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revision 1.11 by molod, Wed Apr 5 03:35:15 2006 UTC revision 1.18 by jmc, Tue Mar 26 15:10:46 2013 UTC
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1  \subsection{GMREDI: Gent/McWiliams/Redi SGS Eddy Parameterization}  \subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization}
2  \label{sec:pkg:gmredi}  \label{sec:pkg:gmredi}
3  \begin{rawhtml}  \begin{rawhtml}
4  <!-- CMIREDIR:gmredi: -->  <!-- CMIREDIR:gmredi: -->
5  \end{rawhtml}  \end{rawhtml}
6    
7  There are two parts to the Redi/GM parameterization of geostrophic  There are two parts to the Redi/GM parameterization of geostrophic
8  eddies. The first aims to mix tracer properties along isentropes  eddies. The first, the Redi scheme \citep{re82}, aims to mix tracer properties
9  (neutral surfaces) by means of a diffusion operator oriented along the  along isentropes (neutral surfaces) by means of a diffusion operator oriented
10  local isentropic surface (Redi). The second part, adiabatically  along the local isentropic surface.
11    The second part, GM \citep{gen-mcw:90,gen-eta:95}, adiabatically
12  re-arranges tracers through an advective flux where the advecting flow  re-arranges tracers through an advective flux where the advecting flow
13  is a function of slope of the isentropic surfaces (GM).  is a function of slope of the isentropic surfaces.
14    
15  The first GCM implementation of the Redi scheme was by Cox 1987 in the  The first GCM implementation of the Redi scheme was by \cite{Cox87} in the
16  GFDL ocean circulation model. The original approach failed to  GFDL ocean circulation model. The original approach failed to
17  distinguish between isopycnals and surfaces of locally referenced  distinguish between isopycnals and surfaces of locally referenced
18  potential density (now called neutral surfaces) which are proper  potential density (now called neutral surfaces) which are proper
# Line 26  stream-functions expressed in terms of t Line 27  stream-functions expressed in terms of t
27  to the boundary condition of zero value on upper and lower  to the boundary condition of zero value on upper and lower
28  boundaries. The horizontal bolus velocities are then the vertical  boundaries. The horizontal bolus velocities are then the vertical
29  derivative of these functions. Here in lies a problem highlighted by  derivative of these functions. Here in lies a problem highlighted by
30  Griffies et al., 1997: the bolus velocities involve multiple  \cite{gretal:98}: the bolus velocities involve multiple
31  derivatives on the potential density field, which can consequently  derivatives on the potential density field, which can consequently
32  give rise to noise. Griffies et al. point out that the GM bolus fluxes  give rise to noise. Griffies et al. point out that the GM bolus fluxes
33  can be identically written as a skew flux which involves fewer  can be identically written as a skew flux which involves fewer
# Line 46  where $\kappa_\rho$ is the along isopycn Line 47  where $\kappa_\rho$ is the along isopycn
47  $\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of  $\bf{K}_{Redi}$ is a rank 2 tensor that projects the gradient of
48  $\tau$ onto the isopycnal surface. The unapproximated projection tensor is:  $\tau$ onto the isopycnal surface. The unapproximated projection tensor is:
49  \begin{equation}  \begin{equation}
50  \bf{K}_{Redi} = \left(  \bf{K}_{Redi} = \frac{1}{1 + |S|^2} \left(
51  \begin{array}{ccc}  \begin{array}{ccc}
52  1 + S_x& S_x S_y & S_x \\  1 + S_y^2& -S_x S_y & S_x \\
