s_phys_pkgs/text/gmredi.tex

\subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization}
\label{sec:pkg:gmredi}
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There are two parts to the Redi/GM parameterization of geostrophic
eddies. The first, the Redi scheme \citep{re82}, aims to mix tracer properties along isentropes
(neutral surfaces) by means of a diffusion operator oriented along the
local isentropic surface. The second part, GM \citep{gen-mcw:90,gen-eta:95}, adiabatically
re-arranges tracers through an advective flux where the advecting flow
is a function of slope of the isentropic surfaces.

The first GCM implementation of the Redi scheme was by \cite{Cox87} in the
GFDL ocean circulation model. The original approach failed to
distinguish between isopycnals and surfaces of locally referenced
potential density (now called neutral surfaces) which are proper
isentropes for the ocean. As will be discussed later, it also appears
that the Cox implementation is susceptible  to a computational mode.
Due to this mode, the Cox scheme requires a background lateral
diffusion to be present to conserve the integrity of the model fields.

The GM parameterization was then added to the GFDL code in the form of
a non-divergent bolus velocity. The method defines two
stream-functions expressed in terms of the isoneutral slopes subject
to the boundary condition of zero value on upper and lower
boundaries. The horizontal bolus velocities are then the vertical
derivative of these functions. Here in lies a problem highlighted by
\cite{gretal:98}: the bolus velocities involve multiple
derivatives on the potential density field, which can consequently
give rise to noise. Griffies et al. point out that the GM bolus fluxes
can be identically written as a skew flux which involves fewer
differential operators. Further, combining the skew flux formulation
and Redi scheme, substantial cancellations take place to the point
that the horizontal fluxes are unmodified from the lateral diffusion
parameterization.

\subsubsection{Redi scheme: Isopycnal diffusion}

The Redi scheme diffuses tracers along isopycnals and introduces a
term in the tendency (rhs) of such a tracer (here $\tau$) of the form:
\begin{equation}
\bf{\nabla} \cdot \kappa_\rho \bf{K}_{Redi}  \bf{\nabla} \tau
\end{equation}
where $\kappa_\rho$ is the along isopycnal diffusivity and
$\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of
$\tau$ onto the isopycnal surface. The unapproximated projection tensor is:
\begin{equation}
\bf{K}_{Redi} = \left(
\begin{array}{ccc}
1 + S_x& S_x S_y & S_x \\
S_x S_y  & 1 + S_y & S_y \\
S_x & S_y & |S|^2 \\
\end{array}
\right)
\end{equation}
Here, $S_x = -\partial_x \sigma / \partial_z \sigma$ and $S_y =
-\partial_y \sigma / \partial_z \sigma$ are the components of the
isoneutral slope.

The first point to note is that a typical slope in the ocean interior
is small, say of the order $10^{-4}$. A maximum slope might be of
order $10^{-2}$ and only exceeds such in unstratified regions where
the slope is ill defined. It is therefore justifiable, and
customary, to make the small slope approximation, $|S| << 1$. The Redi
projection tensor then becomes:
\begin{equation}
\bf{K}_{Redi} = \left(
\begin{array}{ccc}
1 & 0 & S_x \\
0 & 1 & S_y \\
S_x & S_y & |S|^2 \\
\end{array}
\right)
\end{equation}


\subsubsection{GM parameterization}

The GM parameterization aims to represent the ``advective'' or
``transport'' effect of geostrophic eddies by means of a ``bolus''
velocity, $\bf{u}^\star$. The divergence of this advective flux is added
to the tracer tendency equation (on the rhs):
\begin{equation}
- \bf{\nabla} \cdot \tau \bf{u}^\star
\end{equation}

The bolus velocity $\bf{u}^\star$ is defined as the rotational of a
streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$:
\begin{equation}
\bf{u}^\star = \nabla \times \bf{F}^\star =
\left( \begin{array}{c}
- \partial_z  F_y^\star \\
+ \partial_z  F_x^\star \\
\partial_x F_y^\star - \partial_y F_x^\star
\end{array} \right),
\end{equation}
and thus is automatically non-divergent. In the GM parameterization, the streamfunction is
specified in terms of the isoneutral slopes $S_x$ and $S_y$:
\begin{eqnarray}
F_x^\star & = & -\kappa_{GM} S_y \\
F_y^\star & = &  \kappa_{GM} S_x
\end{eqnarray}
with boundary conditions $F_x^\star=F_y^\star=0$ on upper and lower boundaries.
In the end, the bolus transport in the GM parameterization is given by:
\begin{equation}
\bf{u}^\star = \left(
\begin{array}{c}
u^\star \\
v^\star \\
w^\star
\end{array}
\right) = \left(
\begin{array}{c}
- \partial_z (\kappa_{GM} S_x) \\
- \partial_z (\kappa_{GM} S_y) \\
\partial_x  (\kappa_{GM} S_x) + \partial_y (\kappa_{GM} S_y)
\end{array}
\right)
\end{equation}

This is the form of the GM parameterization as applied by Donabasaglu,
1997, in MOM versions 1 and 2.

