s_phys_pkgs/text/gmredi.tex

\section{Gent/McWiliams/Redi SGS Eddy parameterization}
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There are two parts to the Redi/GM parameterization of geostrophic
eddies. The first aims to mix tracer properties along isentropes
(neutral surfaces) by means of a diffusion operator oriented along the
local isentropic surface (Redi). The second part, adiabatically
re-arranges tracers through an advective flux where the advecting flow
is a function of slope of the isentropic surfaces (GM).

The first GCM implementation of the Redi scheme was by Cox 1987 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
Griffies et al., 1997: 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.

\subsection{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}


\subsection{GM parameterization}

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

The bolus velocity is defined as:
\begin{eqnarray}
u^* & = & - \partial_z F_x \\
v^* & = & - \partial_z F_y \\
w^* & = & \partial_x F_x + \partial_y F_y
\end{eqnarray}
where $F_x$ and $F_y$ are stream-functions with boundary conditions
$F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the
bolus velocity in terms of these stream-functions is that they are
automatically non-divergent ($\partial_x u^* + \partial_y v^* = -
\partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and
$F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$:
\begin{eqnarray}
F_x & = & \kappa_{GM} S_x \\
F_y & = & \kappa_{GM} S_y
\end{eqnarray}
This is the form of the GM parameterization as applied by Donabasaglu,
1997, in MOM versions 1 and 2.

\subsection{Griffies Skew Flux}

Griffies 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}^* \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}^* \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^* \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} }


\subsection{Variable $\kappa_{GM}$}

Visbeck et al., 1996, 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}


\subsection{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 (Large et al., 97) 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{part6/tapers.eps}}
\end{center}
\caption{Taper functions used in GKW99 and DM95.}
\label{fig:tapers}
\end{figure}

\begin{figure}
\begin{center}
\resizebox{5.0in}{3.0in}{\includegraphics{part6/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
Cox, 1987, 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 Gerdes et al., 1999, (\cite{gkw:99})
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 GKW tapering scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.

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

The tapering scheme used by Danabasoglu and McWilliams, 1995,
\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 DM tapering scheme is activated in the model by setting {\bf
GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.

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

The tapering used in Large et al., 1997, \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(S) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \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 LDD 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}

