# 1.3.3. Ocean¶

In the ocean we interpret:

(1.19)$r=z\text{ is the height}$
(1.20)$\dot{r}=\frac{Dz}{Dt}=w\text{ is the vertical velocity}$
(1.21)$\phi=\frac{p}{\rho _{c}}\text{ is the pressure}$
(1.22)$b(\theta ,S,r)=\frac{g}{\rho _{c}} \left( \vphantom{\dot{W}} \rho (\theta,S,r) - \rho_{c}\right) \text{ is the buoyancy}$

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:

(1.23)$w=0~\text{at }r=R_{\rm fixed}\text{ (ocean bottom)}$
(1.24)$w=\frac{D\eta }{Dt}\text{ at }r=R_{\rm moving}=\eta \text{ (ocean surface)}$

where $$\eta$$ is the elevation of the free surface.

Then equations (1.1)- (1.6) yield a consistent set of oceanic equations which, for convenience, are written out in $$z-$$coordinates in Section 1.5.1 - see eqs. (1.98) to (1.103).