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1  \section{Gent/McWiliams/Redi SGS Eddy parameterization}  \subsection{GMREDI: Gent/McWiliams/Redi SGS Eddy Parameterization}
2    \label{sec:pkg:gmredi}
3    \begin{rawhtml}
4    <!-- CMIREDIR:gmredi: -->
5    \end{rawhtml}
6    
7  There are two parts to the Redi/GM parameterization of geostrophic  There are two parts to the Redi/GM parameterization of geostrophic
8  eddies. The first aims to mix tracer properties along isentropes  eddies. The first aims to mix tracer properties along isentropes
# Line 31  and Redi scheme, substantial cancellatio Line 35  and Redi scheme, substantial cancellatio
35  that the horizontal fluxes are unmodified from the lateral diffusion  that the horizontal fluxes are unmodified from the lateral diffusion
36  parameterization.  parameterization.
37    
38  \subsection{Redi scheme: Isopycnal diffusion}  \subsubsection{Redi scheme: Isopycnal diffusion}
39    
40  The Redi scheme diffuses tracers along isopycnals and introduces a  The Redi scheme diffuses tracers along isopycnals and introduces a
41  term in the tendency (rhs) of such a tracer (here $\tau$) of the form:  term in the tendency (rhs) of such a tracer (here $\tau$) of the form:
# Line 71  S_x & S_y & |S|^2 \\ Line 75  S_x & S_y & |S|^2 \\
75  \end{equation}  \end{equation}
76    
77    
78  \subsection{GM parameterization}  \subsubsection{GM parameterization}
79    
80  The GM parameterization aims to parameterise the ``advective'' or  The GM parameterization aims to parameterise the ``advective'' or
81  ``transport'' effect of geostrophic eddies by means of a ``bolus''  ``transport'' effect of geostrophic eddies by means of a ``bolus''
# Line 100  F_y & = & \kappa_{GM} S_y Line 104  F_y & = & \kappa_{GM} S_y
104  This is the form of the GM parameterization as applied by Donabasaglu,  This is the form of the GM parameterization as applied by Donabasaglu,
105  1997, in MOM versions 1 and 2.  1997, in MOM versions 1 and 2.
106    
107  \subsection{Griffies Skew Flux}  \subsubsection{Griffies Skew Flux}
108    
109  Griffies notes that the discretisation of bolus velocities involves  Griffies notes that the discretisation of bolus velocities involves
110  multiple layers of differencing and interpolation that potentially  multiple layers of differencing and interpolation that potentially
# Line 167  In the instance that $\kappa_{GM} = \kap Line 171  In the instance that $\kappa_{GM} = \kap
171  \end{array}  \end{array}
172  \right)  \right)
173  \end{equation}  \end{equation}
174  which differs from the variable laplacian diffusion tensor by only  which differs from the variable Laplacian diffusion tensor by only
175  two non-zero elements in the $z$-row.  two non-zero elements in the $z$-row.
176    
177  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
# Line 187  $S_y$: {\bf SlopeY} (argument on exit) Line 191  $S_y$: {\bf SlopeY} (argument on exit)
191    
192    
193    
194  \subsection{Variable $\kappa_{GM}$}  \subsubsection{Variable $\kappa_{GM}$}
195    
196  Visbeck et al., 1996, suggest making the eddy coefficient,  Visbeck et al., 1996, suggest making the eddy coefficient,
197  $\kappa_{GM}$, a function of the Eady growth rate,  $\kappa_{GM}$, a function of the Eady growth rate,
# Line 213  Substituting into the formula for $\kapp Line 217  Substituting into the formula for $\kapp
217  \end{displaymath}  \end{displaymath}
218    
219    
220  \subsection{Tapering and stability}  \subsubsection{Tapering and stability}
221    
222  Experience with the GFDL model showed that the GM scheme has to be  Experience with the GFDL model showed that the GM scheme has to be
223  matched to the convective parameterization. This was originally  matched to the convective parameterization. This was originally
224  expressed in connection with the introduction of the KPP boundary  expressed in connection with the introduction of the KPP boundary
225  layer scheme (Large et al., 97) but infact, as subsequent experience  layer scheme (Large et al., 97) but in fact, as subsequent experience
226  with the MIT model has found, is necessary for any convective  with the MIT model has found, is necessary for any convective
227  parameterization.  parameterization.
