Discrete enstrophy conservation

Vorticity Term with ENS scheme (ln_dynvor_ens=.true.)

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:

where is the volume element. Indeed, using (C.13), the discrete form of the right hand side of (C.14) can be transformed as follow:

Since and operators commute: , and introducing the horizontal divergence , it becomes:
| |||

The later equality is obtain only when the flow is horizontally non-divergent, =0.

Vorticity Term with EEN scheme (ln_dynvor_een=.true.)

With the EEN scheme, the vorticity terms are represented as:

where the indices and take the following value: or and or , and the vorticity triads, , defined at -point, are given by:

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