| 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} |
| 174 |
|
{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F}) |
| 175 |
|
|
| 176 |
|
$\sigma_x$: {\bf SlopeX} (argument on entry) |
| 177 |
|
|
| 178 |
|
$\sigma_y$: {\bf SlopeY} (argument on entry) |
| 179 |
|
|
| 180 |
|
$\sigma_z$: {\bf SlopeY} (argument) |
| 181 |
|
|
| 182 |
|
$S_x$: {\bf SlopeX} (argument on exit) |
| 183 |
|
|
| 184 |
|
$S_y$: {\bf SlopeY} (argument on exit) |
| 185 |
|
|
| 186 |
|
\end{minipage} } |
| 187 |
|
|
| 188 |
|
|
| 189 |
|
|
| 190 |
\subsection{Variable $\kappa_{GM}$} |
\subsection{Variable $\kappa_{GM}$} |
| 191 |
|
|
| 192 |
Visbeck et al., 1996, suggest making the eddy coefficient, |
Visbeck et al., 1996, suggest making the eddy coefficient, |
| 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 |
|
|
| 236 |
|
|
| 237 |
\end{minipage} } |
\end{minipage} } |
| 238 |
|
|
| 239 |
|
\begin{figure} |
| 240 |
|
\begin{center} |
| 241 |
|
\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
| 242 |
|
\end{center} |
| 243 |
|
\caption{Taper functions used in GKW91 and DM95.} |
| 244 |
|
\label{fig:tapers} |
| 245 |
|
\end{figure} |
| 246 |
|
|
| 247 |
|
\begin{figure} |
| 248 |
|
\begin{center} |
| 249 |
|
\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}} |
| 250 |
|
\end{center} |
| 251 |
|
\caption{Effective slope as a function of ``true'' slope using Cox |
| 252 |
|
slope clipping, GKW91 limiting and DM95 limiting.} |
| 253 |
|
\label{fig:effective_slopes} |
| 254 |
|
\end{figure} |
| 255 |
|
|
| 256 |
|
|
| 257 |
\subsubsection{Slope clipping} |
\subsubsection{Slope clipping} |
| 258 |
|
|
| 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: |
| 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. |
| 360 |
|
|
| 361 |
|
|
| 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} |
|
\end{figure} |
|
|
|
|
|
\begin{figure} |
|
|
%\includegraphics{slopelimits.eps} |
|
|
\caption{Effective slope as a function of ``true'' slope using a) Cox |
|
|
slope clipping, b) GKW91 limiting, c) DM95 limiting and d) LDD97 |
|
|
limiting.} |
|
| 373 |
\end{figure} |
\end{figure} |
| 374 |
|
|
| 375 |
|
|
|
%\begin{figure} |
|
|
%\includegraphics{coxslope.eps} |
|
|
%\includegraphics{gkw91slope.eps} |
|
|
%\includegraphics{dm95slope.eps} |
|
|
%\includegraphics{ldd97slope.eps} |
|
|
%\caption{Effective slope magnitude at 100~m depth evaluated using a) |
|
|
%Cox slope clipping, b) GKW91 limiting, c) DM95 limiting and d) LDD97 |
|
|
%limiting.} |
|
|
%\end{figure} |
|
|
|
|
|
\section{Discretisation and code} |
|
|
|
|
|
This is the old documentation.....has to be brought upto date with MITgcm. |
|
|
|
|
|
|
|
|
The Gent-McWilliams-Redi parameterization is implemented through the |
|
|
package ``gmredi''. There are two necessary calls to ``gmredi'' |
|
|
routines other than initialization; 1) to calculate the slope tensor |
|
|
as a function of the current model state ({\bf gmredi\_calc\_tensor}) |
|
|
and 2) evaluation of the lateral and vertical fluxes due to gradients |
|
|
along isopycnals or bolus transport ({\bf gmredi\_xtransport}, {\bf |
|
|
gmredi\_ytransport} and {\bf gmredi-rtransport}). |
|
|
|
|
|
Each element of the tensor is discretised to be adiabatic and so that |
|
|
there would be no flux if the gmredi operator is applied to buoyancy. |
|
|
