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revision 1.2 by adcroft, Fri Sep 28 14:09:56 2001 UTC revision 1.5 by adcroft, Tue Nov 13 19:01:42 2001 UTC
# Line 167  In the instance that $\kappa_{GM} = \kap Line 167  In the instance that $\kappa_{GM} = \kap
167  \end{array}  \end{array}
168  \right)  \right)
169  \end{equation}  \end{equation}
170  which differs from the variable laplacian diffusion tensor by only  which differs from the variable Laplacian diffusion tensor by only
171  two non-zero elements in the $z$-row.  two non-zero elements in the $z$-row.
172    
173  \fbox{ \begin{minipage}{4.75in}  \fbox{ \begin{minipage}{4.75in}
# Line 218  Substituting into the formula for $\kapp Line 218  Substituting into the formula for $\kapp
218  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
219  matched to the convective parameterization. This was originally  matched to the convective parameterization. This was originally
220  expressed in connection with the introduction of the KPP boundary  expressed in connection with the introduction of the KPP boundary
221  layer scheme (Large et al., 97) but infact, as subsequent experience  layer scheme (Large et al., 97) but in fact, as subsequent experience
222  with the MIT model has found, is necessary for any convective  with the MIT model has found, is necessary for any convective
223  parameterization.  parameterization.
224    
# Line 240  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum Line 240  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
240  \begin{center}  \begin{center}
241  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}
242  \end{center}  \end{center}
243  \caption{Taper functions used in GKW91 and DM95.}  \caption{Taper functions used in GKW99 and DM95.}
244  \label{fig:tapers}  \label{fig:tapers}
245  \end{figure}  \end{figure}
246    
# Line 261  homogenized, unstable or nearly unstable Line 261  homogenized, unstable or nearly unstable
261  such regions can be either infinite, very large with a sign reversal  such regions can be either infinite, very large with a sign reversal
262  or simply very large. From a numerical point of view, large slopes  or simply very large. From a numerical point of view, large slopes
263  lead to large variations in the tensor elements (implying large bolus  lead to large variations in the tensor elements (implying large bolus
264  flow) and can be numerically unstable. This was first reognized by  flow) and can be numerically unstable. This was first recognized by
265  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing
266  tensor. Here, the slope magnitude is simply restricted by an upper  tensor. Here, the slope magnitude is simply restricted by an upper
267  limit:  limit:
# Line 296  a) using the GM scheme with clipping and Line 296  a) using the GM scheme with clipping and
296  diffusion). The classic result of dramatically reduced mixed layers is  diffusion). The classic result of dramatically reduced mixed layers is
297  evident. Indeed, the deep convection sites to just one or two points  evident. Indeed, the deep convection sites to just one or two points
298  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,
299  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
300  parameterization. Limiting the slopes also breaks the adiabatic nature  parameterization. Limiting the slopes also breaks the adiabatic nature
301  of the GM/Redi parameterization, re-introducing diabatic fluxes in  of the GM/Redi parameterization, re-introducing diabatic fluxes in
302  regions where the limiting is in effect.  regions where the limiting is in effect.
303    
304  \subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}  \subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
305    
306  The tapering scheme used in Gerdes et al., 1991, (\cite{gkw91})  The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})
307  addressed two issues with the clipping method: the introduction of  addressed two issues with the clipping method: the introduction of
308  large vertical fluxes in addition to convective adjustment fluxes is  large vertical fluxes in addition to convective adjustment fluxes is
309  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 328  GM\_tap\-er\_scheme = 'gkw91'} in {\em d
328  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}  \subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
329    
330  The tapering scheme used by Danabasoglu and McWilliams, 1995,  The tapering scheme used by Danabasoglu and McWilliams, 1995,
331  \cite{DM95}, followed a similar procedure but used a different  \cite{dm:95}, followed a similar procedure but used a different
332  tapering function, $f_1(S)$:  tapering function, $f_1(S)$:
333  \begin{equation}  \begin{equation}
334  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 344  GM\_tap\-er\_scheme = 'dm95'} in {\em da
344    
345  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}  \subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}
346    
347  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
348  DM95 tapering scheme, but also tapers the scheme with an additional  DM95 tapering scheme, but also tapers the scheme with an additional
349  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
350  reduced near the surface:  reduced near the surface:
# Line 362  GM\_tap\-er\_scheme = 'ldd97'} in {\em d Line 362  GM\_tap\-er\_scheme = 'ldd97'} in {\em d
362    
363    
364  \begin{figure}  \begin{figure}
365    \begin{center}
366  %\includegraphics{mixedlayer-cox.eps}  %\includegraphics{mixedlayer-cox.eps}
367  %\includegraphics{mixedlayer-diff.eps}  %\includegraphics{mixedlayer-diff.eps}
368    Figure missing.
369    \end{center}
370  \caption{Mixed layer depth using GM parameterization with a) Cox slope  \caption{Mixed layer depth using GM parameterization with a) Cox slope
371  clipping and for comparison b) using horizontal constant diffusion.}  clipping and for comparison b) using horizontal constant diffusion.}
372  \ref{fig-mixedlayer}  \label{fig-mixedlayer}
373  \end{figure}  \end{figure}
374    
375    
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