Note that in the MITgcm, the variables containing the GM bolus streamfunction are:
\begin{equation}
\left(
\begin{array}{c}
GM\_PsiX \\
GM\_PsiY
\end{array}
\right) = \left(
\begin{array}{c}
\kappa_{GM} S_x \\
\kappa_{GM} S_y
\end{array}
\right)= \left(
\begin{array}{c}
F_y^\star \\
-F_x^\star
\end{array}
\right).
\end{equation}
  
\subsubsection{Griffies Skew Flux}

\cite{gr:98} notes that the discretisation of bolus velocities involves
multiple layers of differencing and interpolation that potentially
lead to noisy fields and computational modes. He pointed out that the
bolus flux can be re-written in terms of a non-divergent flux and a
skew-flux:
\begin{eqnarray*}
\bf{u}^\star \tau
& = &
\left( \begin{array}{c}
- \partial_z ( \kappa_{GM} S_x ) \tau \\
- \partial_z ( \kappa_{GM} S_y ) \tau \\
(\partial_x \kappa_{GM} S_x + \partial_y \kappa_{GM} S_y)\tau
\end{array} \right)
\\
& = &
\left( \begin{array}{c}
- \partial_z ( \kappa_{GM} S_x \tau) \\
- \partial_z ( \kappa_{GM} S_y \tau) \\
\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y \tau)
\end{array} \right)
+ \left( \begin{array}{c}
 \kappa_{GM} S_x \partial_z \tau \\
 \kappa_{GM} S_y \partial_z \tau \\
- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y \partial_y \tau
\end{array} \right)
\end{eqnarray*}
The first vector is non-divergent and thus has no effect on the tracer
field and can be dropped. The remaining flux can be written:
\begin{equation}
\bf{u}^\star \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau
\end{equation}
where
\begin{equation}
\bf{K}_{GM} =
\left(
\begin{array}{ccc}
0 & 0 & -S_x \\
0 & 0 & -S_y \\
S_x & S_y & 0
\end{array}
\right)
\end{equation}
is an anti-symmetric tensor.

This formulation of the GM parameterization involves fewer derivatives
than the original and also involves only terms that already appear in
the Redi mixing scheme. Indeed, a somewhat fortunate cancellation
becomes apparent when we use the GM parameterization in conjunction
with the Redi isoneutral mixing scheme:
\begin{equation}
\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
- u^\star \tau = 
( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau
\end{equation}
In the instance that $\kappa_{GM} = \kappa_{\rho}$ then
\begin{equation}
\kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} =
\kappa_\rho
\left( \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
2 S_x & 2 S_y & |S|^2 
\end{array}
\right)
\end{equation}
which differs from the variable Laplacian diffusion tensor by only
two non-zero elements in the $z$-row.

\fbox{ \begin{minipage}{4.75in}
{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F})

$\sigma_x$: {\bf SlopeX} (argument on entry)

$\sigma_y$: {\bf SlopeY} (argument on entry)

$\sigma_z$: {\bf SlopeY} (argument)

$S_x$: {\bf SlopeX} (argument on exit)

$S_y$: {\bf SlopeY} (argument on exit)

\end{minipage} }


\subsubsection{Variable $\kappa_{GM}$}

\cite{visbeck:97} suggest making the eddy coefficient,
$\kappa_{GM}$, a function of the Eady growth rate,
$|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant,
$\alpha$, and a length-scale $L$:
\begin{displaymath}
\kappa_{GM} = \alpha L^2 \overline{ \frac{|f|}{\sqrt{Ri}} }^z
\end{displaymath}
where the Eady growth rate has been depth averaged (indicated by the
over-line). A local Richardson number is defined $Ri = N^2 / (\partial
u/\partial z)^2$ which, when combined with thermal wind gives:
\begin{displaymath}
\frac{1}{Ri} = \frac{(\frac{\partial u}{\partial z})^2}{N^2} =
\frac{ ( \frac{g}{f \rho_o} | {\bf \nabla} \sigma | )^2 }{N^2} =
\frac{ M^4 }{ |f|^2 N^2 }
\end{displaymath}
where $M^2$ is defined $M^2 = \frac{g}{\rho_o} |{\bf \nabla} \sigma|$.
Substituting into the formula for $\kappa_{GM}$ gives:
\begin{displaymath}
\kappa_{GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z =
\alpha L^2 \overline{ \frac{M^2}{N^2} N }^z =
\alpha L^2 \overline{ |S| N }^z
\end{displaymath}


\subsubsection{Tapering and stability}

Experience with the GFDL model showed that the GM scheme has to be
matched to the convective parameterization. This was originally
expressed in connection with the introduction of the KPP boundary
layer scheme \citep{lar-eta:94} but in fact, as subsequent experience
with the MIT model has found, is necessary for any convective
parameterization.

