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)¶\[b(\theta ,S,r)=\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho
_{c}\right) \text{ is the buoyancy}\]
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: \(R_{fixed}(x,y)=-H(x,y)\).
The surface of the ocean is given by: \(R_{moving}=\eta\)
The position of the resting free surface of the ocean is given by \(R_{o}=Z_{o}=0\).
Boundary conditions are:
(1.23)¶\[w=0~\text{at }r=R_{fixed}\text{ (ocean bottom)}\]
(1.24)¶\[w=\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)}\]
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).