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Subsections

6.3.5 Tapering and stability

Experience with the GFDL model showed that the GM scheme has to be matched to the convective parameterization. This was originally expressed in connection with the introduction of the KPP boundary layer scheme (Large et al., 97) but in fact, as subsequent experience with the MIT model has found, is necessary for any convective parameterization.

\fbox{ \begin{minipage}{4.75in} {\em S/R GMREDI\_SLOPE\_LIMIT} ({\em pkg/gmredi/... ...rgument) \par $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argument) \par \end{minipage} }

Figure 6.1: Taper functions used in GKW99 and DM95.
\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}

Figure 6.2: Effective slope as a function of ``true'' slope using Cox slope clipping, GKW91 limiting and DM95 limiting.
\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}}

6.3.5.1 Slope clipping

Deep convection sites and the mixed layer are indicated by homogenized, unstable or nearly unstable stratification. The slopes in such regions can be either infinite, very large with a sign reversal or simply very large. From a numerical point of view, large slopes lead to large variations in the tensor elements (implying large bolus flow) and can be numerically unstable. This was first recognized by Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing tensor. Here, the slope magnitude is simply restricted by an upper limit:

$\displaystyle \vert\nabla \sigma\vert$ $\displaystyle =$ $\displaystyle \sqrt{ \sigma_x^2 + \sigma_y^2 }$ (6.14)
$\displaystyle S_{lim}$ $\displaystyle =$ $\displaystyle - \frac{\vert\nabla \sigma\vert}{ S_{max} } \;\;\;\;\;\;\;\;$   $\displaystyle \mbox{where $S_{max}$\ is a parameter}$ (6.15)
$\displaystyle \sigma_z^\star$ $\displaystyle =$ $\displaystyle \min( \sigma_z , S_{lim} )$ (6.16)
$\displaystyle {[s_x,s_y]}$ $\displaystyle =$ $\displaystyle - \frac{ [\sigma_x,\sigma_y] }{\sigma_z^\star}$ (6.17)

Notice that this algorithm assumes stable stratification through the ``min'' function. In the case where the fluid is well stratified ( $ \sigma_z < S_{lim}$) then the slopes evaluate to:

$\displaystyle {[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z}$ (6.18)

while in the limited regions ( $ \sigma_z > S_{lim}$) the slopes become:

$\displaystyle {[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{\vert\nabla \sigma\vert/S_{max}}$ (6.19)

so that the slope magnitude is limited $ \sqrt{s_x^2 + s_y^2} = S_{max}$.

The slope clipping scheme is activated in the model by setting GM_taper_scheme = 'clipping' in data.gmredi.

Even using slope clipping, it is normally the case that the vertical diffusion term (with coefficient $ \kappa_\rho{\bf K}_{33} = \kappa_\rho S_{max}^2$) is large and must be time-stepped using an implicit procedure (see section on discretisation and code later). Fig. 6.3 shows the mixed layer depth resulting from a) using the GM scheme with clipping and b) no GM scheme (horizontal diffusion). The classic result of dramatically reduced mixed layers is evident. Indeed, the deep convection sites to just one or two points each and are much shallower than we might prefer. This, it turns out, is due to the over zealous re-stratification due to the bolus transport parameterization. Limiting the slopes also breaks the adiabatic nature of the GM/Redi parameterization, re-introducing diabatic fluxes in regions where the limiting is in effect.

6.3.5.2 Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991

The tapering scheme used in Gerdes et al., 1999, ([42]) addressed two issues with the clipping method: the introduction of large vertical fluxes in addition to convective adjustment fluxes is avoided by tapering the GM/Redi slopes back to zero in low-stratification regions; the adjustment of slopes is replaced by a tapering of the entire GM/Redi tensor. This means the direction of fluxes is unaffected as the amplitude is scaled.

The scheme inserts a tapering function, $ f_1(S)$, in front of the GM/Redi tensor:

$\displaystyle f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{\vert S\vert}\right)^2 \right]$ (6.20)

where $ S_{max}$ is the maximum slope you want allowed. Where the slopes, $ \vert S\vert<S_{max}$ then $ f_1(S) = 1$ and the tensor is un-tapered but where $ \vert S\vert \ge S_{max}$ then $ f_1(S)$ scales down the tensor so that the effective vertical diffusivity term $ \kappa f_1(S) \vert S\vert^2 = \kappa S_{max}^2$.

The GKW tapering scheme is activated in the model by setting GM_taper_scheme = 'gkw91' in data.gmredi.

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Next: 6.3.6 Tapering: Danabasoglu and Up: 6.3 Gent/McWiliams/Redi SGS Eddy Previous: 6.3.4 Variable   Contents
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