\fbox{ \begin{minipage}{4.75in}
{\em S/R GMREDI\_SLOPE\_LIMIT} ({\em
pkg/gmredi/gmredi\_slope\_limit.F})

$\sigma_x, s_x$: {\bf SlopeX} (argument)

$\sigma_y, s_y$: {\bf SlopeY} (argument)

$\sigma_z$: {\bf dSigmadRReal} (argument)

$z_\sigma^{*}$: {\bf dRdSigmaLtd} (argument)

\end{minipage} }

\begin{figure}
\begin{center}
\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}}
\end{center}
\caption{Taper functions used in GKW91 and DM95.}
\label{fig:tapers}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/effective_slopes.eps}}
\end{center}
\caption{Effective slope as a function of ``true'' slope using Cox
slope clipping, GKW91 limiting and DM95 limiting.}
\label{fig:effective_slopes}
\end{figure}


\subsubsection*{Slope clipping}

Deep convection sites and the mixed layer are indicated by
homogenized, unstable or nearly unstable stratification. The slopes in
such regions can be either infinite, very large with a sign reversal
or simply very large. From a numerical point of view, large slopes
lead to large variations in the tensor elements (implying large bolus
flow) and can be numerically unstable. This was first recognized by
\cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing
tensor. Here, the slope magnitude is simply restricted by an upper
limit:
\begin{eqnarray}
|\nabla \sigma| & = & \sqrt{ \sigma_x^2 + \sigma_y^2 } \\
S_{lim} & = & - \frac{|\nabla \sigma|}{ S_{max} }
\;\;\;\;\;\;\;\; \mbox{where $S_{max}$ is a parameter} \\
\sigma_z^\star & = & \min( \sigma_z , S_{lim} ) \\
{[s_x,s_y]} & = & - \frac{ [\sigma_x,\sigma_y] }{\sigma_z^\star}
\end{eqnarray}
Notice that this algorithm assumes stable stratification through the
``min'' function. In the case where the fluid is well stratified ($\sigma_z < S_{lim}$) then the slopes evaluate to:
\begin{equation}
{[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z}
\end{equation}
while in the limited regions ($\sigma_z > S_{lim}$) the slopes become:
\begin{equation}
{[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{|\nabla \sigma|/S_{max}}
\end{equation}
so that the slope magnitude is limited $\sqrt{s_x^2 + s_y^2} =
S_{max}$.

The slope clipping scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'clipping'} in {\em data.gmredi}.

Even using slope clipping, it is normally the case that the vertical
diffusion term (with coefficient $\kappa_\rho{\bf K}_{33} =
\kappa_\rho S_{max}^2$) is large and must be time-stepped using an
implicit procedure (see section on discretisation and code later).
Fig. \ref{fig-mixedlayer} shows the mixed layer depth resulting from
a) using the GM scheme with clipping and b) no GM scheme (horizontal
diffusion). The classic result of dramatically reduced mixed layers is
evident. Indeed, the deep convection sites to just one or two points
each and are much shallower than we might prefer. This, it turns out,
is due to the over zealous re-stratification due to the bolus transport
parameterization. Limiting the slopes also breaks the adiabatic nature
of the GM/Redi parameterization, re-introducing diabatic fluxes in
regions where the limiting is in effect.

\subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}

The tapering scheme used in \cite{gkw:91}
addressed two issues with the clipping method: the introduction of
large vertical fluxes in addition to convective adjustment fluxes is
avoided by tapering the GM/Redi slopes back to zero in
low-stratification regions; the adjustment of slopes is replaced by a
tapering of the entire GM/Redi tensor. This means the direction of
fluxes is unaffected as the amplitude is scaled.

The scheme inserts a tapering function, $f_1(S)$, in front of the
GM/Redi tensor:
\begin{equation}
f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{|S|}\right)^2 \right]
\end{equation}
where $S_{max}$ is the maximum slope you want allowed. Where the
slopes, $|S|<S_{max}$ then $f_1(S) = 1$ and the tensor is un-tapered
but where $|S| \ge S_{max}$ then $f_1(S)$ scales down the tensor so
that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 =
\kappa S_{max}^2$.