53  S_x S_y  & 1 + S_y & S_y \\  -S_x S_y  & 1 + S_x^2 & S_y \\
54  S_x & S_y & |S|^2 \\  S_x & S_y & |S|^2 \\
55  \end{array}  \end{array}
56  \right)  \right)
# Line 77  S_x & S_y & |S|^2 \\ Line 78  S_x & S_y & |S|^2 \\
78    
79  \subsubsection{GM parameterization}  \subsubsection{GM parameterization}
80    
81  The GM parameterization aims to parameterise the ``advective'' or  The GM parameterization aims to represent the ``advective'' or
82  ``transport'' effect of geostrophic eddies by means of a ``bolus''  ``transport'' effect of geostrophic eddies by means of a ``bolus''
83  velocity, $\bf{u}^*$. The divergence of this advective flux is added  velocity, $\bf{u}^\star$. The divergence of this advective flux is added
84  to the tracer tendency equation (on the rhs):  to the tracer tendency equation (on the rhs):
85  \begin{equation}  \begin{equation}
86  - \bf{\nabla} \cdot \tau \bf{u}^*  - \bf{\nabla} \cdot \tau \bf{u}^\star
87  \end{equation}  \end{equation}
88    
89  The bolus velocity is defined as:  The bolus velocity $\bf{u}^\star$ is defined as the rotational of a
90  \begin{eqnarray}  streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$:
91  u^* & = & - \partial_z F_x \\  \begin{equation}
92  v^* & = & - \partial_z F_y \\  \bf{u}^\star = \nabla \times \bf{F}^\star =
93  w^* & = & \partial_x F_x + \partial_y F_y  \left( \begin{array}{c}
94  \end{eqnarray}  - \partial_z  F_y^\star \\
95  where $F_x$ and $F_y$ are stream-functions with boundary conditions  + \partial_z  F_x^\star \\
96  $F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the  \partial_x F_y^\star - \partial_y F_x^\star
97  bolus velocity in terms of these stream-functions is that they are  \end{array} \right),
98  automatically non-divergent ($\partial_x u^* + \partial_y v^* = -  \end{equation}
99  \partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and  and thus is automatically non-divergent. In the GM parameterization, the streamfunction is
100  $F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$:  specified in terms of the isoneutral slopes $S_x$ and $S_y$:
101  \begin{eqnarray}  \begin{eqnarray}
102  F_x & = & \kappa_{GM} S_x \\  F_x^\star & = & -\kappa_{GM} S_y \\
103  F_y & = & \kappa_{GM} S_y  F_y^\star & = &  \kappa_{GM} S_x
104  \end{eqnarray}  \end{eqnarray}
105    with boundary conditions $F_x^\star=F_y^\star=0$ on upper and lower boundaries.
106    In the end, the bolus transport in the GM parameterization is given by:
107    \begin{equation}
108    \bf{u}^\star = \left(
109    \begin{array}{c}
110    u^\star \\
111    v^\star \\
112    w^\star
113    \end{array}
114    \right) = \left(
115    \begin{array}{c}
116    - \partial_z (\kappa_{GM} S_x) \\
117    - \partial_z (\kappa_{GM} S_y) \\
118    \partial_x  (\kappa_{GM} S_x) + \partial_y (\kappa_{GM} S_y)
119    \end{array}
120    \right)
121    \end{equation}
122    
123  This is the form of the GM parameterization as applied by Donabasaglu,  This is the form of the GM parameterization as applied by Donabasaglu,
124  1997, in MOM versions 1 and 2.  1997, in MOM versions 1 and 2.
125    
126    Note that in the MITgcm, the variables containing the GM bolus streamfunction are:
127    \begin{equation}
128    \left(
129    \begin{array}{c}
130    GM\_PsiX \\
131    GM\_PsiY
132    \end{array}
133    \right) = \left(
134    \begin{array}{c}
135    \kappa_{GM} S_x \\
136    \kappa_{GM} S_y
137    \end{array}
138    \right)= \left(
139    \begin{array}{c}
140    F_y^\star \\
141    -F_x^\star
142    \end{array}
143    \right).
