4.2.4 Tracer equations

Equations (4.5) and (4.6) are the equations for the active tracers potential temperature $\theta$ and salinity s. Potential temperature is used rather than in situ temperature because it more closely approximates a conservative variable. However, the paper by McDougall and Jackett (1998) provide motivation to employ a modified version of potential temperature which even more closely approximates a conservative variable than the traditional definition of potential temperature. Note that in an adiabatic ocean for which nonlinear equation of state effects are ignored, both salinity and potential temperature are materially conserved active tracers.

The flux vector ${\bf F}$ takes on one of a variety of forms depending on the choice of subgrid scale parameterizations. For example, an older choice is to use horizontal and vertical diffusion

$${\bf F}_{h}(T) - A_{h} \, \nabla_{h} T$$ = (4.17)

$$-\kappa_{h} T_{z},$$ F^z(T) = (4.18)

where A_h is the horizontal diffusivity $(\mbox{cm}^{2}\,\mbox{s}^{-1})$ and $\kappa_{h}$ $(\mbox{cm}^{2}\,\mbox{s}^{-1})$ is the vertical diffusivity. The ``h'' subscript on the diffusivities is historical, and it stands for ``heat.'' As discussed in Section B.3.0.5 of Appendix B, there is no problem with the use of vertically aligned tracer diffusion as a framework for parameterizing dianeutral processes. Yet there is a fundamental problem with horizontally aligned diffusive fluxes. The problem is that the ocean has a strong tendency to diffuse along, rather than across, directions of constant locally referenced potential density (the neutral directions as described by McDougall 1987), rather than constant depth. The differences can be nontrivial for certain regions of the ocean, especially in western boundary currents (Veronis 1975). For this reason, preference is given to use of the isoneutral diffusion tensor (see Section 34.1) rather than horizontal diffusion. Another increasingly common choice is to add the Gent-McWilliams eddy induced transport, which can be formulated as a skew-diffusion (see Section 34.1.6). Given these two choices, along with vertical diffusion, the flux for an arbitrary tracer T is then given by

$${\bf F}_{h}(T) - A_{I} \, \nabla_{h} T - (A_{I} - \kappa) {\bf S} \, T_{z}$$ = (4.19)

$$-(A_{I} + \kappa) \, {\bf S} \cdot \nabla_{h}T - (\kappa_{h} + A_{I} S^{2}) \, T_{z},$$ F^z(T) = (4.20)

where

$${\bf S} = - \left(\frac{\nabla_{h} \rho}{\rho_{z}}\right)$$ (4.21)

is the isoneutral slope vector with magnitude S, A_I is the isoneutral diffusivity, and $\kappa$ is the ``thickness diffusivity.'' Note that $\kappa$ parameterizes the mixing of thickness only in the case when $\kappa$ is constant. The distinction is discussed by McDougall (1998). Taking $A_{I} = \kappa$ is common, and it simplifies the horizontal tracer flux tremendously.

The hydrostatic approximation necessitates the use of a parameterization of vertical overturning associated with gravitationally unstable water columns. This parameterization is often represented in the model by a convective adjustment algorithm, as described in Section 32.1. Other means to gravitationally stabilize the column are through the use of a very large value of the vertical diffusivity, thus enhancing the vertical flux. More on vertical convection is discussed in Section 32.1.

Finally, the model allows for the input of various tracer source terms, which may represent, for example, a radioactive source for a passive tracer. These terms are not explicitly represented in the equations written above for purposes of brevity.