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).