The discrete form of the total energy conservation, (C.3), is given by:
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).
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.
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:
With the EEN scheme, the vorticity terms are represented as:
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:
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 :
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:
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:
Second, as soon as the chosen -coordinate depends on time, an extra constraint arises on the time derivative of the volume at - and -points:
Blah blah required on the the step representation of bottom topography.....
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:
Let introduce the pressure at -point such that . The right-hand-side of the above equation can be transformed as follows:
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