- The continuous skew flux formulation
- The discrete skew flux formulation

- Treatment of the triads at the boundaries
- Limiting of the slopes within the interior
- Tapering within the surface mixed layer
- Streamfunction diagnostics

Eddy induced advection formulated as a skew flux

The continuous skew flux formulation

When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added. The associated velocity is the so called eddy induced velocity, the formulation of which depends on the slopes of iso- neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used here are referenced to the geopotential surfaces, (9.10) is used in -coordinate, and the sum (9.10) + (9.11) in or -coordinates.

The eddy induced velocity is given by:

with the eddy induced velocity coefficient, and and the slopes between the iso-neutral and the geopotential surfaces.

The traditional way to implement this additional advection is to add it to the Eulerian velocity prior to computing the tracer advection. This is implemented if key_ traldf_eiv is set in the default implementation, where ln_traldf_grif is set false. This allows us to take advantage of all the advection schemes offered for the tracers (see §5.1) and not just a order advection scheme. This is particularly useful for passive tracers where positivity of the advection scheme is of paramount importance.

However, when ln_traldf_grif is set true, NEMO instead implements eddy induced advection according to the so-called skew form [Griffies, 1998]. It is based on a transformation of the advective fluxes using the non-divergent nature of the eddy induced velocity. For example in the (i,k) plane, the tracer advective fluxes per unit area in space can be transformed as follows:

and since the eddy induced velocity field is non-divergent, we end up with the skew form of the eddy induced advective fluxes per unit area in space:

The total fluxes per unit physical area are then

Note that Eq. (D.41) takes the same form whatever the vertical coordinate, though of course the slopes which define the in (D.39b) are relative to geopotentials. The tendency associated with eddy induced velocity is then simply the convergence of the fluxes (D.40, D.41), so

It naturally conserves the tracer content, as it is expressed in flux form. Since it has the same divergence as the advective form it also preserves the tracer variance.

Such a discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes. It also ensures the following two key properties.

while the associated vertical skew-flux gives a variance change summed over the -points (above) and (below) of

Inspection of the definitions (D.43b, D.43c) shows that these two variance changes (D.44, D.45) sum to zero. Hence the two fluxes associated with each triad make no net contribution to the variance budget.

For the change in gravitational PE driven by the -flux is

Substituting from (D.43c), gives
| ||

(D.43) |

using the definition of the triad slope , (D.8) to express in terms of .

Where the coordinates slope, the -flux gives a PE change

(using (D.43b)) and so the total PE change (D.46) + (D.47) associated with the triad fluxes is

Where the fluid is stable, with , this PE change is negative.

Treatment of the triads at the boundaries

Limiting of the slopes within the interior

Tapering within the surface mixed layer

The justification for this linear slope tapering is that, for that is constant or varies only in the horizontal (the most commonly used options in NEMO: see §9.1), it is equivalent to a horizontal eiv (eddy-induced velocity) that is uniform within the mixed layer (D.39a). This ensures that the eiv velocities do not restratify the mixed layer [Danabasoglu et al., 2008, Tréguier et al., 1997]. Equivantly, in terms of the skew-flux formulation we use here, the linear slope tapering within the mixed-layer gives a linearly varying vertical flux, and so a tracer convergence uniform in depth (the horizontal flux convergence is relatively insignificant within the mixed-layer).

Streamfunction diagnostics

The streamfunction is calculated similarly at points. The eddy-induced velocities are then calculated from the straightforward discretisation of (D.39a):

Gurvan Madec and the NEMO Team

NEMO European Consortium2017-02-17