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, $i.e.$ (9.10) is used in $z$-coordinate, and the sum (9.10) + (9.11) in $z^*$ or $s$-coordinates.

The eddy induced velocity is given by:

$$\begin{equation}\begin{split}u^* & = - \frac{1}{e_{3}}\; \partial... ... \; \tilde{r}_1, \psi_2 & = A_{e} \; \tilde{r}_2, \end{split} \end{equation}$$

with $A_{e}$ the eddy induced velocity coefficient, and $\tilde{r}_1$ and $\tilde{r}_2$ 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 $2^{nd}$ 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 $ijk$ 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 $ijk$ space:

$$\textbf{F}_\mathrm{eiv}^T = \begin{pmatrix}{+ e_{2} \psi_1 \; \partial_k T} { - e_{2} \psi_1 \; \partial_i T} \end{pmatrix}$$ (D.37)

The total fluxes per unit physical area are then

$$\begin{split}f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T ... ...artial_i T + e_{1} \psi_2 \partial_j T \right\}. \end{split}$$ (D.38)

Note that Eq.  (D.41) takes the same form whatever the vertical coordinate, though of course the slopes $\tilde{r}_i$ which define the $\psi_i$ 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

$$\frac{\partial T}{\partial t}= -\frac{1}{e_1 e_2 e_3 } \lef... ...k} \left( e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right) \right]$$ (D.39)

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.

The discrete skew flux formulation

The skew fluxes in (D.41, D.40), like the off-diagonal terms (D.3, D.4) of the small angle diffusion tensor, are best expressed in terms of the triad slopes, as in Fig. D.1 and Eqs (D.6, D.7); but now in terms of the triad slopes $\tilde{\mathbb{R}}$ relative to geopotentials instead of the $\mathbb{R}$ relative to coordinate surfaces. The discrete form of (D.40) using the slopes (D.8) and defining $A_e$ at $T$-points is then given by:

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.

No change in tracer variance

The discretization conserves tracer variance, $i.e.$ it does not include a diffusive component but is a `pure' advection term. This can be seen by considering the fluxes associated with a given triad slope $_i^k{\mathbb{R}}_{i_p}^{k_p} (T)$. For, following §D.2.5 and (D.16), the associated horizontal skew-flux $_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)$ drives a net rate of change of variance, summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of

$$_i^k{\mathbb{S}_u}_{i_p}^{k_p} (T) \delta_{i+ i_p}[T^k],$$ (D.41)

while the associated vertical skew-flux gives a variance change summed over the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of

$$_i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \delta_{k+ k_p}[T^i].$$ (D.42)

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.

Reduction in gravitational PE

The vertical density flux associated with the vertical skew-flux always has the same sign as the vertical density gradient; thus, so long as the fluid is stable (the vertical density gradient is negative) the vertical density flux is negative (downward) and hence reduces the gravitational PE.

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

$$g {e_{3w}}_{ i}^{ k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) =g {e_{3w}}_{ i}^{ k+k_p}\left[-\alpha _i^k {\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p} (S) \right].$$

$${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$$

$$=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}... ...ta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{ i + i_p}^{ k} }$$

$$=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k \left({_i^k\mathbb{R}_{... ...delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{ i}^{ k+k_p}},$$ (D.43)

using the definition of the triad slope $\rtriad{R}$, (D.8) to express $-\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]$ in terms of $-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]$.

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

$$\begin{multline} g \delta_{i+i_p}[z_T^k] \left[ -\alpha _i^k {\:}_i^k {\mathb... ...[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{ i}^{ k+k_p}}, \end{multline}$$

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

$$\begin{multline} g{e_{3w}}_{ i}^{ k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) +... ...[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{ i}^{ k+k_p}}. \end{multline}$$

Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative.

Treatment of the triads at the boundaries

Triad slopes R used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries in exactly the same way as are the triad slopes R used for the iso-neutral diffusive fluxes, as described in §D.2.8 and Fig. D.3. Thus surface layer triads $\triadt{i}{1}{R}{1/2}{-1/2}$ and $\triadt{i+1}{1}{R}{-1/2}{-1/2}$ are masked, and both near bottom triad slopes $\triadt{i}{k}{R}{1/2}{1/2}$ and $\triadt{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked, i.e. the $i,k+1$ $u$-point is masked. The namelist parameter ln_botmix_grif has no effect on the eddy-induced skew-fluxes.

Limiting of the slopes within the interior

Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, exactly as in calculating the iso-neutral diffusion, §D.2.9. Each individual triad R is so limited.

Tapering within the surface mixed layer

The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads R) are always tapered linearly from their value immediately below the mixed layer to zero at the surface (D.32a), as described in §D.2.10. This is option (c) of Fig. 9.2. This linear tapering for the slopes used to calculate the eddy-induced fluxes is unaffected by the value of ln_triad_iso.

The justification for this linear slope tapering is that, for $A_e$ 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

Where the namelist parameter ln_traldf_gdia=true, diagnosed mean eddy-induced velocities are output. Each time step, streamfunctions are calculated in the $i$-$k$ and $j$-$k$ planes at $uw$ (integer +1/2 $i$, integer $j$, integer +1/2 $k$) and $vw$ (integer $i$, integer +1/2 $j$, integer +1/2 $k$) points (see Table 4.1) respectively. We follow [Griffies, 2004] and calculate the streamfunction at a given $uw$-point from the surrounding four triads according to:

$${\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p, k_p}} {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}.$$ (D.44)

The streamfunction $\psi_1$ is calculated similarly at $vw$ points. The eddy-induced velocities are then calculated from the straightforward discretisation of (D.39a):

$$\begin{split}{u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k... ...j-1/2}^{k+1/2} {\psi_2}_{j-1/2}^{k+1/2} \right\}, \end{split}$$ (D.45)