144    \end{equation}
145      
146  \subsubsection{Griffies Skew Flux}  \subsubsection{Griffies Skew Flux}
147    
148  Griffies notes that the discretisation of bolus velocities involves  \cite{gr:98} notes that the discretisation of bolus velocities involves
149  multiple layers of differencing and interpolation that potentially  multiple layers of differencing and interpolation that potentially
150  lead to noisy fields and computational modes. He pointed out that the  lead to noisy fields and computational modes. He pointed out that the
151  bolus flux can be re-written in terms of a non-divergent flux and a  bolus flux can be re-written in terms of a non-divergent flux and a
152  skew-flux:  skew-flux:
153  \begin{eqnarray*}  \begin{eqnarray*}
154  \bf{u}^* \tau  \bf{u}^\star \tau
155  & = &  & = &
156  \left( \begin{array}{c}  \left( \begin{array}{c}
157  - \partial_z ( \kappa_{GM} S_x ) \tau \\  - \partial_z ( \kappa_{GM} S_x ) \tau \\
# Line 124  skew-flux: Line 163  skew-flux:
163  \left( \begin{array}{c}  \left( \begin{array}{c}
164  - \partial_z ( \kappa_{GM} S_x \tau) \\  - \partial_z ( \kappa_{GM} S_x \tau) \\
165  - \partial_z ( \kappa_{GM} S_y \tau) \\  - \partial_z ( \kappa_{GM} S_y \tau) \\
166  \partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y) \tau)  \partial_x ( \kappa_{GM} S_x \tau) + \partial_y ( \kappa_{GM} S_y \tau)
167  \end{array} \right)  \end{array} \right)
168  + \left( \begin{array}{c}  + \left( \begin{array}{c}
169   \kappa_{GM} S_x \partial_z \tau \\   \kappa_{GM} S_x \partial_z \tau \\
170   \kappa_{GM} S_y \partial_z \tau \\   \kappa_{GM} S_y \partial_z \tau \\
171  - \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y) \partial_y \tau  - \kappa_{GM} S_x \partial_x \tau - \kappa_{GM} S_y \partial_y \tau
172  \end{array} \right)  \end{array} \right)
173  \end{eqnarray*}  \end{eqnarray*}
174  The first vector is non-divergent and thus has no effect on the tracer  The first vector is non-divergent and thus has no effect on the tracer
175  field and can be dropped. The remaining flux can be written:  field and can be dropped. The remaining flux can be written:
176  \begin{equation}  \begin{equation}
177  \bf{u}^* \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau  \bf{u}^\star \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau
178  \end{equation}  \end{equation}
179  where  where
180  \begin{equation}  \begin{equation}
# Line 157  becomes apparent when we use the GM para Line 196  becomes apparent when we use the GM para
196  with the Redi isoneutral mixing scheme:  with the Redi isoneutral mixing scheme:
197  \begin{equation}  \begin{equation}
198  \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau  \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
199  - u^* \tau =  - u^\star \tau =
200  ( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau  ( \kappa_\rho \bf{K}_{Redi} + \kappa_{GM} \bf{K}_{GM} ) \bf{\nabla} \tau
201  \end{equation}  \end{equation}
202  In the instance that $\kappa_{GM} = \kappa_{\rho}$ then  In the instance that $\kappa_{GM} = \kappa_{\rho}$ then
# Line 193  $S_y$: {\bf SlopeY} (argument on exit) Line 232  $S_y$: {\bf SlopeY} (argument on exit)
232    
233  \subsubsection{Variable $\kappa_{GM}$}  \subsubsection{Variable $\kappa_{GM}$}
234    
235  Visbeck et al., 1996, suggest making the eddy coefficient,  \cite{visbeck:97} suggest making the eddy coefficient,
236  $\kappa_{GM}$, a function of the Eady growth rate,  $\kappa_{GM}$, a function of the Eady growth rate,
237  $|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant,  $|f|/\sqrt{Ri}$. The formula involves a non-dimensional constant,
238  $\alpha$, and a length-scale $L$:  $\alpha$, and a length-scale $L$:
# Line 222  Substituting into the formula for $\kapp Line 261  Substituting into the formula for $\kapp
261  Experience with the GFDL model showed that the GM scheme has to be  Experience with the GFDL model showed that the GM scheme has to be
262  matched to the convective parameterization. This was originally  matched to the convective parameterization. This was originally
263  expressed in connection with the introduction of the KPP boundary  expressed in connection with the introduction of the KPP boundary
264  layer scheme (Large et al., 97) but in fact, as subsequent experience  layer scheme \citep{lar-eta:94} but in fact, as subsequent experience
265  with the MIT model has found, is necessary for any convective  with the MIT model has found, is necessary for any convective
266  parameterization.  parameterization.
