Continuity Equation in $ s-$coordinates

Using (A.2) and the fact that the horizontal scale factors $ e_1$ and $ e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($ i$,$ j$,$ z$) coordinate system is transformed as follows in order to obtain its expression in the curvilinear $ s-$coordinate system:

\begin{subequations}\begin{align*}{\begin{array}{*{20}l} \nabla \cdot {\rm {\bf ...
...u\;\sigma _1 - v\;\sigma _2 \right] \end{array} } \end{align*}\end{subequations}

Here, $ w$ is the vertical velocity relative to the $ z-$coordinate system. Introducing the dia-surface velocity component, $ \omega$, defined as the volume flux across the moving $ s$-surfaces per unit horizontal area:

$\displaystyle \omega = w - w_s - \sigma _1  u - \sigma _2  v  $ (A.5)

with $ w_s$ given by (A.3), we obtain the expression for the divergence of the velocity in the curvilinear $ s-$coordinate system:
\begin{subequations}\begin{align*}{\begin{array}{*{20}l} \nabla \cdot {\rm {\bf ...
... \frac{\partial e_3}{\partial t}  \end{array} } \end{align*}\end{subequations}

As a result, the continuity equation (2.1c) in the $ s-$coordinates is:

$\displaystyle \frac{1}{e_3 } \frac{\partial e_3}{\partial t} + \frac{1}{e_1  e...
...\right\vert _s } \right] +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0$ (A.7)

A additional term has appeared that take into account the contribution of the time variation of the vertical coordinate to the volume budget.

Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17