The GKW91 tapering scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.

\subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}

The tapering scheme used by \cite{dm:95} followed a similar procedure but used a different
tapering function, $f_1(S)$:
\begin{equation}
f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right)
\end{equation}
where $S_c = 0.004$ is a cut-off slope and $S_d=0.001$ is a scale over
which the slopes are smoothly tapered. Functionally, the operates in
the same way as the GKW91 scheme but has a substantially lower
cut-off, turning off the GM/Redi SGS parameterization for weaker
slopes.

The DM95 tapering scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.

\subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997}

The tapering used in \cite{ldd:97} is based on the
DM95 tapering scheme, but also tapers the scheme with an additional
function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are
reduced near the surface:
\begin{equation}
f_2(z) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \frac{\pi}{2})\right)
\end{equation}
where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with
$c=2$~m~s$^{-1}$.  This tapering with height was introduced to fix
some spurious interaction with the mixed-layer KPP parameterization.

The LDD97 tapering scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.


\begin{figure}
\begin{center}
%\includegraphics{mixedlayer-cox.eps}
%\includegraphics{mixedlayer-diff.eps}
Figure missing.
\end{center}
\caption{Mixed layer depth using GM parameterization with a) Cox slope
clipping and for comparison b) using horizontal constant diffusion.}
\label{fig-mixedlayer}
\end{figure}

\subsubsection{Package Reference}
\label{sec:pkg:gmredi:diagnostics}

{\footnotesize
\begin{verbatim}
------------------------------------------------------------------------
<-Name->|Levs|<-parsing code->|<--  Units   -->|<- Tile (max=80c) 
------------------------------------------------------------------------
GM_VisbK|  1 |SM P    M1      |m^2/s           |Mixing coefficient from Visbeck etal parameterization
GM_Kux  | 15 |UU P 177MR      |m^2/s           |K_11 element (U.point, X.dir) of GM-Redi tensor
GM_Kvy  | 15 |VV P 176MR      |m^2/s           |K_22 element (V.point, Y.dir) of GM-Redi tensor
GM_Kuz  | 15 |UU   179MR      |m^2/s           |K_13 element (U.point, Z.dir) of GM-Redi tensor
GM_Kvz  | 15 |VV   178MR      |m^2/s           |K_23 element (V.point, Z.dir) of GM-Redi tensor
GM_Kwx  | 15 |UM   181LR      |m^2/s           |K_31 element (W.point, X.dir) of GM-Redi tensor
GM_Kwy  | 15 |VM   180LR      |m^2/s           |K_32 element (W.point, Y.dir) of GM-Redi tensor
GM_Kwz  | 15 |WM P    LR      |m^2/s           |K_33 element (W.point, Z.dir) of GM-Redi tensor
GM_PsiX | 15 |UU   184LR      |m^2/s           |GM Bolus transport stream-function : X component
GM_PsiY | 15 |VV   183LR      |m^2/s           |GM Bolus transport stream-function : Y component
GM_KuzTz| 15 |UU   186MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: X component
GM_KvzTz| 15 |VV   185MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: Y component
\end{verbatim}
}

\subsubsection{Experiments and tutorials that use gmredi}
\label{sec:pkg:gmredi:experiments}

\begin{itemize}
\item{Global Ocean tutorial, in tutorial\_global\_oce\_latlon verification directory,
described in section \ref{sec:eg-global} }
\item{ Front Relax experiment, in front\_relax verification directory.}
\item{ Ideal 2D Ocean experiment, in ideal\_2D\_oce verification directory.}
\end{itemize}

