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1  \section{Gent/McWiliams/Redi SGS Eddy parameterization}  \section{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 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 218  Substituting into the formula for $\kapp Line 222  Substituting into the formula for $\kapp
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 261  homogenized, unstable or nearly unstable Line 265  homogenized, unstable or nearly unstable
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}  \subsubsection{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 328  GM\_tap\-er\_scheme = 'gkw91'} in {\em d Line 332  GM\_tap\-er\_scheme = 'gkw91'} in {\em d
332  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}  \subsection{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 344  GM\_tap\-er\_scheme = 'dm95'} in {\em da Line 348  GM\_tap\-er\_scheme = 'dm95'} in {\em da
348    
349  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}  \subsection{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    \subsection{Package Reference}
380    % DO NOT EDIT HERE
381    
382    
383    
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