267    
# Line 242  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum Line 281  $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
281    
282  \begin{figure}  \begin{figure}
283  \begin{center}  \begin{center}
284  \resizebox{5.0in}{3.0in}{\includegraphics{part6/tapers.eps}}  \resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}}
285  \end{center}  \end{center}
286  \caption{Taper functions used in GKW99 and DM95.}  \caption{Taper functions used in GKW91 and DM95.}
287  \label{fig:tapers}  \label{fig:tapers}
288  \end{figure}  \end{figure}
289    
290  \begin{figure}  \begin{figure}
291  \begin{center}  \begin{center}
292  \resizebox{5.0in}{3.0in}{\includegraphics{part6/effective_slopes.eps}}  \resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/effective_slopes.eps}}
293  \end{center}  \end{center}
294  \caption{Effective slope as a function of ``true'' slope using Cox  \caption{Effective slope as a function of ``true'' slope using Cox
295  slope clipping, GKW91 limiting and DM95 limiting.}  slope clipping, GKW91 limiting and DM95 limiting.}
# Line 258  slope clipping, GKW91 limiting and DM95 Line 297  slope clipping, GKW91 limiting and DM95
297  \end{figure}  \end{figure}
298    
299    
300  Slope clipping:  \subsubsection*{Slope clipping}
301    
302  Deep convection sites and the mixed layer are indicated by  Deep convection sites and the mixed layer are indicated by
303  homogenized, unstable or nearly unstable stratification. The slopes in  homogenized, unstable or nearly unstable stratification. The slopes in
# Line 266  such regions can be either infinite, ver Line 305  such regions can be either infinite, ver
305  or simply very large. From a numerical point of view, large slopes  or simply very large. From a numerical point of view, large slopes
306  lead to large variations in the tensor elements (implying large bolus  lead to large variations in the tensor elements (implying large bolus
307  flow) and can be numerically unstable. This was first recognized by  flow) and can be numerically unstable. This was first recognized by
308  Cox, 1987, who implemented ``slope clipping'' in the isopycnal mixing  \cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing
309  tensor. Here, the slope magnitude is simply restricted by an upper  tensor. Here, the slope magnitude is simply restricted by an upper
310  limit:  limit:
311  \begin{eqnarray}  \begin{eqnarray}
# Line 305  parameterization. Limiting the slopes al Line 344  parameterization. Limiting the slopes al
344  of the GM/Redi parameterization, re-introducing diabatic fluxes in  of the GM/Redi parameterization, re-introducing diabatic fluxes in
345  regions where the limiting is in effect.  regions where the limiting is in effect.
346    
347  Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991:  \subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
348    
349  The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})  The tapering scheme used in \cite{gkw:91}
350  addressed two issues with the clipping method: the introduction of  addressed two issues with the clipping method: the introduction of
351  large vertical fluxes in addition to convective adjustment fluxes is  large vertical fluxes in addition to convective adjustment fluxes is
352  avoided by tapering the GM/Redi slopes back to zero in  avoided by tapering the GM/Redi slopes back to zero in
# Line 326  but where $|S| \ge S_{max}$ then $f_1(S) Line 365  but where $|S| \ge S_{max}$ then $f_1(S)
365  that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 =  that the effective vertical diffusivity term $\kappa f_1(S) |S|^2 =
366  \kappa S_{max}^2$.  \kappa S_{max}^2$.
