| 1 |
\subsection{GMREDI: Gent/McWiliams/Redi SGS Eddy Parameterization} |
\subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization} |
| 2 |
\label{sec:pkg:gmredi} |
\label{sec:pkg:gmredi} |
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\begin{rawhtml} |
\begin{rawhtml} |
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<!-- CMIREDIR:gmredi: --> |
<!-- CMIREDIR:gmredi: --> |
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\end{rawhtml} |
\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, the Redi scheme \citep{re82}, aims to mix tracer properties along isentropes |
| 9 |
(neutral surfaces) by means of a diffusion operator oriented along the |
(neutral surfaces) by means of a diffusion operator oriented along the |
| 10 |
local isentropic surface (Redi). The second part, adiabatically |
local isentropic surface. The second part, GM \citep{gen-mcw:90,gen-eta:95}, adiabatically |
| 11 |
re-arranges tracers through an advective flux where the advecting flow |
re-arranges tracers through an advective flux where the advecting flow |
| 12 |
is a function of slope of the isentropic surfaces (GM). |
is a function of slope of the isentropic surfaces. |
| 13 |
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|
| 14 |
The first GCM implementation of the Redi scheme was by Cox 1987 in the |
The first GCM implementation of the Redi scheme was by \cite{Cox87} in the |
| 15 |
GFDL ocean circulation model. The original approach failed to |
GFDL ocean circulation model. The original approach failed to |
| 16 |
distinguish between isopycnals and surfaces of locally referenced |
distinguish between isopycnals and surfaces of locally referenced |
| 17 |
potential density (now called neutral surfaces) which are proper |
potential density (now called neutral surfaces) which are proper |
| 26 |
to the boundary condition of zero value on upper and lower |
to the boundary condition of zero value on upper and lower |
| 27 |
boundaries. The horizontal bolus velocities are then the vertical |
boundaries. The horizontal bolus velocities are then the vertical |
| 28 |
derivative of these functions. Here in lies a problem highlighted by |
derivative of these functions. Here in lies a problem highlighted by |
| 29 |
Griffies et al., 1997: the bolus velocities involve multiple |
\cite{gretal:98}: the bolus velocities involve multiple |
| 30 |
derivatives on the potential density field, which can consequently |
derivatives on the potential density field, which can consequently |
| 31 |
give rise to noise. Griffies et al. point out that the GM bolus fluxes |
give rise to noise. Griffies et al. point out that the GM bolus fluxes |
| 32 |
can be identically written as a skew flux which involves fewer |
can be identically written as a skew flux which involves fewer |
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|
|
| 78 |
\subsubsection{GM parameterization} |
\subsubsection{GM parameterization} |
| 79 |
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|
| 80 |
The GM parameterization aims to parameterise the ``advective'' or |
The GM parameterization aims to represent the ``advective'' or |
| 81 |
``transport'' effect of geostrophic eddies by means of a ``bolus'' |
``transport'' effect of geostrophic eddies by means of a ``bolus'' |
| 82 |
velocity, $\bf{u}^*$. The divergence of this advective flux is added |
velocity, $\bf{u}^\star$. The divergence of this advective flux is added |
| 83 |
to the tracer tendency equation (on the rhs): |
to the tracer tendency equation (on the rhs): |
| 84 |
\begin{equation} |
\begin{equation} |
| 85 |
- \bf{\nabla} \cdot \tau \bf{u}^* |
- \bf{\nabla} \cdot \tau \bf{u}^\star |
| 86 |
\end{equation} |
\end{equation} |
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|
| 88 |
The bolus velocity is defined as: |
The bolus velocity $\bf{u}^\star$ is defined as the rotational of a |
| 89 |
\begin{eqnarray} |
streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$: |
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u^* & = & - \partial_z F_x \\ |
\begin{equation} |
| 91 |
v^* & = & - \partial_z F_y \\ |
\bf{u}^\star = \nabla \times \bf{F}^\star = |