\subsection{Package Reference}
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1	adcroft	1.1	\section{Gent/McWiliams/Redi SGS Eddy parameterization}
2	edhill	1.7	\begin{rawhtml}
3			<!-- CMIREDIR:gmredi: -->
4			\end{rawhtml}
5	adcroft	1.1
6			There are two parts to the Redi/GM parameterization of geostrophic
7			eddies. The first aims to mix tracer properties along isentropes
8			(neutral surfaces) by means of a diffusion operator oriented along the
9			local isentropic surface (Redi). The second part, adiabatically
10			re-arranges tracers through an advective flux where the advecting flow
11			is a function of slope of the isentropic surfaces (GM).
12
13			The first GCM implementation of the Redi scheme was by Cox 1987 in the
14			GFDL ocean circulation model. The original approach failed to
15			distinguish between isopycnals and surfaces of locally referenced
16			potential density (now called neutral surfaces) which are proper
17			isentropes for the ocean. As will be discussed later, it also appears
18			that the Cox implementation is susceptible to a computational mode.
19			Due to this mode, the Cox scheme requires a background lateral
20			diffusion to be present to conserve the integrity of the model fields.
21
22			The GM parameterization was then added to the GFDL code in the form of
23			a non-divergent bolus velocity. The method defines two
24			stream-functions expressed in terms of the isoneutral slopes subject
25			to the boundary condition of zero value on upper and lower
26			boundaries. The horizontal bolus velocities are then the vertical
27			derivative of these functions. Here in lies a problem highlighted by
28			Griffies et al., 1997: the bolus velocities involve multiple
29			derivatives on the potential density field, which can consequently
30			give rise to noise. Griffies et al. point out that the GM bolus fluxes
31			can be identically written as a skew flux which involves fewer
32			differential operators. Further, combining the skew flux formulation
33			and Redi scheme, substantial cancellations take place to the point
34			that the horizontal fluxes are unmodified from the lateral diffusion
35			parameterization.
36
37			\subsection{Redi scheme: Isopycnal diffusion}
38
39			The Redi scheme diffuses tracers along isopycnals and introduces a
40			term in the tendency (rhs) of such a tracer (here $\tau$) of the form:
41			\begin{equation}
42			\bf{\nabla} \cdot \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
43			\end{equation}
44			where $\kappa_\rho$ is the along isopycnal diffusivity and
45			$\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of
46			$\tau$ onto the isopycnal surface. The unapproximated projection tensor is:
47			\begin{equation}
48			\bf{K}_{Redi} = \left(
49			\begin{array}{ccc}
50			1 + S_x& S_x S_y & S_x \\
51			S_x S_y & 1 + S_y & S_y \\
52			S_x & S_y & \|S\|^2 \\
53			\end{array}
54			\right)
55			\end{equation}
56			Here, $S_x = -\partial_x \sigma / \partial_z \sigma$ and $S_y =
57			-\partial_y \sigma / \partial_z \sigma$ are the components of the
58			isoneutral slope.
59
60			The first point to note is that a typical slope in the ocean interior
61			is small, say of the order $10^{-4}$. A maximum slope might be of
62			order $10^{-2}$ and only exceeds such in unstratified regions where
63			the slope is ill defined. It is therefore justifiable, and
64			customary, to make the small slope approximation, $\|S\| << 1$. The Redi
65			projection tensor then becomes:
66			\begin{equation}
67			\bf{K}_{Redi} = \left(
68			\begin{array}{ccc}
69			1 & 0 & S_x \\
70			0 & 1 & S_y \\
71			S_x & S_y & \|S\|^2 \\
72			\end{array}
73			\right)
74			\end{equation}
75
76
77			\subsection{GM parameterization}
78
79			The GM parameterization aims to parameterise the ``advective'' or
80			``transport'' effect of geostrophic eddies by means of a ``bolus''
81			velocity, $\bf{u}^*$. The divergence of this advective flux is added
82			to the tracer tendency equation (on the rhs):
83			\begin{equation}
84			- \bf{\nabla} \cdot \tau \bf{u}^*
85			\end{equation}
86
87			The bolus velocity is defined as:
88			\begin{eqnarray}
89			u^* & = & - \partial_z F_x \\