To acheive this we have to consider both these constraints for each |
|
|
row of the tensor, each row corresponding to a 'u', 'v' or 'w' point |
|
|
on the model grid. |
|
|
|
|
|
The code that implements the Redi/GM/Griffies schemes involves an |
|
|
original core routine {\bf inc\_tracer()} that is used to calculate |
|
|
the tendency in the tracers (namely, salt and potential temperature) |
|
|
and a new routine {\bf RediTensor()} that calculates the tensor |
|
|
components and $\kappa_{GM}$. |
|
|
|
|
|
\subsection{subroutine RediTensor()} |
|
|
|
|
|
{\small |
|
|
\begin{verbatim} |
|
|
subroutine RediTensor(Temp,Salt,Kredigm,K31,K32,K33, nIter,DumpFlag) |
|
|
|---in--| |-------out-------| |
|
|
! Input |
|
|
real Temp(Nx,Ny,Nz) ! Potential temperature |
|
|
real Salt(Nx,Ny,Nz) ! Salinity |
|
|
! Output |
|
|
real Kredigm(Nx,Ny,Nz) ! Redi/GM eddy coefficient |
|
|
real K31(Nx,Ny,Nz) ! Redi/GM (3,1) tensor component |
|
|
real K32(Nx,Ny,Nz) ! Redi/GM (3,2) tensor component |
|
|
real K33(Nx,Ny,Nz) ! Redi/GM (3,3) tensor component |
|
|
! Auxiliary input |
|
|
integer nIter ! interation/time-step number |
|
|
logical DumpFlag ! flag to indicate routine should ``dump'' |
|
|
\end{verbatim} |
|
|
} |
|
|
|
|
|
The subroutine {\bf RediTensor()} is called from {\bf model()} with |
|
|
input arguments $T$ and $S$. It returns the 3D-arrays {\tt Kredigm}, |
|
|
{\t K31}, {\tt K32} and {\tt K33} which represent $\kappa_{GM}$ (at |
|
|
$T/S$ points) and the three components of the bottom row in the |
|
|
Redi/GM tensor; $2 S_x$, $2 S_y$ and $|S|^2$ respectively, all at $W$ |
|
|
points. |
|
|
|
|
|
The discretisations and algorithm within {\bf RediTensor()} are as |
|
|
follows. The routine first calculates the locally reference potential |
|
|
density $\sigma_\theta$ from $T$ and $S$ and calculates the potential |
|
|
density gradients in subroutine {\bf gradSigma()}: |
|
|
|
|
|
\centerline{\begin{tabular}{ccl} |
|
|
& & \\ |
|
|
Array & Grid-point & Definition \\ |
|
|
{\tt SigX} & U & |
|
|
$\sigma_x = \frac{1}{\Delta x} \delta_x \sigma|_{z(k)}$ |
|
|
\\ |
|
|
{\tt SigY} & V & |
|
|
$\sigma_y = \frac{1}{\Delta y} \delta_y \sigma|_{z(k)}$ |
|
|
\\ |
|
|
{\tt SigZ} & W & |
|
|
$\sigma_z = \frac{1}{\Delta z} |
|
|
[ \sigma|_{z(k)}(k-1/2) - \sigma|_{z(k)}(k+1/2) ]$ |
|
|
\\ |
|
|
\\ |
|
|
\end{tabular}} |
|
|
|
|
|
Note that $\sigma_z$ is the static stability because the potential |
|
|
densities are referenced to the same reference level ($W$-level). |
|
|
|
|
|
The next step calculates the three tensor components {\tt K13}, {\tt |
|
|
K23} and {\tt K33} in subroutine {\bf KtensorWface()}. First, the |
|
|
lateral gradients $\sigma_x$ and $\sigma_y$ are interpolated to the |
|
|
$W$ points and stored in intermediate variables: |
|
|
\begin{eqnarray*} |
|
|
\mbox{\tt Sx} & = & \overline{ \overline{ \sigma_x }^x }^z \\ |
|
|
\mbox{\tt Sy} & = & \overline{ \overline{ \sigma_y }^y }^z |
|
|
\end{eqnarray*} |
|
|
Next, the magnitude of ${\bf \nabla}_z \sigma$ is stored in an intermediate |
|
|
variable: |
|
|
\begin{displaymath} |
|
|
\mbox{\tt Sxy2} = \sqrt{ {\tt Sx}^2 + {\tt Sy}^2 } |
|
|
\end{displaymath} |
|
|
The stratification ($\sigma_z$) is ``checked'' such that the slope |
|
|
vector has magnitude less than or equal to {\tt Smax} and stored in |
|
|