228    
# Line 240  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum Line 244  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
244  \begin{center}  \begin{center}
245  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}
246  \end{center}  \end{center}
247  \caption{Taper functions used in GKW91 and DM95.}  \caption{Taper functions used in GKW99 and DM95.}
248  \label{fig:tapers}  \label{fig:tapers}
249  \end{figure}  \end{figure}
250    
# Line 254  slope clipping, GKW91 limiting and DM95 Line 258  slope clipping, GKW91 limiting and DM95
258  \end{figure}  \end{figure}
259    
260    
261  \subsubsection{Slope clipping}  Slope clipping:
262    
263  Deep convection sites and the mixed layer are indicated by  Deep convection sites and the mixed layer are indicated by
264  homogenized, unstable or nearly unstable stratification. The slopes in  homogenized, unstable or nearly unstable stratification. The slopes in
265  such regions can be either infinite, very large with a sign reversal  such regions can be either infinite, very large with a sign reversal
266  or simply very large. From a numerical point of view, large slopes  or simply very large. From a numerical point of view, large slopes
267  lead to large variations in the tensor elements (implying large bolus  lead to large variations in the tensor elements (implying large bolus
268  flow) and can be numerically unstable. This was first reognized by  flow) and can be numerically unstable. This was first recognized by
269  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing
270  tensor. Here, the slope magnitude is simply restricted by an upper  tensor. Here, the slope magnitude is simply restricted by an upper
271  limit:  limit:
# Line 296  a) using the GM scheme with clipping and Line 300  a) using the GM scheme with clipping and
300  diffusion). The classic result of dramatically reduced mixed layers is  diffusion). The classic result of dramatically reduced mixed layers is
301  evident. Indeed, the deep convection sites to just one or two points  evident. Indeed, the deep convection sites to just one or two points
302  each and are much shallower than we might prefer. This, it turns out,  each and are much shallower than we might prefer. This, it turns out,
303  is due to the over zealous restratification due to the bolus transport  is due to the over zealous re-stratification due to the bolus transport
304  parameterization. Limiting the slopes also breaks the adiabatic nature  parameterization. Limiting the slopes also breaks the adiabatic nature
305  of the GM/Redi parameterization, re-introducing diabatic fluxes in  of the GM/Redi parameterization, re-introducing diabatic fluxes in
306  regions where the limiting is in effect.  regions where the limiting is in effect.
307    
308  \subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}  Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991:
309    
310  The tapering scheme used in Gerdes et al., 1991, (\cite{gkw91})  The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})
311  addressed two issues with the clipping method: the introduction of  addressed two issues with the clipping method: the introduction of
312  large vertical fluxes in addition to convective adjustment fluxes is  large vertical fluxes in addition to convective adjustment fluxes is
313  avoided by tapering the GM/Redi slopes back to zero in  avoided by tapering the GM/Redi slopes back to zero in
# Line 325  that the effective vertical diffusivity Line 329  that the effective vertical diffusivity
329  The GKW tapering scheme is activated in the model by setting {\bf  The GKW tapering scheme is activated in the model by setting {\bf
330  GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.  GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.
331    
332  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}  \subsubsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
333    
334  The tapering scheme used by Danabasoglu and McWilliams, 1995,  The tapering scheme used by Danabasoglu and McWilliams, 1995,
335  \cite{DM95}, followed a similar procedure but used a different  \cite{dm:95}, followed a similar procedure but used a different
336  tapering function, $f_1(S)$:  tapering function, $f_1(S)$:
337  \begin{equation}  \begin{equation}
338  f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right)  f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right)
# Line 342  slopes. Line 346  slopes.
346  The DM tapering scheme is activated in the model by setting {\bf  The DM tapering scheme is activated in the model by setting {\bf
347  GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.  GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.
348    
349  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}  \subsubsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}
350    
351  The tapering used in Large et al., 1997, \cite{ldd97}, is based on the  The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the
352  DM95 tapering scheme, but also tapers the scheme with an additional  DM95 tapering scheme, but also tapers the scheme with an additional
353  function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are  function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are
354  reduced near the surface:  reduced near the surface:
# Line 362  GM\_tap\-er\_scheme = 'ldd97'} in {\em d Line 366  GM\_tap\-er\_scheme = 'ldd97'} in {\em d
366    
367    
368  \begin{figure}  \begin{figure}
369    \begin{center}
370  %\includegraphics{mixedlayer-cox.eps}  %\includegraphics{mixedlayer-cox.eps}
371  %\includegraphics{mixedlayer-diff.eps}  %\includegraphics{mixedlayer-diff.eps}
372    Figure missing.
373    \end{center}
374  \caption{Mixed layer depth using GM parameterization with a) Cox slope  \caption{Mixed layer depth using GM parameterization with a) Cox slope
375  clipping and for comparison b) using horizontal constant diffusion.}  clipping and for comparison b) using horizontal constant diffusion.}
376  \ref{fig-mixedlayer}  \label{fig-mixedlayer}
377  \end{figure}  \end{figure}
378    
379    \subsubsection{Package Reference}
380    % DO NOT EDIT HERE
381    
382    
383    
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