% DO NOT EDIT HERE


1	dfer	1.16	\subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization}
2	edhill	1.8	\label{sec:pkg:gmredi}
3	edhill	1.7	\begin{rawhtml}
4			<!-- CMIREDIR:gmredi: -->
5			\end{rawhtml}
6	adcroft	1.1
7			There are two parts to the Redi/GM parameterization of geostrophic
8	dfer	1.16	eddies. The first, the Redi scheme \citep{re82}, aims to mix tracer properties along isentropes
9	adcroft	1.1	(neutral surfaces) by means of a diffusion operator oriented along the
10	dfer	1.16	local isentropic surface. The second part, GM \citep{gen-mcw:90,gen-eta:95}, adiabatically
11	adcroft	1.1	re-arranges tracers through an advective flux where the advecting flow
12	dfer	1.16	is a function of slope of the isentropic surfaces.
13	adcroft	1.1
14	dfer	1.16	The first GCM implementation of the Redi scheme was by \cite{Cox87} in the
15	adcroft	1.1	GFDL ocean circulation model. The original approach failed to
16			distinguish between isopycnals and surfaces of locally referenced
17			potential density (now called neutral surfaces) which are proper
18			isentropes for the ocean. As will be discussed later, it also appears
19			that the Cox implementation is susceptible to a computational mode.
20			Due to this mode, the Cox scheme requires a background lateral
21			diffusion to be present to conserve the integrity of the model fields.
22
23			The GM parameterization was then added to the GFDL code in the form of
24			a non-divergent bolus velocity. The method defines two
25			stream-functions expressed in terms of the isoneutral slopes subject
26			to the boundary condition of zero value on upper and lower
27			boundaries. The horizontal bolus velocities are then the vertical
28			derivative of these functions. Here in lies a problem highlighted by
29	dfer	1.16	\cite{gretal:98}: the bolus velocities involve multiple
30	adcroft	1.1	derivatives on the potential density field, which can consequently
31			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
33			differential operators. Further, combining the skew flux formulation
34			and Redi scheme, substantial cancellations take place to the point
35			that the horizontal fluxes are unmodified from the lateral diffusion
36			parameterization.
37
38	molod	1.10	\subsubsection{Redi scheme: Isopycnal diffusion}
39	adcroft	1.1
40			The Redi scheme diffuses tracers along isopycnals and introduces a
41			term in the tendency (rhs) of such a tracer (here $\tau$) of the form:
42			\begin{equation}
43			\bf{\nabla} \cdot \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
44			\end{equation}
45			where $\kappa_\rho$ is the along isopycnal diffusivity and
46			$\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of
47			$\tau$ onto the isopycnal surface. The unapproximated projection tensor is:
48			\begin{equation}
49			\bf{K}_{Redi} = \left(
50			\begin{array}{ccc}
51			1 + S_x& S_x S_y & S_x \\
52			S_x S_y & 1 + S_y & S_y \\
53			S_x & S_y & \|S\|^2 \\
54			\end{array}
55			\right)
56			\end{equation}
57			Here, $S_x = -\partial_x \sigma / \partial_z \sigma$ and $S_y =
58			-\partial_y \sigma / \partial_z \sigma$ are the components of the
59			isoneutral slope.
60
61			The first point to note is that a typical slope in the ocean interior
62			is small, say of the order $10^{-4}$. A maximum slope might be of
63			order $10^{-2}$ and only exceeds such in unstratified regions where
64			the slope is ill defined. It is therefore justifiable, and
65			customary, to make the small slope approximation, $\|S\| << 1$. The Redi
66			projection tensor then becomes:
67			\begin{equation}
68			\bf{K}_{Redi} = \left(
69			\begin{array}{ccc}
70			1 & 0 & S_x \\
71			0 & 1 & S_y \\
72			S_x & S_y & \|S\|^2 \\
73			\end{array}
74			\right)
75			\end{equation}
76
77
78	molod	1.10	\subsubsection{GM parameterization}
79	adcroft	1.1
80	dfer	1.16	The GM parameterization aims to represent the ``advective'' or
81	adcroft	1.1	``transport'' effect of geostrophic eddies by means of a ``bolus''
82	dfer	1.16	velocity, $\bf{u}^\star$. The divergence of this advective flux is added
83	adcroft	1.1	to the tracer tendency equation (on the rhs):
84			\begin{equation}
85	dfer	1.16	- \bf{\nabla} \cdot \tau \bf{u}^\star
86	adcroft	1.1	\end{equation}
87
88	dfer	1.16	The bolus velocity $\bf{u}^\star$ is defined as the rotational of a
89			streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$:
90			\begin{equation}
91			\bf{u}^\star = \nabla \times \bf{F}^\star =
92			\left( \begin{array}{c}
