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-revision 1.15 by jmc,
Mon Aug 30 23:09:21 2010 UTC
+revision 1.16 by dfer,
Mon Jun 27 02:08:35 2011 UTC
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- \subsection{GMREDI: Gent/McWiliams/Redi SGS Eddy Parameterization}
+ \subsection{GMREDI: Gent-McWilliams/Redi SGS Eddy Parameterization}
  \label{sec:pkg:gmredi}
  \begin{rawhtml}
  <!-- CMIREDIR:gmredi: -->
  \end{rawhtml}
  There are two parts to the Redi/GM parameterization of geostrophic
- eddies. The first aims to mix tracer properties along isentropes
+ eddies. The first, the Redi scheme \citep{re82}, aims to mix tracer properties along isentropes
  (neutral surfaces) by means of a diffusion operator oriented along the
- local isentropic surface (Redi). The second part, adiabatically
+ 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 (GM).
+ is a function of slope of the isentropic surfaces.
- 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
  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
 Line 26 
 stream-functions expressed in terms of t
  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
+ \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
 Line 77 
 S_x & S_y & |S|^2 \\
  \subsubsection{GM parameterization}
- The GM parameterization aims to parameterise the ``advective'' or
+ The GM parameterization aims to represent the ``advective'' or
  ``transport'' effect of geostrophic eddies by means of a ``bolus''
- velocity, $\bf{u}^*$. The divergence of this advective flux is added
+ 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}^*
+ - \bf{\nabla} \cdot \tau \bf{u}^\star
  \end{equation}
- The bolus velocity is defined as:
+ The bolus velocity $\bf{u}^\star$ is defined as the rotational of a
- \begin{eqnarray}
+ streamfunction $\bf{F}^\star$=$(F_x^\star,F_y^\star,0)$:
- u^* & = & - \partial_z F_x \\
+ \begin{equation}
- v^* & = & - \partial_z F_y \\
+ \bf{u}^\star = \nabla \times \bf{F}^\star =
- w^* & = & \partial_x F_x + \partial_y F_y
+ \left( \begin{array}{c}
- \end{eqnarray}
+ - \partial_z  F_y^\star \\
- where $F_x$ and $F_y$ are stream-functions with boundary conditions
+ + \partial_z  F_x^\star \\
- $F_x=F_y=0$ on upper and lower boundaries. The virtue of casting the
+ \partial_x F_y^\star - \partial_y F_x^\star
- bolus velocity in terms of these stream-functions is that they are
+ \end{array} \right),
- automatically non-divergent ($\partial_x u^* + \partial_y v^* = -
+ \end{equation}
- \partial_{xz} F_x - \partial_{yz} F_y = - \partial_z w^*$). $F_x$ and
+ and thus is automatically non-divergent. In the GM parameterization, the streamfunction is
- $F_y$ are specified in terms of the isoneutral slopes $S_x$ and $S_y$:
+ specified in terms of the isoneutral slopes $S_x$ and $S_y$:
  \begin{eqnarray}
- F_x & = & \kappa_{GM} S_x \\
+ F_x^\star & = & -\kappa_{GM} S_y \\
- F_y & = & \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,
 , 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}
- Griffies notes that the discretisation of bolus velocities involves
+ \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}^* \tau
+ \bf{u}^\star \tau
  & = &
  \left( \begin{array}{c}
  - \partial_z ( \kappa_{GM} S_x ) \tau \\
-Line 124 
 skew-flux:
+Line 162 
 skew-flux:
  \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)
+ \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
+ - \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
+ \bf{u}^\star \tau = - \kappa_{GM} \bf{K}_{GM} \bf{\nabla} \tau
  \end{equation}
  where
  \begin{equation}
-Line 157 
 becomes apparent when we use the GM para
+Line 195 
 becomes apparent when we use the GM para
  with the Redi isoneutral mixing scheme:
  \begin{equation}
  \kappa_\rho \bf{K}_{Redi} \bf{\nabla} \tau
- - u^* \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
-Line 193 
 $S_y$: {\bf SlopeY} (argument on exit)
+Line 231 
 $S_y$: {\bf SlopeY} (argument on exit)
  \subsubsection{Variable $\kappa_{GM}$}
- Visbeck et al., 1996, suggest making the eddy coefficient,
+ \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 $L$:
-Line 222 
 Substituting into the formula for $\kapp
+Line 260 
 Substituting into the formula for $\kapp
  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
+ layer scheme \citep{lar-eta:94} but in fact, as subsequent experience
  with the MIT model has found, is necessary for any convective
  parameterization.
-Line 244 
 $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
+Line 282 
 $z_\sigma^{*}$: {\bf dRdSigmaLtd} (argum
  \begin{center}
  \resizebox{5.0in}{3.0in}{\includegraphics{s_phys_pkgs/figs/tapers.eps}}
  \end{center}
- \caption{Taper functions used in GKW99 and DM95.}
+ \caption{Taper functions used in GKW91 and DM95.}
  \label{fig:tapers}
  \end{figure}
-Line 258 
 slope clipping, GKW91 limiting and DM95
+Line 296 
 slope clipping, GKW91 limiting and DM95
  \end{figure}
- Slope clipping:
+ \subsubsection*{Slope clipping}
  Deep convection sites and the mixed layer are indicated by
  homogenized, unstable or nearly unstable stratification. The slopes in
-Line 266 
 such regions can be either infinite, ver
+Line 304 
 such regions can be either infinite, ver
  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
+ \cite{Cox87} who implemented ``slope clipping'' in the isopycnal mixing
  tensor. Here, the slope magnitude is simply restricted by an upper
  limit:
  \begin{eqnarray}
-Line 305 
 parameterization. Limiting the slopes al
+Line 343 
 parameterization. Limiting the slopes al
  of the GM/Redi parameterization, re-introducing diabatic fluxes in
  regions where the limiting is in effect.
- Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991:
+ \subsubsection*{Tapering: Gerdes, Koberle and Willebrand, Clim. Dyn. 1991}
- The tapering scheme used in Gerdes et al., 1999, (\cite{gkw:99})
+ 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
-Line 326 
 but where $|S| \ge S_{max}$ then $f_1(S)
+Line 364 
 but where $|S| \ge S_{max}$ then $f_1(S)
  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
+ 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}
+ \subsubsection*{Tapering: Danabasoglu and McWilliams, J. Clim. 1995}
- 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
  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)
-Line 343 
 the same way as the GKW91 scheme but has
+Line 380 
 the same way as the GKW91 scheme but has
  cut-off, turning off the GM/Redi SGS parameterization for weaker
  slopes.
- The DM tapering scheme is activated in the model by setting {\bf
+ 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}
+ \subsubsection*{Tapering: Large, Danabasoglu and Doney, JPO 1997}
- The tapering used in Large et al., 1997, \cite{ldd:97}, is based on the
+ The tapering used in \cite{ldd:97} is based on the
  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)
+ 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 LDD tapering scheme is activated in the model by setting {\bf
+ The LDD97 tapering scheme is activated in the model by setting {\bf
  GM\_tap\-er\_scheme = 'ldd97'} in {\em data.gmredi}.

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