In the ocean we interpret:
r=z\text{ is the height}
\dot{r}=\frac{Dz}{Dt}=w\text{ is the vertical velocity}
\phi=\frac{p}{\rho _{c}}\text{ is the pressure}
b(\theta ,S,r)=\frac{g}{\rho _{c}} \left( \vphantom{\dot{W}} \rho (\theta,S,r) - \rho_{c}\right)
\text{ is the buoyancy}
where \rho_{c} is a fixed reference density of water and g is the acceleration due to gravity.
In the above:
At the bottom of the ocean: R_{\rm fixed}(x,y)=-H(x,y).
The surface of the ocean is given by: R_{\rm moving}=\eta
The position of the resting free surface of the ocean is given by R_{o}=Z_{o}=0.
Boundary conditions are:
w=0~\text{at }r=R_{\rm fixed}\text{ (ocean bottom)}
w=\frac{D\eta }{Dt}\text{ at }r=R_{\rm moving}=\eta \text{ (ocean surface)}
where \eta is the elevation of the free surface.
Then equations :eq:`horiz-mtm`- :eq:`humidity-salt` yield a consistent set of oceanic equations which, for convenience, are written out in z-coordinates in :numref:`ocean_appendix` - see eqs. :eq:`eq-ocean-mom` to :eq:`eq-ocean-salt`.