93			- \partial_z F_y^\star \\
94			+ \partial_z F_x^\star \\
95			\partial_x F_y^\star - \partial_y F_x^\star
96			\end{array} \right),
97			\end{equation}
98			and thus is automatically non-divergent. In the GM parameterization, the streamfunction is
99			specified in terms of the isoneutral slopes $S_x$ and $S_y$:
100	adcroft	1.1	\begin{eqnarray}
101	dfer	1.16	F_x^\star & = & -\kappa_{GM} S_y \\
102			F_y^\star & = & \kappa_{GM} S_x
103	adcroft	1.1	\end{eqnarray}
104	dfer	1.16	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) \\
116			- \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
122	adcroft	1.1	This is the form of the GM parameterization as applied by Donabasaglu,
123			1997, in MOM versions 1 and 2.
124
125	dfer	1.16	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}
132			\right) = \left(
133			\begin{array}{c}
134			\kappa_{GM} S_x \\
135			\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
145	molod	1.10	\subsubsection{Griffies Skew Flux}
146	adcroft	1.1
147	dfer	1.16	\cite{gr:98} notes that the discretisation of bolus velocities involves
148	adcroft	1.1	multiple layers of differencing and interpolation that potentially
149			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
151			skew-flux:
152			\begin{eqnarray*}
153	dfer	1.16	\bf{u}^\star \tau
154	adcroft	1.1	& = &
155			\left( \begin{array}{c}
156			- \partial_z ( \kappa_{GM} S_x ) \tau \\
157			- \partial_z ( \kappa_{GM} S_y ) \tau \\
158			(\partial_x \kappa_{GM} S_x + \partial_y \kappa_{GM} S_y)\tau
159			\end{array} \right)
160			\\
161			& = &
162			\left( \begin{array}{c}
163			- \partial_z ( \kappa_{GM} S_x \tau) \\
164			- \partial_z ( \kappa_{GM} S_y \tau) \\
165	dfer	1.16	\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y \tau)
166	adcroft	1.1	\end{array} \right)
167			+ \left( \begin{array}{c}
168			\kappa_{GM} S_x \partial_z \tau \\
169			\kappa_{GM} S_y \partial_z \tau \\
170	dfer	1.16	- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y \partial_y \tau
171	adcroft	1.1	\end{array} \right)
172			\end{eqnarray*}
173			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:
175			\begin{equation}
176	dfer	1.16	\bf{u}^\star \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau
177	adcroft	1.1	\end{equation}
178			where
179			\begin{equation}
180			\bf{K}_{GM} =
181			\left(
182			\begin{array}{ccc}
183			0 & 0 & -S_x \\
184			0 & 0 & -S_y \\
185			S_x & S_y & 0
186			\end{array}
187			\right)
188			\end{equation}
189			is an anti-symmetric tensor.
190
191			This formulation of the GM parameterization involves fewer derivatives
192			than the original and also involves only terms that already appear in
193			the Redi mixing scheme. Indeed, a somewhat fortunate cancellation
194			becomes apparent when we use the GM parameterization in conjunction
195			with the Redi isoneutral mixing scheme:
196			\begin{equation}
197			\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
198	dfer	1.16	- u^\star \tau =
199	adcroft	1.1	( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau
200			\end{equation}
201			In the instance that $\kappa_{GM} = \kappa_{\rho}$ then
202			\begin{equation}
203			\kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} =
204			\kappa_\rho
205			\left( \begin{array}{ccc}
206			1 & 0 & 0 \\
207			0 & 1 & 0 \\
208			2 S_x & 2 S_y & \|S\|^2
209			\end{array}
210			\right)
211			\end{equation}
212	cnh	1.3	which differs from the variable Laplacian diffusion tensor by only
213	adcroft	1.1	two non-zero elements in the $z$-row.
214
215	adcroft	1.2	\fbox{ \begin{minipage}{4.75in}
216			{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F})
217
218			$\sigma_x$: {\bf SlopeX} (argument on entry)
219
220			$\sigma_y$: {\bf SlopeY} (argument on entry)
221
222			$\sigma_z$: {\bf SlopeY} (argument)
223
224			$S_x$: {\bf SlopeX} (argument on exit)
225
226			$S_y$: {\bf SlopeY} (argument on exit)
227
228			\end{minipage} }
229
230
231
232	molod	1.10	\subsubsection{Variable $\kappa_{GM}$}
233	adcroft	1.1
234	dfer	1.16	\cite{visbeck:97} suggest making the eddy coefficient,
235	adcroft	1.1	$\kappa_{GM}$, a function of the Eady growth rate,
236			$\|f\|/\sqrt{Ri}$. The formula involves a non-dimensional constant,
237			$\alpha$, and a length-scale $L$:
238			\begin{displaymath}
239			\kappa_{GM} = \alpha L^2 \overline{ \frac{\|f\|}{\sqrt{Ri}} }^z
240			\end{displaymath}
241			where the Eady growth rate has been depth averaged (indicated by the
242			over-line). A local Richardson number is defined $Ri = N^2 / (\partial