90			v^* & = & - \partial_z F_y \\
91			w^* & = & \partial_x F_x + \partial_y F_y
92			\end{eqnarray}
93			where $F_x$ and $F_y$ are stream-functions with boundary conditions
94			$F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the
95			bolus velocity in terms of these stream-functions is that they are
96			automatically non-divergent ($\partial_x u^* + \partial_y v^* = -
97			\partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and
98			$F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$:
99			\begin{eqnarray}
100			F_x & = & \kappa_{GM} S_x \\
101			F_y & = & \kappa_{GM} S_y
102			\end{eqnarray}
103			This is the form of the GM parameterization as applied by Donabasaglu,
104			1997, in MOM versions 1 and 2.
105
106			\subsection{Griffies Skew Flux}
107
108			Griffies notes that the discretisation of bolus velocities involves
109			multiple layers of differencing and interpolation that potentially
110			lead to noisy fields and computational modes. He pointed out that the
111			bolus flux can be re-written in terms of a non-divergent flux and a
112			skew-flux:
113			\begin{eqnarray*}
114			\bf{u}^* \tau
115			& = &
116			\left( \begin{array}{c}
117			- \partial_z ( \kappa_{GM} S_x ) \tau \\
118			- \partial_z ( \kappa_{GM} S_y ) \tau \\
119			(\partial_x \kappa_{GM} S_x + \partial_y \kappa_{GM} S_y)\tau
120			\end{array} \right)
121			\\
122			& = &
123			\left( \begin{array}{c}
124			- \partial_z ( \kappa_{GM} S_x \tau) \\
125			- \partial_z ( \kappa_{GM} S_y \tau) \\
126			\partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y) \tau)
127			\end{array} \right)
128			+ \left( \begin{array}{c}
129			\kappa_{GM} S_x \partial_z \tau \\
130			\kappa_{GM} S_y \partial_z \tau \\
131			- \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y) \partial_y \tau
132			\end{array} \right)
133			\end{eqnarray*}
134			The first vector is non-divergent and thus has no effect on the tracer
135			field and can be dropped. The remaining flux can be written:
136			\begin{equation}
137			\bf{u}^* \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau
138			\end{equation}
139			where
140			\begin{equation}
141			\bf{K}_{GM} =
142			\left(
143			\begin{array}{ccc}
144			0 & 0 & -S_x \\
145			0 & 0 & -S_y \\
146			S_x & S_y & 0
147			\end{array}
148			\right)
149			\end{equation}
150			is an anti-symmetric tensor.
151
152			This formulation of the GM parameterization involves fewer derivatives
153			than the original and also involves only terms that already appear in
154			the Redi mixing scheme. Indeed, a somewhat fortunate cancellation
155			becomes apparent when we use the GM parameterization in conjunction
156			with the Redi isoneutral mixing scheme:
157			\begin{equation}
158			\kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
159			- u^* \tau =
160			( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau
161			\end{equation}
162			In the instance that $\kappa_{GM} = \kappa_{\rho}$ then
163			\begin{equation}
164			\kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} =
165			\kappa_\rho
166			\left( \begin{array}{ccc}
167			1 & 0 & 0 \\
168			0 & 1 & 0 \\
169			2 S_x & 2 S_y & \|S\|^2
170			\end{array}
171			\right)
172			\end{equation}
173	cnh	1.3	which differs from the variable Laplacian diffusion tensor by only
174	adcroft	1.1	two non-zero elements in the $z$-row.
175
176	adcroft	1.2	\fbox{ \begin{minipage}{4.75in}
177			{\em S/R GMREDI\_CALC\_TENSOR} ({\em pkg/gmredi/gmredi\_calc\_tensor.F})
178
179			$\sigma_x$: {\bf SlopeX} (argument on entry)
180
181			$\sigma_y$: {\bf SlopeY} (argument on entry)
182
183			$\sigma_z$: {\bf SlopeY} (argument)
184
185			$S_x$: {\bf SlopeX} (argument on exit)
186
187			$S_y$: {\bf SlopeY} (argument on exit)
188
189			\end{minipage} }
190
191
192
193	adcroft	1.1	\subsection{Variable $\kappa_{GM}$}
194
195			Visbeck et al., 1996, suggest making the eddy coefficient,
196			$\kappa_{GM}$, a function of the Eady growth rate,
197			$\|f\|/\sqrt{Ri}$. The formula involves a non-dimensional constant,
198			$\alpha$, and a length-scale $L$:
199			\begin{displaymath}
200			\kappa_{GM} = \alpha L^2 \overline{ \frac{\|f\|}{\sqrt{Ri}} }^z
201			\end{displaymath}
