# The chain rule for coordinates

In order to establish the set of Primitive Equation in curvilinear -coordinates ( 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 and thus , and we introduce an arbitrary vertical coordinate . Let us define a new vertical scale factor by (which now depends on ) and the horizontal slope of surfaces by :

 and (A.1)

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

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

 (A.3)

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