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1.3.3 Ocean

In the ocean we interpret:

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

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:


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

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 Appendix Ocean - see eqs(1.99) to (1.104).


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