| 92 |
w^* & = & \partial_x F_x + \partial_y F_y |
\left( \begin{array}{c} |
| 93 |
\end{eqnarray} |
- \partial_z F_y^\star \\ |
| 94 |
where $F_x$ and $F_y$ are stream-functions with boundary conditions |
+ \partial_z F_x^\star \\ |
| 95 |
$F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the |
\partial_x F_y^\star - \partial_y F_x^\star |
| 96 |
bolus velocity in terms of these stream-functions is that they are |
\end{array} \right), |
| 97 |
automatically non-divergent ($\partial_x u^* + \partial_y v^* = - |
\end{equation} |
| 98 |
\partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and |
and thus is automatically non-divergent. In the GM parameterization, the streamfunction is |
| 99 |
$F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$: |
specified in terms of the isoneutral slopes $S_x$ and $S_y$: |
| 100 |
\begin{eqnarray} |
\begin{eqnarray} |
| 101 |
F_x & = & \kappa_{GM} S_x \\ |
F_x^\star & = & -\kappa_{GM} S_y \\ |
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F_y & = & \kappa_{GM} S_y |
F_y^\star & = & \kappa_{GM} S_x |
| 103 |
\end{eqnarray} |
\end{eqnarray} |
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|
with boundary conditions $F_x^\star=F_y^\star=0$ on upper and lower boundaries. |
| 105 |
|
In the end, the bolus transport in the GM parameterization is given by: |
| 106 |
|
\begin{equation} |
| 107 |
|
\bf{u}^\star = \left( |
| 108 |
|
\begin{array}{c} |
| 109 |
|
u^\star \\ |
| 110 |
|
v^\star \\ |
| 111 |
|
w^\star |
| 112 |
|
\end{array} |
| 113 |
|
\right) = \left( |
| 114 |
|
\begin{array}{c} |
| 115 |
|
- \partial_z (\kappa_{GM} S_x) \\ |
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|
- \partial_z (\kappa_{GM} S_y) \\ |
| 117 |
|
\partial_x (\kappa_{GM} S_x) + \partial_y (\kappa_{GM} S_y) |
| 118 |
|
\end{array} |
| 119 |
|
\right) |
| 120 |
|
\end{equation} |
| 121 |
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|
| 122 |
This is the form of the GM parameterization as applied by Donabasaglu, |
This is the form of the GM parameterization as applied by Donabasaglu, |
| 123 |
1997, in MOM versions 1 and 2. |
1997, in MOM versions 1 and 2. |
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|
| 125 |
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Note that in the MITgcm, the variables containing the GM bolus streamfunction are: |
| 126 |
|
\begin{equation} |
| 127 |
|
\left( |
| 128 |
|
\begin{array}{c} |
| 129 |
|
GM\_PsiX \\ |
| 130 |
|
GM\_PsiY |
| 131 |
|
\end{array} |
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|
\right) = \left( |
| 133 |
|
\begin{array}{c} |
| 134 |
|
\kappa_{GM} S_x \\ |
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|
\kappa_{GM} S_y |
| 136 |
|
\end{array} |
| 137 |
|
\right)= \left( |
| 138 |
|
\begin{array}{c} |
| 139 |
|
F_y^\star \\ |
| 140 |
|
-F_x^\star |
| 141 |
|
\end{array} |
| 142 |
|
\right). |
| 143 |
|
\end{equation} |
| 144 |
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|
| 145 |
\subsubsection{Griffies Skew Flux} |
\subsubsection{Griffies Skew Flux} |
| 146 |
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|
| 147 |
Griffies notes that the discretisation of bolus velocities involves |
\cite{gr:98} notes that the discretisation of bolus velocities involves |
| 148 |
multiple layers of differencing and interpolation that potentially |
multiple layers of differencing and interpolation that potentially |
| 149 |
lead to noisy fields and computational modes. He pointed out that the |
lead to noisy fields and computational modes. He pointed out that the |
| 150 |
bolus flux can be re-written in terms of a non-divergent flux and a |
bolus flux can be re-written in terms of a non-divergent flux and a |
| 151 |
skew-flux: |
skew-flux: |
| 152 |
\begin{eqnarray*} |
\begin{eqnarray*} |
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\bf{u}^* \tau |
\bf{u}^\star \tau |
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& = & |
& = & |
| 155 |
\left( \begin{array}{c} |
\left( \begin{array}{c} |
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- \partial_z ( \kappa_{GM} S_x ) \tau \\ |
- \partial_z ( \kappa_{GM} S_x ) \tau \\ |
| 162 |
\left( \begin{array}{c} |