an intermediate variable: |
|
|
\begin{displaymath} |
|
|
\mbox{\tt Sz} = \max ( \sigma_z , - \mbox{\tt Sxy2/Smax} ) |
|
|
\end{displaymath} |
|
|
This guarantees stability and at the same time retains the lateral |
|
|
orientation of the slope vector. The tensor components are then calculated: |
|
|
\begin{eqnarray*} |
|
|
\mbox{\tt K13} & = & -2 {\tt Sx/Sz} \\ |
|
|
\mbox{\tt K23} & = & -2 {\tt Sx/Sz} \\ |
|
|
\mbox{\tt K33} & = & ({\tt Sx/Sz})^2 + ({\tt Sy/Sz})^2 |
|
|
\end{eqnarray*} |
|
|
|
|
|
Finally, {\tt Kredigm} ($\kappa_{GM}$) is calculated in subroutine |
|
|
{\bf GMRediCoefficient()}. First, all the gradients are interpolated |
|
|
to the $T/S$ points and stored in intermediate variables: |
|
|
\begin{eqnarray*} |
|
|
\mbox{\tt Sx} & = & \overline{ \sigma_x }^x \\ |
|
|
\mbox{\tt Sy} & = & \overline{ \sigma_y }^y \\ |
|
|
\mbox{\tt Sz} & = & \overline{ \sigma_z }^z |
|
|
\end{eqnarray*} |
|
|
Again, a nominal stratification is found by ``check'' the magnitude of |
|
|
the slope vector but here is converted to a Brunt-Vasala frequency: |
|
|
\begin{eqnarray*} |
|
|
{\tt M2} & = & \sqrt{ {\tt Sx}^2 + {\tt Sy}^2} \\ |
|
|
{\tt N2} & = & - \frac{g}{\rho_o} \max ( {\tt Sz} , -{\tt M2 / Smax} |
|
|
\end{eqnarray*} |
|
|
The magnitude of the slope is then $|S| = {\tt M2}/{\tt N2}$. The Eady |
|
|
growth rate is defined as $|f|/\sqrt(Ri) = |S| N$ and is calculated |
|
|
as: |
|
|
\begin{displaymath} |
|
|
{\tt FrRi} = \frac{\tt M2}{\tt N2} ( - \frac{g}{\rho} {\tt Sz} ) |
|
|
\end{displaymath} |
|
|
The Eady growth rate is then averaged over the upper layers (about |
|
|
1100m) and $\kappa_{GM}$ specified from this 2D-variable: |
|
|
\begin{displaymath} |
|
|
{\tt Kredigm} = 0.02 * (200d3 **2) * {\tt FrRi} |
|
|
\end{displaymath} |
|
|
|
|
|
\subsection{subroutine inc\_tracer()} |
|
|
|
|
|
{\bf inc\-tracer()} is called from {\bf model()} and has {\em four |
|
|
new} arguments: |
|
|
\begin{verbatim} |
|
|
subroutine inc_tracer( ...,Kredigm,K31,K32,K33, ... ) |
|
|
real Kredigm(Nx,Ny,Nz) ! Eddy coefficient |
|
|
real K31(Nx,Ny,Nz) ! (3,1) tensor coefficient |
|
|
real K32(Nx,Ny,Nz) ! (3,2) tensor coefficient |
|
|
real K33(Nx,Ny,Nz) ! (3,3) tensor coefficient |
|
|
\end{verbatim} |
|
|
|
|
|
Within the routine, the lateral fluxes, {\tt fluxWest} and {\tt |
|
|
fluxSouth}, in the Redi/GM/Griffies scheme are very similar to the |
|
|
conventional horizontal diffusion terms except that the diffusion |
|
|
coefficient is a function of space and must be interpolated from the |
|
|
$T/S$ points: |
|
|
\begin{eqnarray*} |
|
|
{\tt fluxWest}(\tau) & = & \ldots + |
|
|
\overline{\tt Kredigm}^x \partial_x \tau \\ |
|
|
{\tt fluxSouth}(\tau) & = & \ldots + |
|
|
\overline{\tt Kredigm}^y \partial_y \tau |
|
|
\end{eqnarray*} |
|
|
|
|
|
The Redi/GM/Griffies scheme adds three terms to the vertical flux |
|
|
({\tt fluxUpper}) in the tracer equation. It is discretise simply: |
|
|
\begin{displaymath} |
|
|
{\tt fluxUpper}(\tau) = \ldots + \overline{\tt Kredigm}^z |
|
|
\left( |
|
|
{\tt K13} \overline{\partial_x \tau}^{xz} + |
|
|
{\tt K23} \overline{\partial_y \tau}^{yz} + |
|
|
{\tt K33} \partial_z \tau |
|
|
\right) |
|
|
\end{displaymath} |
|
|
On boundaries, {\tt fluxUpper} is set to zero. |
|
| 376 |
|
|
| 377 |
|
|
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