In the ENS scheme, the vorticity term is descretized as follows:
The scheme does not allow but the conservation of the total kinetic energy but the conservation of , the potential enstrophy for a horizontally non-divergent flow ( when =0). Indeed, using the symmetry or skew symmetry properties of the operators (Eqs (4.12) and (4.11)), it can be shown that:
Since and operators commute: , and introducing the horizontal divergence , it becomes:
With the EEN scheme, the vorticity terms are represented as:
This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ( ).
Let consider one of the vorticity triad, for example , similar manipulation can be done for the 3 others. The discrete form of the right hand side of (C.14) applied to this triad only can be transformed as follow:
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17