\left( \begin{array}{c} |
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- \partial_z ( \kappa_{GM} S_x \tau) \\ |
- \partial_z ( \kappa_{GM} S_x \tau) \\ |
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- \partial_z ( \kappa_{GM} S_y \tau) \\ |
- \partial_z ( \kappa_{GM} S_y \tau) \\ |
| 165 |
\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y) \tau) |
\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y \tau) |
| 166 |
\end{array} \right) |
\end{array} \right) |
| 167 |
+ \left( \begin{array}{c} |
+ \left( \begin{array}{c} |
| 168 |
\kappa_{GM} S_x \partial_z \tau \\ |
\kappa_{GM} S_x \partial_z \tau \\ |
| 169 |
\kappa_{GM} S_y \partial_z \tau \\ |
\kappa_{GM} S_y \partial_z \tau \\ |
| 170 |
- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y) \partial_y \tau |
- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y \partial_y \tau |
| 171 |
\end{array} \right) |
\end{array} \right) |
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\end{eqnarray*} |
\end{eqnarray*} |
| 173 |
The first vector is non-divergent and thus has no effect on the tracer |
The first vector is non-divergent and thus has no effect on the tracer |
| 174 |
field and can be dropped. The remaining flux can be written: |
field and can be dropped. The remaining flux can be written: |
| 175 |
\begin{equation} |
\begin{equation} |
| 176 |
\bf{u}^* \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau |
\bf{u}^\star \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau |
| 177 |
\end{equation} |
\end{equation} |
| 178 |
where |
where |
| 179 |
\begin{equation} |
\begin{equation} |
| 195 |
with the Redi isoneutral mixing scheme: |
with the Redi isoneutral mixing scheme: |
| 196 |
\begin{equation} |
\begin{equation} |
| 197 |
\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau |
\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau |
| 198 |
- u^* \tau = |
- u^\star \tau = |
| 199 |
( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau |
( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau |
| 200 |
\end{equation} |
\end{equation} |
| 201 |
In the instance that $\kappa_{GM} = \kappa_{\rho}$ then |
In the instance that $\kappa_{GM} = \kappa_{\rho}$ then |
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|
| 232 |
\subsubsection{Variable $\kappa_{GM}$} |
\subsubsection{Variable $\kappa_{GM}$} |
| 233 |
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|
| 234 |
Visbeck et al., 1996, suggest making the eddy coefficient, |
\cite{visbeck:97} suggest making the eddy coefficient, |
| 235 |
$\kappa_{GM}$, a function of the Eady growth rate, |
$\kappa_{GM}$, a function of the Eady growth rate, |
| 236 |
$|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant, |
$|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant, |
| 237 |
$\alpha$, and a length-scale $L$: |
$\alpha$, and a length-scale $L$: |
| 260 |
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 |
| 261 |
matched to the convective parameterization. This was originally |
matched to the convective parameterization. This was originally |
| 262 |
expressed in connection with the introduction of the KPP boundary |
expressed in connection with the introduction of the KPP boundary |
| 263 |
layer scheme (Large et al., 97) but in fact, as subsequent experience |
layer scheme \citep{lar-eta:94} but in fact, as subsequent experience |
| 264 |
with the MIT model has found, is necessary for any convective |
with the MIT model has found, is necessary for any convective |
| 265 |
parameterization. |
parameterization. |
| 266 |
|
|
| 280 |
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|
| 281 |
\begin{figure} |
\begin{figure} |
| 282 |
\begin{center} |
\begin{center} |
| 283 |
\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}} |
\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}} |
| 284 |
\end{center} |
\end{center} |
| 285 |
\caption{Taper functions used in GKW99 and DM95.} |
\caption{Taper functions used in GKW91 and DM95.} |
| 286 |
\label{fig:tapers} |
\label{fig:tapers} |
| 287 |
\end{figure} |
\end{figure} |
| 288 |
|
|
| 289 |
\begin{figure} |
\begin{figure} |
| 290 |
\begin{center} |
\begin{center} |
