Subsections

# Discrete total energy conservation : vector invariant form

## Total energy conservation

The discrete form of the total energy conservation, (C.3), is given by:

which in vector invariant forms, it leads to:

 (C.7)

Substituting the discrete expression of the time derivative of the velocity either in vector invariant, leads to the discrete equivalent of the four equations (C.6).

## Vorticity term (coriolis + vorticity part of the advection)

Let , located at -points, be either the relative ( ), or the planetary ( ), or the total potential vorticity ( ). Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy.

### Vorticity Term with ENE scheme (ln_dynvor_ene=.true.)

For the ENE scheme, the two components of the vorticity term are given by :

This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. Indeed, the kinetic energy tendency associated to the vorticity term and averaged over the ocean domain can be transformed as follows:

In other words, the domain averaged kinetic energy does not change due to the vorticity term.

### 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:

 (C.9)

This formulation does conserve the total kinetic energy. Indeed,

 Expending the summation on and , it becomes: The summation is done over all and indices, it is therefore possible to introduce a shift of either in or direction in some of the term of the summation (first term of the first and second lines, second term of the second and fourth lines). By doning so, we can regroup all the terms of the summation by triad at a (,) point. In other words, we regroup all the terms in the neighbourhood that contain a triad at the same (,) indices. It becomes:

### Gradient of Kinetic Energy / Vertical Advection

The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE :

Indeed, using successively (4.11) ( the skew symmetry property of the operator) and the continuity equation, then (4.11) again, then the commutativity of operators and , and finally (4.12) ( the symmetry property of the operator) applied in the horizontal and vertical directions, it becomes:

 KEG Assuming that and , or at least that the time derivative of these two equations is satisfied, it becomes: The first term provides the discrete expression for the vertical advection of momentum (ZAD), while the second term corresponds exactly to (C.7), therefore: ZAD

There is two main points here. First, the satisfaction of this property links the choice of the discrete formulation of the vertical advection and of the horizontal gradient of KE. Choosing one imposes the other. For example KE can also be discretized as . This leads to the following expression for the vertical advection:

a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. Nine velocity points have to be used instead of 3. This is the reason why it has not been chosen.

Second, as soon as the chosen -coordinate depends on time, an extra constraint arises on the time derivative of the volume at - and -points:

which is (over-)satified by defining the vertical scale factor as follows:

 (C.10) (C.11)

Blah blah required on the the step representation of bottom topography.....

## Pressure Gradient Term

When the equation of state is linear ( when an advection-diffusion equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces:

This property can be satisfied in a discrete sense for both - and -coordinates. Indeed, defining the depth of a -point, , as the sum of the vertical scale factors at -points starting from the surface, the work of pressure forces can be written as:

 Using successively (4.11), the skew symmetry property of the operator, (6.4), the continuity equation, (6.20), the hydrostatic equation in the -coordinate, and , which comes from the definition of , it becomes:

The first term is exactly the first term of the right-hand-side of (C.7). It remains to demonstrate that the last term, which is obviously a discrete analogue of is equal to the last term of (C.7). In other words, the following property must be satisfied:

Let introduce the pressure at -point such that . The right-hand-side of the above equation can be transformed as follows:

therefore, the balance to be satisfied is:

which is a purely vertical balance:

Defining

Note that this property strongly constrains the discrete expression of both the depth of points and of the term added to the pressure gradient in the -coordinate. Nevertheless, it is almost never satisfied since a linear equation of state is rarely used.

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