the buoyancy term is missing here rho g khat

The first equation set is the most general, so gravity would be within ${\bf F}^u$ but is not yet specified. A few paragraphs down I begin "Geophysical fluids such as the ocean and atmosphere are rotating and stratified..." and start to make assumptions and call out specific terms. I have the sentence "The gravitational force $g {\bf k}$ is included in ${\bf F}^u$".

oh sorry, totally missed that. Thanks for the explanation

I realize that one could include gravity in ${\bf F}^u$ but I would much prefer that we not do so. In my view, it makes a lot of the subsequent derivation clearer if we include $\nabla_{3D} \Phi$ here from the start. I have been pushing for this the whole time and will continue to do so.

Is there a need to work with other types of salinity? If not, I would prefer to just use $S$ for absolute salinity, rather than $S_A$. Likewise, we work with conservative rather than potential temperature, and conservative temperature simply uses $\Theta$.