243			u/\partial z)^2$ which, when combined with thermal wind gives:
244			\begin{displaymath}
245			\frac{1}{Ri} = \frac{(\frac{\partial u}{\partial z})^2}{N^2} =
246			\frac{ ( \frac{g}{f \rho_o} \| {\bf \nabla} \sigma \| )^2 }{N^2} =
247			\frac{ M^4 }{ \|f\|^2 N^2 }
248			\end{displaymath}
249			where $M^2$ is defined $M^2 = \frac{g}{\rho_o} \|{\bf \nabla} \sigma\|$.
250			Substituting into the formula for $\kappa_{GM}$ gives:
251			\begin{displaymath}
252			\kappa_{GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z =
253			\alpha L^2 \overline{ \frac{M^2}{N^2} N }^z =
254			\alpha L^2 \overline{ \|S\| N }^z
255			\end{displaymath}
256
257
258	molod	1.10	\subsubsection{Tapering and stability}
259	adcroft	1.1
260			Experience with the GFDL model showed that the GM scheme has to be
261			matched to the convective parameterization. This was originally
262			expressed in connection with the introduction of the KPP boundary
263	dfer	1.16	layer scheme \citep{lar-eta:94} but in fact, as subsequent experience
264	adcroft	1.1	with the MIT model has found, is necessary for any convective
265			parameterization.
266
267			\fbox{ \begin{minipage}{4.75in}
268			{\em S/R GMREDI\_SLOPE\_LIMIT} ({\em
269			pkg/gmredi/gmredi\_slope\_limit.F})
270
271			$\sigma_x, s_x$: {\bf SlopeX} (argument)
272
273			$\sigma_y, s_y$: {\bf SlopeY} (argument)
274
275			$\sigma_z$: {\bf dSigmadRReal} (argument)
276
277			$z_\sigma^{*}$: {\bf dRdSigmaLtd} (argument)
278
279			\end{minipage} }
280
281	adcroft	1.2	\begin{figure}
282			\begin{center}
283	jmc	1.14	\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}}
284	adcroft	1.2	\end{center}
285	dfer	1.16	\caption{Taper functions used in GKW91 and DM95.}
286	adcroft	1.2	\label{fig:tapers}
287			\end{figure}
288
289			\begin{figure}
290			\begin{center}
291	jmc	1.14	\resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/effective_slopes.eps}}
292	adcroft	1.2	\end{center}
293			\caption{Effective slope as a function of ``true'' slope using Cox
294			slope clipping, GKW91 limiting and DM95 limiting.}
295			\label{fig:effective_slopes}
296			\end{figure}
297
298	adcroft	1.1
299	dfer	1.16	\subsubsection*{Slope clipping}
300	adcroft	1.1
301			Deep convection sites and the mixed layer are indicated by
302			homogenized, unstable or nearly unstable stratification. The slopes in
303			such regions can be either infinite, very large with a sign reversal
304			or simply very large. From a numerical point of view, large slopes
305			lead to large variations in the tensor elements (implying large bolus
306	cnh	1.3	flow) and can be numerically unstable. This was first recognized by
307	dfer	1.16	\cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing
308	adcroft	1.1	tensor. Here, the slope magnitude is simply restricted by an upper
309			limit:
310			\begin{eqnarray}
311			\|\nabla \sigma\| & = & \sqrt{ \sigma_x^2 + \sigma_y^2 } \\
312			S_{lim} & = & - \frac{\|\nabla \sigma\|}{ S_{max} }
313			\;\;\;\;\;\;\;\; \mbox{where $S_{max}$ is a parameter} \\
314			\sigma_z^\star & = & \min( \sigma_z , S_{lim} ) \\
315			{[s_x,s_y]} & = & - \frac{ [\sigma_x,\sigma_y] }{\sigma_z^\star}
316			\end{eqnarray}
317			Notice that this algorithm assumes stable stratification through the
318			``min'' function. In the case where the fluid is well stratified ($\sigma_z < S_{lim}$) then the slopes evaluate to:
319			\begin{equation}
320			{[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z}
321			\end{equation}
322			while in the limited regions ($\sigma_z > S_{lim}$) the slopes become:
323			\begin{equation}
324			{[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{\|\nabla \sigma\|/S_{max}}
325			\end{equation}
326			so that the slope magnitude is limited $\sqrt{s_x^2 + s_y^2} =
327			S_{max}$.
328
329			The slope clipping scheme is activated in the model by setting {\bf
330			GM\_tap\-er\_scheme = 'clipping'} in {\em data.gmredi}.
331
332			Even using slope clipping, it is normally the case that the vertical
333			diffusion term (with coefficient $\kappa_\rho{\bf K}_{33} =
334			\kappa_\rho S_{max}^2$) is large and must be time-stepped using an
335			implicit procedure (see section on discretisation and code later).
336			Fig. \ref{fig-mixedlayer} shows the mixed layer depth resulting from
337			a) using the GM scheme with clipping and b) no GM scheme (horizontal
338			diffusion). The classic result of dramatically reduced mixed layers is
