The chain rule for $ s-$coordinates

In order to establish the set of Primitive Equation in curvilinear $ s$-coordinates ($ i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian Eulerian (ALE) coordinate in the vertical), we start from the set of equations established in §2.3.2 for the special case $ k=z$ and thus $ e_3=1$, and we introduce an arbitrary vertical coordinate $ a = a(i,j,z,t)$. Let us define a new vertical scale factor by $ e_3 = \partial z / \partial s$ (which now depends on $ (i,j,z,t)$) and the horizontal slope of $ s-$surfaces by :

$\displaystyle \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right\vert _s$   and$\displaystyle \quad \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right\vert _s$ (A.1)

The chain rule to establish the model equations in the curvilinear $ s-$coordinate system is:

\begin{equation*}\begin{aligned}&\left. {\frac{\partial \bullet }{\partial t}} \...
...rac{1}{e_3 }\frac{\partial \bullet }{\partial s}  \end{aligned}\end{equation*}

In particular applying the time derivative chain rule to $ z$ provides the expression for $ w_s$, the vertical velocity of the $ s-$surfaces referenced to a fix z-coordinate:

$\displaystyle w_s = \left. \frac{\partial z }{\partial t} \right\vert _s = \fra...
...tial s} \; \frac{\partial s}{\partial t} = e_3   \frac{\partial s}{\partial t}$ (A.3)

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