367    
368  The GKW tapering scheme is activated in the model by setting {\bf  The GKW91 tapering scheme is activated in the model by setting {\bf
369  GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.  GM\_tap\-er\_scheme = 'gkw91'} in {\em data.gmredi}.
370    
371  \subsubsection{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}  \subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
372    
373  The tapering scheme used by Danabasoglu and McWilliams, 1995,  The tapering scheme used by \cite{dm:95} followed a similar procedure but used a different
 \cite{dm:95}, followed a similar procedure but used a different  
374  tapering function, $f_1(S)$:  tapering function, $f_1(S)$:
375  \begin{equation}  \begin{equation}
376  f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right)  f_1(S) = \frac{1}{2} \left( 1+\tanh \left[ \frac{S_c - |S|}{S_d} \right] \right)
# Line 343  the same way as the GKW91 scheme but has Line 381  the same way as the GKW91 scheme but has
381  cut-off, turning off the GM/Redi SGS parameterization for weaker  cut-off, turning off the GM/Redi SGS parameterization for weaker
382  slopes.  slopes.
383    
384  The DM tapering scheme is activated in the model by setting {\bf  The DM95 tapering scheme is activated in the model by setting {\bf
385  GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.  GM\_tap\-er\_scheme = 'dm95'} in {\em data.gmredi}.
386    
387  \subsubsection{Tapering: Large, Danabasoglu and Doney, JPO 1997}  \subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997}
388    
389  The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the  The tapering used in \cite{lar-eta:97} is based on the
390  DM95 tapering scheme, but also tapers the scheme with an additional  DM95 tapering scheme, but also tapers the scheme with an additional
391  function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are  function of height, $f_2(z)$, so that the GM/Redi SGS fluxes are
392  reduced near the surface:  reduced near the surface:
393  \begin{equation}  \begin{equation}
394  f_2(S) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \pi/2)\right)  f_2(z) = \frac{1}{2} \left( 1 + \sin(\pi \frac{z}{D} - \frac{\pi}{2})\right)
395  \end{equation}  \end{equation}
396  where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with  where $D = L_\rho |S|$ is a depth-scale and $L_\rho=c/f$ with
397  $c=2$~m~s$^{-1}$.  This tapering with height was introduced to fix  $c=2$~m~s$^{-1}$.  This tapering with height was introduced to fix
398  some spurious interaction with the mixed-layer KPP parameterization.  some spurious interaction with the mixed-layer KPP parameterization.
399    
400  The LDD tapering scheme is activated in the model by setting {\bf  The LDD97 tapering scheme is activated in the model by setting {\bf
401  GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.  GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.
402    
403    
# Line 379  clipping and for comparison b) using hor Line 417  clipping and for comparison b) using hor
417  \subsubsection{Package Reference}  \subsubsection{Package Reference}
418  \label{sec:pkg:gmredi:diagnostics}  \label{sec:pkg:gmredi:diagnostics}
419    
420    {\footnotesize
421  \begin{verbatim}  \begin{verbatim}
422  ------------------------------------------------------------------------  ------------------------------------------------------------------------
423  <-Name->|Levs|<-parsing code->|<--  Units   -->|<- Tile (max=80c)  <-Name->|Levs|<-parsing code->|<--  Units   -->|<- Tile (max=80c)
# Line 396  GM_PsiY | 15 |VV   183LR      |m^2/s Line 435  GM_PsiY | 15 |VV   183LR      |m^2/s
435  GM_KuzTz| 15 |UU   186MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: X component  GM_KuzTz| 15 |UU   186MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: X component
436  GM_KvzTz| 15 |VV   185MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: Y component  GM_KvzTz| 15 |VV   185MR      |degC.m^3/s      |Redi Off-diagonal Tempetature flux: Y component
437  \end{verbatim}  \end{verbatim}
438    }
439    
440    \subsubsection{Experiments and tutorials that use gmredi}
441    \label{sec:pkg:gmredi:experiments}
442    
443    \begin{itemize}
444    \item{Global Ocean tutorial, in tutorial\_global\_oce\_latlon verification directory,
445    described in section \ref{sec:eg-global} }
446    \item{ Front Relax experiment, in front\_relax verification directory.}
447    \item{ Ideal 2D Ocean experiment, in ideal\_2D\_oce verification directory.}
448    \end{itemize}
449    
 \subsubsection{Package Reference}  
450  % DO NOT EDIT HERE  % DO NOT EDIT HERE
451    
452    
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