339			evident. Indeed, the deep convection sites to just one or two points
340			each and are much shallower than we might prefer. This, it turns out,
341	cnh	1.3	is due to the over zealous re-stratification due to the bolus transport
342	adcroft	1.1	parameterization. Limiting the slopes also breaks the adiabatic nature
343			of the GM/Redi parameterization, re-introducing diabatic fluxes in
344			regions where the limiting is in effect.
345
346	dfer	1.16	\subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
347	adcroft	1.1
348	dfer	1.16	The tapering scheme used in \cite{gkw:91}
349	adcroft	1.1	addressed two issues with the clipping method: the introduction of
350			large vertical fluxes in addition to convective adjustment fluxes is
351			avoided by tapering the GM/Redi slopes back to zero in
352			low-stratification regions; the adjustment of slopes is replaced by a
353			tapering of the entire GM/Redi tensor. This means the direction of
354			fluxes is unaffected as the amplitude is scaled.
355
356			The scheme inserts a tapering function, $f_1(S)$, in front of the
357			GM/Redi tensor:
358			\begin{equation}
359			f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{\|S\|}\right)^2 \right]
360			\end{equation}
361			where $S_{max}$ is the maximum slope you want allowed. Where the
362			slopes, $\|S\|<S_{max}$ then $f_1(S) = 1$ and the tensor is un-tapered
363			but where $\|S\| \ge S_{max}$ then $f_1(S)$ scales down the tensor so
364			that the effective vertical diffusivity term $\kappa f_1(S) \|S\|^2 =
365			\kappa S_{max}^2$.
366
367	dfer	1.16	The GKW91 tapering scheme is activated in the model by setting {\bf
368	adcroft	1.1	GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.
369
370	dfer	1.16	\subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
371	adcroft	1.1
372	dfer	1.16	The tapering scheme used by \cite{dm:95} followed a similar procedure but used a different
373	adcroft	1.1	tapering function, $f_1(S)$:
374			\begin{equation}
375			f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - \|S\|}{S_d} \right] \right)
376			\end{equation}
377			where $S_c = 0.004$ is a cut-off slope and $S_d=0.001$ is a scale over
378			which the slopes are smoothly tapered. Functionally, the operates in
379			the same way as the GKW91 scheme but has a substantially lower
380			cut-off, turning off the GM/Redi SGS parameterization for weaker
381			slopes.
382
383	dfer	1.16	The DM95 tapering scheme is activated in the model by setting {\bf
384	adcroft	1.1	GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.
385
386	dfer	1.16	\subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997}
387	adcroft	1.1
388	dfer	1.16	The tapering used in \cite{ldd:97} is based on the
389	adcroft	1.1	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
391			reduced near the surface:
392			\begin{equation}
393	dfer	1.16	f_2(z) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \frac{\pi}{2})\right)
394	adcroft	1.1	\end{equation}
395			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
397			some spurious interaction with the mixed-layer KPP parameterization.
398
399	dfer	1.16	The LDD97 tapering scheme is activated in the model by setting {\bf
400	adcroft	1.1	GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.
401
402
403
404	adcroft	1.2
405	adcroft	1.1	\begin{figure}
406	adcroft	1.4	\begin{center}
407	adcroft	1.1	%\includegraphics{mixedlayer-cox.eps}
408			%\includegraphics{mixedlayer-diff.eps}
409	adcroft	1.4	Figure missing.
410			\end{center}
411	adcroft	1.1	\caption{Mixed layer depth using GM parameterization with a) Cox slope
412			clipping and for comparison b) using horizontal constant diffusion.}
413	adcroft	1.4	\label{fig-mixedlayer}
414	adcroft	1.1	\end{figure}
415
416	molod	1.10	\subsubsection{Package Reference}
417	molod	1.11	\label{sec:pkg:gmredi:diagnostics}
418
419	edhill	1.12	{\footnotesize
420	molod	1.11	\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	edhill	1.12	}
438	molod	1.11
439	molod	1.13	\subsubsection{Experiments and tutorials that use gmredi}
440			\label{sec:pkg:gmredi:experiments}
441
442			\begin{itemize}
443			\item{Global Ocean tutorial, in tutorial\_global\_oce\_latlon verification directory,
444	jmc	1.15	described in section \ref{sec:eg-global} }
445	molod	1.13	\item{ Front Relax experiment, in front\_relax verification directory.}
446			\item{ Ideal 2D Ocean experiment, in ideal\_2D\_oce verification directory.}
447			\end{itemize}
448
449	cnh	1.6	% DO NOT EDIT HERE
450	adcroft	1.1
451
452