Here we only consider the first component of the momentum equation, the generalization to the second one being straightforward.
Total derivative in vector invariant form
Let us consider (2.13), the first component of the momentum equation in the vector invariant form. Its total coordinate time derivative, can be transformed as follows in order to obtain its expression in the curvilinear coordinate system:
Total derivative in flux form
Let us start from the total time derivative in the curvilinear coordinate system we have just establish. Following the procedure used to establish (2.11), it can be transformed into :
horizontal pressure gradient
The horizontal pressure gradient term can be transformed as follows:
An additional term appears in (A.14) which accounts for the tilt of surfaces with respect to geopotential surfaces.
As in -coordinate, the horizontal pressure gradient can be split in two parts following Marsaleix et al. . Let defined a density anomaly, , by , and a hydrostatic pressure anomaly, , by . The pressure is then given by:
Substituing (A.13) in (A.14) and using the definition of the density anomaly it comes the expression in two parts:
The other terms of the momentum equation
The coriolis and forcing terms as well as the the vertical physics remain unchanged as they involve neither time nor space derivatives. The form of the lateral physics is discussed in appendix B.
Full momentum equation
To sum up, in a curvilinear -coordinate system, the vector invariant momentum equation solved by the model has the same mathematical expression as the one in a curvilinear coordinate, except for the pressure gradient term :
It is important to realize that the change in coordinate system has only concerned the position on the vertical. It has not affected (i,j,k), the orthogonal curvilinear set of unit vectors. (,) are always horizontal velocities so that their evolution is driven by horizontal forces, in particular the pressure gradient. By contrast, is not , the third component of the velocity, but the dia-surface velocity component, the volume flux across the moving -surfaces per unit horizontal area.
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17