where is the along isopycnal diffusivity and is a rank 2 tensor that projects the gradient of onto the isopycnal surface. The unapproximated projection tensor is:
Here, and are the components of the isoneutral slope. The first point to note is that a typical slope in the ocean interior is small, say of the order . A maximum slope might be of order and only exceeds such in unstratified regions where the slope is ill defined. It is therefore justifiable, and customary, to make the small slope approximation, . The Redi projection tensor then becomes:
6.4.1.2 GM parameterizationThe GM parameterization aims to represent the ``advective'' or ``transport'' effect of geostrophic eddies by means of a ``bolus'' velocity, . The divergence of this advective flux is added to the tracer tendency equation (on the rhs):
The bolus velocity is defined as the rotational of a streamfunction = :
and thus is automatically non-divergent. In the GM parameterization, the streamfunction is specified in terms of the isoneutral slopes and :
with boundary conditions on upper and lower boundaries. In the end, the bolus transport in the GM parameterization is given by:
This is the form of the GM parameterization as applied by Donabasaglu, 1997, in MOM versions 1 and 2. Note that in the MITgcm, the variables containing the GM bolus streamfunction are:
6.4.1.3 Griffies Skew Flux
Griffies [1998] notes that the discretisation of bolus velocities involves
multiple layers of differencing and interpolation that potentially
lead to noisy fields and computational modes. He pointed out that the
bolus flux can be re-written in terms of a non-divergent flux and a
skew-flux:
The first vector is non-divergent and thus has no effect on the tracer field and can be dropped. The remaining flux can be written:
where
is an anti-symmetric tensor. This formulation of the GM parameterization involves fewer derivatives than the original and also involves only terms that already appear in the Redi mixing scheme. Indeed, a somewhat fortunate cancellation becomes apparent when we use the GM parameterization in conjunction with the Redi isoneutral mixing scheme:
In the instance that then
which differs from the variable Laplacian diffusion tensor by only two non-zero elements in the -row.
6.4.1.4 VariableVisbeck et al. [1997] suggest making the eddy coefficient, , a function of the Eady growth rate, . The formula involves a non-dimensional constant, , and a length-scale :
where the Eady growth rate has been depth averaged (indicated by the over-line). A local Richardson number is defined which, when combined with thermal wind gives:
where is defined . Substituting into the formula for gives:
6.4.1.5 Tapering and stabilityExperience 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., 1994] but in fact, as subsequent experience with the MIT model has found, is necessary for any convective parameterization.
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:
Notice that this algorithm assumes stable stratification through the ``min'' function. In the case where the fluid is well stratified ( ) then the slopes evaluate to:
while in the limited regions ( ) the slopes become:
so that the slope magnitude is limited . 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 ) is large and must be time-stepped using an implicit procedure (see section on discretisation and code later). Fig. 6.8 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.
Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991The tapering scheme used in Gerdes et al. [1991] 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, , in front of the GM/Redi tensor:
where is the maximum slope you want allowed. Where the slopes, then and the tensor is un-tapered but where then scales down the tensor so that the effective vertical diffusivity term . The GKW91 tapering scheme is activated in the model by setting GM_taper_scheme = 'gkw91' in data.gmredi.
Tapering: Danabasoglu and McWilliams, J. Clim. 1995The tapering scheme used by Danabasoglu and McWilliams [1995] followed a similar procedure but used a different tapering function, :
where is a cut-off slope and is a scale over which the slopes are smoothly tapered. Functionally, the operates in the same way as the GKW91 scheme but has a substantially lower cut-off, turning off the GM/Redi SGS parameterization for weaker slopes. The DM95 tapering scheme is activated in the model by setting GM_taper_scheme = 'dm95' in data.gmredi.
Tapering: Large, Danabasoglu and Doney, JPO 1997The tapering used in Large et al. [1997] is based on the DM95 tapering scheme, but also tapers the scheme with an additional function of height, , so that the GM/Redi SGS fluxes are reduced near the surface:
where is a depth-scale and with m s . This tapering with height was introduced to fix some spurious interaction with the mixed-layer KPP parameterization. The LDD97 tapering scheme is activated in the model by setting GM_taper_scheme = 'ldd97' in data.gmredi.
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