202			where the Eady growth rate has been depth averaged (indicated by the
203			over-line). A local Richardson number is defined $Ri = N^2 / (\partial
204			u/\partial z)^2$ which, when combined with thermal wind gives:
205			\begin{displaymath}
206			\frac{1}{Ri} = \frac{(\frac{\partial u}{\partial z})^2}{N^2} =
207			\frac{ ( \frac{g}{f \rho_o} \| {\bf \nabla} \sigma \| )^2 }{N^2} =
208			\frac{ M^4 }{ \|f\|^2 N^2 }
209			\end{displaymath}
210			where $M^2$ is defined $M^2 = \frac{g}{\rho_o} \|{\bf \nabla} \sigma\|$.
211			Substituting into the formula for $\kappa_{GM}$ gives:
212			\begin{displaymath}
213			\kappa_{GM} = \alpha L^2 \overline{ \frac{M^2}{N} }^z =
214			\alpha L^2 \overline{ \frac{M^2}{N^2} N }^z =
215			\alpha L^2 \overline{ \|S\| N }^z
216			\end{displaymath}
217
218
219			\subsection{Tapering and stability}
220
221			Experience with the GFDL model showed that the GM scheme has to be
222			matched to the convective parameterization. This was originally
223			expressed in connection with the introduction of the KPP boundary
224	cnh	1.3	layer scheme (Large et al., 97) but in fact, as subsequent experience
225	adcroft	1.1	with the MIT model has found, is necessary for any convective
226			parameterization.
227
228			\fbox{ \begin{minipage}{4.75in}
229			{\em S/R GMREDI\_SLOPE\_LIMIT} ({\em
230			pkg/gmredi/gmredi\_slope\_limit.F})
231
232			$\sigma_x, s_x$: {\bf SlopeX} (argument)
233
234			$\sigma_y, s_y$: {\bf SlopeY} (argument)
235
236			$\sigma_z$: {\bf dSigmadRReal} (argument)
237
238			$z_\sigma^{*}$: {\bf dRdSigmaLtd} (argument)
239
240			\end{minipage} }
241
242	adcroft	1.2	\begin{figure}
243			\begin{center}
244			\resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}
245			\end{center}
246	adcroft	1.5	\caption{Taper functions used in GKW99 and DM95.}
247	adcroft	1.2	\label{fig:tapers}
248			\end{figure}
249
250			\begin{figure}
251			\begin{center}
252			\resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}}
253			\end{center}
254			\caption{Effective slope as a function of ``true'' slope using Cox
255			slope clipping, GKW91 limiting and DM95 limiting.}
256			\label{fig:effective_slopes}
257			\end{figure}
258
259	adcroft	1.1
260			\subsubsection{Slope clipping}
261
262			Deep convection sites and the mixed layer are indicated by
263			homogenized, unstable or nearly unstable stratification. The slopes in
264			such regions can be either infinite, very large with a sign reversal
265			or simply very large. From a numerical point of view, large slopes
266			lead to large variations in the tensor elements (implying large bolus
267	cnh	1.3	flow) and can be numerically unstable. This was first recognized by
268	adcroft	1.1	Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing
269			tensor. Here, the slope magnitude is simply restricted by an upper
270			limit:
271			\begin{eqnarray}
272			\|\nabla \sigma\| & = & \sqrt{ \sigma_x^2 + \sigma_y^2 } \\
273			S_{lim} & = & - \frac{\|\nabla \sigma\|}{ S_{max} }
274			\;\;\;\;\;\;\;\; \mbox{where $S_{max}$ is a parameter} \\
275			\sigma_z^\star & = & \min( \sigma_z , S_{lim} ) \\
276			{[s_x,s_y]} & = & - \frac{ [\sigma_x,\sigma_y] }{\sigma_z^\star}
277			\end{eqnarray}
278			Notice that this algorithm assumes stable stratification through the
279			``min'' function. In the case where the fluid is well stratified ($\sigma_z < S_{lim}$) then the slopes evaluate to:
280			\begin{equation}
281			{[s_x,s_y]} = - \frac{ [\sigma_x,\sigma_y] }{\sigma_z}
282			\end{equation}
283			while in the limited regions ($\sigma_z > S_{lim}$) the slopes become:
284			\begin{equation}
285			{[s_x,s_y]} = \frac{ [\sigma_x,\sigma_y] }{\|\nabla \sigma\|/S_{max}}
286			\end{equation}
287			so that the slope magnitude is limited $\sqrt{s_x^2 + s_y^2} =
288			S_{max}$.
289
290			The slope clipping scheme is activated in the model by setting {\bf
291			GM\_tap\-er\_scheme = 'clipping'} in {\em data.gmredi}.
292
293			Even using slope clipping, it is normally the case that the vertical
294			diffusion term (with coefficient $\kappa_\rho{\bf K}_{33} =
295			\kappa_\rho S_{max}^2$) is large and must be time-stepped using an
296			implicit procedure (see section on discretisation and code later).
