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\section{Gent/McWiliams/Redi SGS Eddy parameterization} |
\subsection{GMREDI: Gent/McWiliams/Redi SGS Eddy Parameterization} |
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\label{sec:pkg:gmredi} |
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\begin{rawhtml} |
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\end{rawhtml} |
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There are two parts to the Redi/GM parameterization of geostrophic |
There are two parts to the Redi/GM parameterization of geostrophic |
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eddies. The first aims to mix tracer properties along isentropes |
eddies. The first aims to mix tracer properties along isentropes |
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that the horizontal fluxes are unmodified from the lateral diffusion |
that the horizontal fluxes are unmodified from the lateral diffusion |
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parameterization. |
parameterization. |
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\subsection{Redi scheme: Isopycnal diffusion} |
\subsubsection{Redi scheme: Isopycnal diffusion} |
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The Redi scheme diffuses tracers along isopycnals and introduces a |
The Redi scheme diffuses tracers along isopycnals and introduces a |
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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: |
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\end{equation} |
\end{equation} |
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\subsection{GM parameterization} |
\subsubsection{GM parameterization} |
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The GM parameterization aims to parameterise the ``advective'' or |
The GM parameterization aims to parameterise the ``advective'' or |
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``transport'' effect of geostrophic eddies by means of a ``bolus'' |
``transport'' effect of geostrophic eddies by means of a ``bolus'' |
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This is the form of the GM parameterization as applied by Donabasaglu, |
This is the form of the GM parameterization as applied by Donabasaglu, |
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1997, in MOM versions 1 and 2. |
1997, in MOM versions 1 and 2. |
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\subsection{Griffies Skew Flux} |
\subsubsection{Griffies Skew Flux} |
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Griffies notes that the discretisation of bolus velocities involves |
Griffies notes that the discretisation of bolus velocities involves |
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multiple layers of differencing and interpolation that potentially |
multiple layers of differencing and interpolation that potentially |
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\end{array} |
\end{array} |
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\right) |
\right) |
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\end{equation} |
\end{equation} |
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which differs from the variable laplacian diffusion tensor by only |
which differs from the variable Laplacian diffusion tensor by only |
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two non-zero elements in the $z$-row. |
two non-zero elements in the $z$-row. |
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\fbox{ \begin{minipage}{4.75in} |
\fbox{ \begin{minipage}{4.75in} |
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\subsection{Variable $\kappa_{GM}$} |
\subsubsection{Variable $\kappa_{GM}$} |
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Visbeck et al., 1996, suggest making the eddy coefficient, |
Visbeck et al., 1996, suggest making the eddy coefficient, |
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$\kappa_{GM}$, a function of the Eady growth rate, |
$\kappa_{GM}$, a function of the Eady growth rate, |
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\end{displaymath} |
\end{displaymath} |
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\subsection{Tapering and stability} |
\subsubsection{Tapering and stability} |
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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 |
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matched to the convective parameterization. This was originally |
matched to the convective parameterization. This was originally |
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expressed in connection with the introduction of the KPP boundary |
expressed in connection with the introduction of the KPP boundary |
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layer scheme (Large et al., 97) but infact, as subsequent experience |
layer scheme (Large et al., 97) but in fact, as subsequent experience |
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with the MIT model has found, is necessary for any convective |
with the MIT model has found, is necessary for any convective |
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parameterization. |
parameterization. |
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\begin{center} |
\begin{center} |
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\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
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\end{center} |
\end{center} |
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\caption{Taper functions used in GKW91 and DM95.} |
\caption{Taper functions used in GKW99 and DM95.} |
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\label{fig:tapers} |
\label{fig:tapers} |
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\end{figure} |
\end{figure} |
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\end{figure} |
\end{figure} |
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\subsubsection{Slope clipping} |
Slope clipping: |
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Deep convection sites and the mixed layer are indicated by |
Deep convection sites and the mixed layer are indicated by |
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homogenized, unstable or nearly unstable stratification. The slopes in |
homogenized, unstable or nearly unstable stratification. The slopes in |
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such regions can be either infinite, very large with a sign reversal |
such regions can be either infinite, very large with a sign reversal |
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or simply very large. From a numerical point of view, large slopes |
or simply very large. From a numerical point of view, large slopes |
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lead to large variations in the tensor elements (implying large bolus |
lead to large variations in the tensor elements (implying large bolus |
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flow) and can be numerically unstable. This was first reognized by |
flow) and can be numerically unstable. This was first recognized by |
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Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing |
Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing |
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tensor. Here, the slope magnitude is simply restricted by an upper |
tensor. Here, the slope magnitude is simply restricted by an upper |
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limit: |
limit: |
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diffusion). The classic result of dramatically reduced mixed layers is |
diffusion). The classic result of dramatically reduced mixed layers is |
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evident. Indeed, the deep convection sites to just one or two points |
evident. Indeed, the deep convection sites to just one or two points |
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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, |
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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 |
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parameterization. Limiting the slopes also breaks the adiabatic nature |
parameterization. Limiting the slopes also breaks the adiabatic nature |
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of the GM/Redi parameterization, re-introducing diabatic fluxes in |
of the GM/Redi parameterization, re-introducing diabatic fluxes in |
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regions where the limiting is in effect. |
regions where the limiting is in effect. |
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\subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991} |
Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991: |
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The tapering scheme used in Gerdes et al., 1991, (\cite{gkw91}) |
The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99}) |
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addressed two issues with the clipping method: the introduction of |
addressed two issues with the clipping method: the introduction of |
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large vertical fluxes in addition to convective adjustment fluxes is |
large vertical fluxes in addition to convective adjustment fluxes is |
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avoided by tapering the GM/Redi slopes back to zero in |
avoided by tapering the GM/Redi slopes back to zero in |
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The GKW tapering scheme is activated in the model by setting {\bf |
The GKW tapering scheme is activated in the model by setting {\bf |
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GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
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\subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
\subsubsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
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The tapering scheme used by Danabasoglu and McWilliams, 1995, |
The tapering scheme used by Danabasoglu and McWilliams, 1995, |
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\cite{DM95}, followed a similar procedure but used a different |
\cite{dm:95}, followed a similar procedure but used a different |
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tapering function, $f_1(S)$: |
tapering function, $f_1(S)$: |
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\begin{equation} |
\begin{equation} |
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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) |
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The DM tapering scheme is activated in the model by setting {\bf |
The DM tapering scheme is activated in the model by setting {\bf |
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GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
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\subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
\subsubsection{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
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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 |
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DM95 tapering scheme, but also tapers the scheme with an additional |
DM95 tapering scheme, but also tapers the scheme with an additional |
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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 |
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reduced near the surface: |
reduced near the surface: |
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\begin{figure} |
\begin{figure} |
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\begin{center} |
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%\includegraphics{mixedlayer-cox.eps} |
%\includegraphics{mixedlayer-cox.eps} |
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%\includegraphics{mixedlayer-diff.eps} |
%\includegraphics{mixedlayer-diff.eps} |
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Figure missing. |
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\end{center} |
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\caption{Mixed layer depth using GM parameterization with a) Cox slope |
\caption{Mixed layer depth using GM parameterization with a) Cox slope |
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clipping and for comparison b) using horizontal constant diffusion.} |
clipping and for comparison b) using horizontal constant diffusion.} |
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\ref{fig-mixedlayer} |
\label{fig-mixedlayer} |
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\end{figure} |
\end{figure} |
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\subsubsection{Package Reference} |
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