s_phys_pkgs/text/gmredi.tex

\section{Gent/McWiliams/Redi SGS Eddy parameterization}

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 GKW91 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., 1991, (\cite{gkw91})
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{DM95}, 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{ldd97}, 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}


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