297			Fig. \ref{fig-mixedlayer} shows the mixed layer depth resulting from
298			a) using the GM scheme with clipping and b) no GM scheme (horizontal
299			diffusion). The classic result of dramatically reduced mixed layers is
300			evident. Indeed, the deep convection sites to just one or two points
301			each and are much shallower than we might prefer. This, it turns out,
302	cnh	1.3	is due to the over zealous re-stratification due to the bolus transport
303	adcroft	1.1	parameterization. Limiting the slopes also breaks the adiabatic nature
304			of the GM/Redi parameterization, re-introducing diabatic fluxes in
305			regions where the limiting is in effect.
306
307			\subsubsection{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
308
309	adcroft	1.5	The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})
310	adcroft	1.1	addressed two issues with the clipping method: the introduction of
311			large vertical fluxes in addition to convective adjustment fluxes is
312			avoided by tapering the GM/Redi slopes back to zero in
313			low-stratification regions; the adjustment of slopes is replaced by a
314			tapering of the entire GM/Redi tensor. This means the direction of
315			fluxes is unaffected as the amplitude is scaled.
316
317			The scheme inserts a tapering function, $f_1(S)$, in front of the
318			GM/Redi tensor:
319			\begin{equation}
320			f_1(S) = \min \left[ 1, \left( \frac{S_{max}}{\|S\|}\right)^2 \right]
321			\end{equation}
322			where $S_{max}$ is the maximum slope you want allowed. Where the
323			slopes, $\|S\|<S_{max}$ then $f_1(S) = 1$ and the tensor is un-tapered
324			but where $\|S\| \ge S_{max}$ then $f_1(S)$ scales down the tensor so
325			that the effective vertical diffusivity term $\kappa f_1(S) \|S\|^2 =
326			\kappa S_{max}^2$.
327
328			The GKW tapering scheme is activated in the model by setting {\bf
329			GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.
330
331			\subsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
332
333			The tapering scheme used by Danabasoglu and McWilliams, 1995,
334	adcroft	1.5	\cite{dm:95}, followed a similar procedure but used a different
335	adcroft	1.1	tapering function, $f_1(S)$:
336			\begin{equation}
337			f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - \|S\|}{S_d} \right] \right)
338			\end{equation}
339			where $S_c = 0.004$ is a cut-off slope and $S_d=0.001$ is a scale over
340			which the slopes are smoothly tapered. Functionally, the operates in
341			the same way as the GKW91 scheme but has a substantially lower
342			cut-off, turning off the GM/Redi SGS parameterization for weaker
343			slopes.
344
345			The DM tapering scheme is activated in the model by setting {\bf
346			GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.
347
348			\subsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}
349
350	adcroft	1.5	The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the
351	adcroft	1.1	DM95 tapering scheme, but also tapers the scheme with an additional
352			function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are
353			reduced near the surface:
354			\begin{equation}
355			f_2(S) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \pi/2)\right)
356			\end{equation}
357			where $D = L_\rho \|S\|$ is a depth-scale and $L_\rho=c/f$ with
358			$c=2$~m~s$^{-1}$. This tapering with height was introduced to fix
359			some spurious interaction with the mixed-layer KPP parameterization.
360
361			The LDD tapering scheme is activated in the model by setting {\bf
362			GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.
363
364
365
366	adcroft	1.2
367	adcroft	1.1	\begin{figure}
368	adcroft	1.4	\begin{center}
369	adcroft	1.1	%\includegraphics{mixedlayer-cox.eps}
370			%\includegraphics{mixedlayer-diff.eps}
371	adcroft	1.4	Figure missing.
372			\end{center}
373	adcroft	1.1	\caption{Mixed layer depth using GM parameterization with a) Cox slope
374			clipping and for comparison b) using horizontal constant diffusion.}
375	adcroft	1.4	\label{fig-mixedlayer}
376	adcroft	1.1	\end{figure}
377
378	cnh	1.6	\subsection{Package Reference}
379			% DO NOT EDIT HERE
380	adcroft	1.1
381
382