| 291 |
\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}} |
\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/effective_slopes.eps}} |
| 292 |
\end{center} |
\end{center} |
| 293 |
\caption{Effective slope as a function of ``true'' slope using Cox |
\caption{Effective slope as a function of ``true'' slope using Cox |
| 294 |
slope clipping, GKW91 limiting and DM95 limiting.} |
slope clipping, GKW91 limiting and DM95 limiting.} |
| 296 |
\end{figure} |
\end{figure} |
| 297 |
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|
| 298 |
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|
| 299 |
Slope clipping: |
\subsubsection*{Slope clipping} |
| 300 |
|
|
| 301 |
Deep convection sites and the mixed layer are indicated by |
Deep convection sites and the mixed layer are indicated by |
| 302 |
homogenized, unstable or nearly unstable stratification. The slopes in |
homogenized, unstable or nearly unstable stratification. The slopes in |
| 304 |
or simply very large. From a numerical point of view, large slopes |
or simply very large. From a numerical point of view, large slopes |
| 305 |
lead to large variations in the tensor elements (implying large bolus |
lead to large variations in the tensor elements (implying large bolus |
| 306 |
flow) and can be numerically unstable. This was first recognized by |
flow) and can be numerically unstable. This was first recognized by |
| 307 |
Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing |
\cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing |
| 308 |
tensor. Here, the slope magnitude is simply restricted by an upper |
tensor. Here, the slope magnitude is simply restricted by an upper |
| 309 |
limit: |
limit: |
| 310 |
\begin{eqnarray} |
\begin{eqnarray} |
| 343 |
of the GM/Redi parameterization, re-introducing diabatic fluxes in |
of the GM/Redi parameterization, re-introducing diabatic fluxes in |
| 344 |
regions where the limiting is in effect. |
regions where the limiting is in effect. |
| 345 |
|
|
| 346 |
Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991: |
\subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991} |
| 347 |
|
|
| 348 |
The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99}) |
The tapering scheme used in \cite{gkw:91} |
| 349 |
addressed two issues with the clipping method: the introduction of |
addressed two issues with the clipping method: the introduction of |
| 350 |
large vertical fluxes in addition to convective adjustment fluxes is |
large vertical fluxes in addition to convective adjustment fluxes is |
| 351 |
avoided by tapering the GM/Redi slopes back to zero in |
avoided by tapering the GM/Redi slopes back to zero in |
| 364 |
that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 = |
that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 = |
| 365 |
\kappa S_{max}^2$. |
\kappa S_{max}^2$. |
| 366 |
|
|
| 367 |
The GKW tapering scheme is activated in the model by setting {\bf |
The GKW91 tapering scheme is activated in the model by setting {\bf |
| 368 |
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}. |
| 369 |
|
|
| 370 |
\subsubsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
\subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995} |
| 371 |
|
|
| 372 |
The tapering scheme used by Danabasoglu and McWilliams, 1995, |
The tapering scheme used by \cite{dm:95} followed a similar procedure but used a different |
|
\cite{dm:95}, followed a similar procedure but used a different |
|
| 373 |
tapering function, $f_1(S)$: |
tapering function, $f_1(S)$: |
| 374 |
\begin{equation} |
\begin{equation} |
| 375 |
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) |
| 380 |
cut-off, turning off the GM/Redi SGS parameterization for weaker |
cut-off, turning off the GM/Redi SGS parameterization for weaker |
| 381 |
slopes. |
slopes. |
| 382 |
|
|
| 383 |
The DM tapering scheme is activated in the model by setting {\bf |
The DM95 tapering scheme is activated in the model by setting {\bf |
| 384 |
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}. |
| 385 |
|
|
| 386 |
\subsubsection{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
\subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997} |
| 387 |
|
|
| 388 |
The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the |
The tapering used in \cite{ldd:97} is based on the |
| 389 |
DM95 tapering scheme, but also tapers the scheme with an additional |
DM95 tapering scheme, but also tapers the scheme with an additional |
| 390 |
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 |
| 391 |
reduced near the surface: |
reduced near the surface: |
| 392 |
\begin{equation} |
\begin{equation} |
| 393 |
f_2(S) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \pi/2)\right) |
f_2(z) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \frac{\pi}{2})\right) |
| 394 |
\end{equation} |
\end{equation} |
| 395 |
where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with |
where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with |
| 396 |
$c=2$~m~s$^{-1}$. This tapering with height was introduced to fix |
$c=2$~m~s$^{-1}$. This tapering with height was introduced to fix |
| 397 |
some spurious interaction with the mixed-layer KPP parameterization. |
some spurious interaction with the mixed-layer KPP parameterization. |
| 398 |
|
|
| 399 |
The LDD tapering scheme is activated in the model by setting {\bf |
The LDD97 tapering scheme is activated in the model by setting {\bf |
| 400 |
GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}. |
GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}. |
| 401 |
|
|
| 402 |
|
|
| 414 |
\end{figure} |
\end{figure} |
| 415 |
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| 416 |
\subsubsection{Package Reference} |
\subsubsection{Package Reference} |
| 417 |
|
\label{sec:pkg:gmredi:diagnostics} |
| 418 |
|
|
| 419 |
|
{\footnotesize |
| 420 |
|
\begin{verbatim} |
| 421 |
|
------------------------------------------------------------------------ |
| 422 |
|
<-Name->|Levs|<-parsing code->|<-- Units -->|<- Tile (max=80c) |
| 423 |
|
------------------------------------------------------------------------ |
| 424 |
|
GM_VisbK| 1 |SM P M1 |m^2/s |Mixing coefficient from Visbeck etal parameterization |
| 425 |
|
GM_Kux | 15 |UU P 177MR |m^2/s |K_11 element (U.point, X.dir) of GM-Redi tensor |
| 426 |
|
GM_Kvy | 15 |VV P 176MR |m^2/s |K_22 element (V.point, Y.dir) of GM-Redi tensor |
| 427 |
|
GM_Kuz | 15 |UU 179MR |m^2/s |K_13 element (U.point, Z.dir) of GM-Redi tensor |
| 428 |
|
GM_Kvz | 15 |VV 178MR |m^2/s |K_23 element (V.point, Z.dir) of GM-Redi tensor |
| 429 |
|
GM_Kwx | 15 |UM 181LR |m^2/s |K_31 element (W.point, X.dir) of GM-Redi tensor |
| 430 |
|
GM_Kwy | 15 |VM 180LR |m^2/s |K_32 element (W.point, Y.dir) of GM-Redi tensor |
| 431 |
|
GM_Kwz | 15 |WM P LR |m^2/s |K_33 element (W.point, Z.dir) of GM-Redi tensor |
| 432 |
|
GM_PsiX | 15 |UU 184LR |m^2/s |GM Bolus transport stream-function : X component |
| 433 |
|
GM_PsiY | 15 |VV 183LR |m^2/s |GM Bolus transport stream-function : Y component |
| 434 |
|
GM_KuzTz| 15 |UU 186MR |degC.m^3/s |Redi Off-diagonal Tempetature flux: X component |
| 435 |
|
GM_KvzTz| 15 |VV 185MR |degC.m^3/s |Redi Off-diagonal Tempetature flux: Y component |
| 436 |
|
\end{verbatim} |
| 437 |
|
} |
| 438 |
|
|
| 439 |
|
\subsubsection{Experiments and tutorials that use gmredi} |
| 440 |
|
\label{sec:pkg:gmredi:experiments} |
| 441 |
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| 442 |
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\begin{itemize} |
| 443 |
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\item{Global Ocean tutorial, in tutorial\_global\_oce\_latlon verification directory, |
| 444 |
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described in section \ref{sec:eg-global} } |
| 445 |
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\item{ Front Relax experiment, in front\_relax verification directory.} |
| 446 |
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\item{ Ideal 2D Ocean experiment, in ideal\_2D\_oce verification directory.} |
| 447 |
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\end{itemize} |
| 448 |
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% DO NOT EDIT HERE |
% DO NOT EDIT HERE |
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