$\vec{u}_{cab} \cdot \nabla \rho = \nabla \theta \cdot \vec{F}_{I}(\theta) \; \, \rho_{a b} V^{a} V^{b},$$ (39.84)

results in a dianeutral component to the velocity which is always directed towards values of increasing locally referenced potential density

$$\vec{u}_{cab} \cdot \nabla \rho \ge 0.$$ (39.85)

Namely, for a stably stratified fluid, the velocity field which balances cabbeling acts in a downward dianeutral direction (McDougall 1991). The other processes, as they are not sign-definite, do not necessarily lead to a particular flow direction for their associated dianeutral velocity.

The diagnostic option cross_flow_netcdf (Section 39.1) allows one to diagnose the component of velocity along and across the neutral directions. In many regions of the ocean, dianeutral advection is presumed to be small. The option local_potential_density_terms allows one to diagnose terms in the locally referenced potential density equation and thus to associate a cause for any dianeutral flow in the model.