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\section{Gent/McWiliams/Redi SGS Eddy parameterization} |
\section{Gent/McWiliams/Redi SGS Eddy parameterization} |
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\begin{rawhtml} |
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<!-- CMIREDIR:gmredi: --> |
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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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\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} |
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{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F}) |
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$\sigma_x$: {\bf SlopeX} (argument on entry) |
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$\sigma_y$: {\bf SlopeY} (argument on entry) |
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$\sigma_z$: {\bf SlopeY} (argument) |
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$S_x$: {\bf SlopeX} (argument on exit) |
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$S_y$: {\bf SlopeY} (argument on exit) |
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\end{minipage} } |
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\subsection{Variable $\kappa_{GM}$} |
\subsection{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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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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\end{minipage} } |
\end{minipage} } |
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\begin{figure} |
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\begin{center} |
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\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
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\end{center} |
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\caption{Taper functions used in GKW99 and DM95.} |
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\label{fig:tapers} |
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\end{figure} |
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\begin{figure} |
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\begin{center} |
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\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}} |
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\end{center} |
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\caption{Effective slope as a function of ``true'' slope using Cox |
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slope clipping, GKW91 limiting and DM95 limiting.} |
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\label{fig:effective_slopes} |
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\end{figure} |
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\subsubsection{Slope clipping} |
\subsubsection{Slope clipping} |
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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} |
\subsubsection{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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\subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
\subsection{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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\subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
\subsection{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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\begin{figure} |
\subsection{Package Reference} |
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%\includegraphics{slopelimits.eps} |
% DO NOT EDIT HERE |
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\caption{Effective slope as a function of ``true'' slope using a) Cox |
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slope clipping, b) GKW91 limiting, c) DM95 limiting and d) LDD97 |
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limiting.} |
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\end{figure} |
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%\begin{figure} |
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%\includegraphics{coxslope.eps} |
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%\includegraphics{gkw91slope.eps} |
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%\includegraphics{dm95slope.eps} |
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%\includegraphics{ldd97slope.eps} |
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%\caption{Effective slope magnitude at 100~m depth evaluated using a) |
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%Cox slope clipping, b) GKW91 limiting, c) DM95 limiting and d) LDD97 |
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%limiting.} |
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%\end{figure} |
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\section{Discretisation and code} |
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This is the old documentation.....has to be brought upto date with MITgcm. |
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The Gent-McWilliams-Redi parameterization is implemented through the |
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package ``gmredi''. There are two necessary calls to ``gmredi'' |
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routines other than initialization; 1) to calculate the slope tensor |
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as a function of the current model state ({\bf gmredi\_calc\_tensor}) |
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and 2) evaluation of the lateral and vertical fluxes due to gradients |
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along isopycnals or bolus transport ({\bf gmredi\_xtransport}, {\bf |
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gmredi\_ytransport} and {\bf gmredi-rtransport}). |
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Each element of the tensor is discretised to be adiabatic and so that |
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there would be no flux if the gmredi operator is applied to buoyancy. |
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To acheive this we have to consider both these constraints for each |
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row of the tensor, each row corresponding to a 'u', 'v' or 'w' point |
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on the model grid. |
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The code that implements the Redi/GM/Griffies schemes involves an |
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original core routine {\bf inc\_tracer()} that is used to calculate |
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the tendency in the tracers (namely, salt and potential temperature) |
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and a new routine {\bf RediTensor()} that calculates the tensor |
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components and $\kappa_{GM}$. |
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\subsection{subroutine RediTensor()} |
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{\small |
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\begin{verbatim} |
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subroutine RediTensor(Temp,Salt,Kredigm,K31,K32,K33, nIter,DumpFlag) |
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|---in--| |-------out-------| |
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! Input |
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real Temp(Nx,Ny,Nz) ! Potential temperature |
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real Salt(Nx,Ny,Nz) ! Salinity |
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! Output |
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real Kredigm(Nx,Ny,Nz) ! Redi/GM eddy coefficient |
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real K31(Nx,Ny,Nz) ! Redi/GM (3,1) tensor component |
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real K32(Nx,Ny,Nz) ! Redi/GM (3,2) tensor component |
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real K33(Nx,Ny,Nz) ! Redi/GM (3,3) tensor component |
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! Auxiliary input |
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integer nIter ! interation/time-step number |
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logical DumpFlag ! flag to indicate routine should ``dump'' |
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\end{verbatim} |
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} |
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The subroutine {\bf RediTensor()} is called from {\bf model()} with |
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input arguments $T$ and $S$. It returns the 3D-arrays {\tt Kredigm}, |
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{\t K31}, {\tt K32} and {\tt K33} which represent $\kappa_{GM}$ (at |
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$T/S$ points) and the three components of the bottom row in the |
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Redi/GM tensor; $2 S_x$, $2 S_y$ and $|S|^2$ respectively, all at $W$ |
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points. |
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The discretisations and algorithm within {\bf RediTensor()} are as |
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follows. The routine first calculates the locally reference potential |
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density $\sigma_\theta$ from $T$ and $S$ and calculates the potential |
