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1  \section{Gent/McWiliams/Redi SGS Eddy parameterization}  \section{Gent/McWiliams/Redi SGS Eddy parameterization}
2    \begin{rawhtml}
3    <!-- CMIREDIR:gmredi: -->
4    \end{rawhtml}
5    
6  There are two parts to the Redi/GM parameterization of geostrophic  There are two parts to the Redi/GM parameterization of geostrophic
7  eddies. The first aims to mix tracer properties along isentropes  eddies. The first aims to mix tracer properties along isentropes
# Line 167  In the instance that $\kappa_{GM} = \kap Line 170  In the instance that $\kappa_{GM} = \kap
170  \end{array}  \end{array}
171  \right)  \right)
172  \end{equation}  \end{equation}
173  which differs from the variable laplacian diffusion tensor by only  which differs from the variable Laplacian diffusion tensor by only
174  two non-zero elements in the $z$-row.  two non-zero elements in the $z$-row.
175    
176  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
# Line 218  Substituting into the formula for $\kapp Line 221  Substituting into the formula for $\kapp
221  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
222  matched to the convective parameterization. This was originally  matched to the convective parameterization. This was originally
223  expressed in connection with the introduction of the KPP boundary  expressed in connection with the introduction of the KPP boundary
224  layer scheme (Large et al., 97) but infact, as subsequent experience  layer scheme (Large et al., 97) but in fact, as subsequent experience
225  with the MIT model has found, is necessary for any convective  with the MIT model has found, is necessary for any convective
226  parameterization.  parameterization.
227    
# Line 240  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum Line 243  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
243  \begin{center}  \begin{center}
244  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}
245  \end{center}  \end{center}
246  \caption{Taper functions used in GKW91 and DM95.}  \caption{Taper functions used in GKW99 and DM95.}
247  \label{fig:tapers}  \label{fig:tapers}
248  \end{figure}  \end{figure}
249    
# Line 261  homogenized, unstable or nearly unstable Line 264  homogenized, unstable or nearly unstable
264  such regions can be either infinite, very large with a sign reversal  such regions can be either infinite, very large with a sign reversal
265  or simply very large. From a numerical point of view, large slopes  or simply very large. From a numerical point of view, large slopes
266  lead to large variations in the tensor elements (implying large bolus  lead to large variations in the tensor elements (implying large bolus
267  flow) and can be numerically unstable. This was first reognized by  flow) and can be numerically unstable. This was first recognized by
268  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing
269  tensor. Here, the slope magnitude is simply restricted by an upper  tensor. Here, the slope magnitude is simply restricted by an upper
270  limit:  limit:
# Line 296  a) using the GM scheme with clipping and Line 299  a) using the GM scheme with clipping and
299  diffusion). The classic result of dramatically reduced mixed layers is  diffusion). The classic result of dramatically reduced mixed layers is
300  evident. Indeed, the deep convection sites to just one or two points  evident. Indeed, the deep convection sites to just one or two points
301  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,
302  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
303  parameterization. Limiting the slopes also breaks the adiabatic nature  parameterization. Limiting the slopes also breaks the adiabatic nature
304  of the GM/Redi parameterization, re-introducing diabatic fluxes in  of the GM/Redi parameterization, re-introducing diabatic fluxes in
305  regions where the limiting is in effect.  regions where the limiting is in effect.
306    
307  \subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}  \subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
308    
309  The tapering scheme used in Gerdes et al., 1991, (\cite{gkw91})  The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})
310  addressed two issues with the clipping method: the introduction of  addressed two issues with the clipping method: the introduction of
311  large vertical fluxes in addition to convective adjustment fluxes is  large vertical fluxes in addition to convective adjustment fluxes is
312  avoided by tapering the GM/Redi slopes back to zero in  avoided by tapering the GM/Redi slopes back to zero in
# Line 328  GM\_tap\-er\_scheme = 'gkw91'} in {\em d Line 331  GM\_tap\-er\_scheme = 'gkw91'} in {\em d
331  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
332    
333  The tapering scheme used by Danabasoglu and McWilliams, 1995,  The tapering scheme used by Danabasoglu and McWilliams, 1995,
334  \cite{DM95}, followed a similar procedure but used a different  \cite{dm:95}, followed a similar procedure but used a different
335  tapering function, $f_1(S)$:  tapering function, $f_1(S)$:
336  \begin{equation}  \begin{equation}
337  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 344  GM\_tap\-er\_scheme = 'dm95'} in {\em da Line 347  GM\_tap\-er\_scheme = 'dm95'} in {\em da
347    
348  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}
349    
350  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
351  DM95 tapering scheme, but also tapers the scheme with an additional  DM95 tapering scheme, but also tapers the scheme with an additional
352  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
353  reduced near the surface:  reduced near the surface:
# Line 362  GM\_tap\-er\_scheme = 'ldd97'} in {\em d Line 365  GM\_tap\-er\_scheme = 'ldd97'} in {\em d
365    
366    
367  \begin{figure}  \begin{figure}
368    \begin{center}
369  %\includegraphics{mixedlayer-cox.eps}  %\includegraphics{mixedlayer-cox.eps}
370  %\includegraphics{mixedlayer-diff.eps}  %\includegraphics{mixedlayer-diff.eps}
371    Figure missing.
372    \end{center}
373  \caption{Mixed layer depth using GM parameterization with a) Cox slope  \caption{Mixed layer depth using GM parameterization with a) Cox slope
374  clipping and for comparison b) using horizontal constant diffusion.}  clipping and for comparison b) using horizontal constant diffusion.}
375  \ref{fig-mixedlayer}  \label{fig-mixedlayer}
376  \end{figure}  \end{figure}
377    
378    \subsection{Package Reference}
379    % DO NOT EDIT HERE
380    
381    
382    
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