The discretization of pimitive equation in -coordinate ( time and space varying vertical coordinate) must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy.
Let us first establish those constraint in the continuous world. The total energy ( kinetic plus potential energies) is conserved :
This equation can be transformed to obtain several sub-equalities. The transformation for the advection term depends on whether the vector invariant form or the flux form is used for the momentum equation. Using (C.2) and introducing (A.15) in (C.3) for the former form and Using (C.1) and introducing (A.16) in (C.3) for the latter form leads to:
where is the gradient along the -surfaces.
The prognostic ocean dynamics equation can be summarized as follows:
Vector invariant form:
(C.6c) is the balance between the conversion KE to PE and PE to KE. Indeed the left hand side of (C.6c) can be transformed as follows:
The left hand side of (C.6c) can be transformed as follows:
introducing the hydrostatic balance in the last term, it becomes:
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17