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density gradients in subroutine {\bf gradSigma()}: |
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\centerline{\begin{tabular}{ccl} |
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& & \\ |
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Array & Grid-point & Definition \\ |
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{\tt SigX} & U & |
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$\sigma_x = \frac{1}{\Delta x} \delta_x \sigma|_{z(k)}$ |
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\\ |
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{\tt SigY} & V & |
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$\sigma_y = \frac{1}{\Delta y} \delta_y \sigma|_{z(k)}$ |
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\\ |
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{\tt SigZ} & W & |
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$\sigma_z = \frac{1}{\Delta z} |
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[ \sigma|_{z(k)}(k-1/2) - \sigma|_{z(k)}(k+1/2) ]$ |
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\\ |
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\\ |
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\end{tabular}} |
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Note that $\sigma_z$ is the static stability because the potential |
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densities are referenced to the same reference level ($W$-level). |
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The next step calculates the three tensor components {\tt K13}, {\tt |
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K23} and {\tt K33} in subroutine {\bf KtensorWface()}. First, the |
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lateral gradients $\sigma_x$ and $\sigma_y$ are interpolated to the |
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$W$ points and stored in intermediate variables: |
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\begin{eqnarray*} |
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\mbox{\tt Sx} & = & \overline{ \overline{ \sigma_x }^x }^z \\ |
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\mbox{\tt Sy} & = & \overline{ \overline{ \sigma_y }^y }^z |
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\end{eqnarray*} |
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Next, the magnitude of ${\bf \nabla}_z \sigma$ is stored in an intermediate |
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variable: |
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\begin{displaymath} |
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\mbox{\tt Sxy2} = \sqrt{ {\tt Sx}^2 + {\tt Sy}^2 } |
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\end{displaymath} |
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The stratification ($\sigma_z$) is ``checked'' such that the slope |
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vector has magnitude less than or equal to {\tt Smax} and stored in |
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an intermediate variable: |
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\begin{displaymath} |
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\mbox{\tt Sz} = \max ( \sigma_z , - \mbox{\tt Sxy2/Smax} ) |
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\end{displaymath} |
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This guarantees stability and at the same time retains the lateral |
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orientation of the slope vector. The tensor components are then calculated: |
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\begin{eqnarray*} |
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\mbox{\tt K13} & = & -2 {\tt Sx/Sz} \\ |
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\mbox{\tt K23} & = & -2 {\tt Sx/Sz} \\ |
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\mbox{\tt K33} & = & ({\tt Sx/Sz})^2 + ({\tt Sy/Sz})^2 |
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\end{eqnarray*} |
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Finally, {\tt Kredigm} ($\kappa_{GM}$) is calculated in subroutine |
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{\bf GMRediCoefficient()}. First, all the gradients are interpolated |
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to the $T/S$ points and stored in intermediate variables: |
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\begin{eqnarray*} |
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\mbox{\tt Sx} & = & \overline{ \sigma_x }^x \\ |
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\mbox{\tt Sy} & = & \overline{ \sigma_y }^y \\ |
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\mbox{\tt Sz} & = & \overline{ \sigma_z }^z |
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\end{eqnarray*} |
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Again, a nominal stratification is found by ``check'' the magnitude of |
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the slope vector but here is converted to a Brunt-Vasala frequency: |
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\begin{eqnarray*} |
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{\tt M2} & = & \sqrt{ {\tt Sx}^2 + {\tt Sy}^2} \\ |
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{\tt N2} & = & - \frac{g}{\rho_o} \max ( {\tt Sz} , -{\tt M2 / Smax} |
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\end{eqnarray*} |
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The magnitude of the slope is then $|S| = {\tt M2}/{\tt N2}$. The Eady |
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growth rate is defined as $|f|/\sqrt(Ri) = |S| N$ and is calculated |
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as: |
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\begin{displaymath} |
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{\tt FrRi} = \frac{\tt M2}{\tt N2} ( - \frac{g}{\rho} {\tt Sz} ) |
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\end{displaymath} |
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The Eady growth rate is then averaged over the upper layers (about |
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1100m) and $\kappa_{GM}$ specified from this 2D-variable: |
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\begin{displaymath} |
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{\tt Kredigm} = 0.02 * (200d3 **2) * {\tt FrRi} |
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\end{displaymath} |
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\subsection{subroutine inc\_tracer()} |
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{\bf inc\-tracer()} is called from {\bf model()} and has {\em four |
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new} arguments: |
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\begin{verbatim} |
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subroutine inc_tracer( ...,Kredigm,K31,K32,K33, ... ) |
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real Kredigm(Nx,Ny,Nz) ! Eddy coefficient |
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real K31(Nx,Ny,Nz) ! (3,1) tensor coefficient |
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real K32(Nx,Ny,Nz) ! (3,2) tensor coefficient |
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real K33(Nx,Ny,Nz) ! (3,3) tensor coefficient |
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\end{verbatim} |
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Within the routine, the lateral fluxes, {\tt fluxWest} and {\tt |
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fluxSouth}, in the Redi/GM/Griffies scheme are very similar to the |
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conventional horizontal diffusion terms except that the diffusion |
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coefficient is a function of space and must be interpolated from the |
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$T/S$ points: |
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\begin{eqnarray*} |
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{\tt fluxWest}(\tau) & = & \ldots + |
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\overline{\tt Kredigm}^x \partial_x \tau \\ |
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{\tt fluxSouth}(\tau) & = & \ldots + |
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\overline{\tt Kredigm}^y \partial_y \tau |
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\end{eqnarray*} |
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The Redi/GM/Griffies scheme adds three terms to the vertical flux |
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({\tt fluxUpper}) in the tracer equation. It is discretise simply: |
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\begin{displaymath} |
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{\tt fluxUpper}(\tau) = \ldots + \overline{\tt Kredigm}^z |
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\left( |
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{\tt K13} \overline{\partial_x \tau}^{xz} + |
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{\tt K23} \overline{\partial_y \tau}^{yz} + |
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{\tt K33} \partial_z \tau |
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\right) |
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\end{displaymath} |
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On boundaries, {\tt fluxUpper} is set to zero. |
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