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 :

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

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

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

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:

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