Index: NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
===================================================================
--- NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex	(revision 11388)
+++ NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex	(revision 11407)
@@ -1329,8 +1329,8 @@
   \label{eq:Eqn_zad_Aimp_partition}
 Cu_{min} &= 0.15 \nonumber \\
-Cu_{max} &= 0.27 \nonumber \\
+Cu_{max} &= 0.3  \nonumber \\
 Cu_{cut} &= 2Cu_{max} - Cu_{min} \nonumber \\
 Fcu    &= 4Cu_{max}*(Cu_{max}-Cu_{min}) \nonumber \\
-Cf &=
+C\kern-0.14em f &=
      \begin{cases}
         0.0                                                        &\text{if $Cu \leq Cu_{min}$} \\
@@ -1340,6 +1340,4 @@
 \end{align}
 
-\noindent With these settings the coefficient ($Cf$) is shown in \autoref{fig:zad_Aimp_coeff}
-
 \begin{figure}[!t]
   \begin{center}
@@ -1347,5 +1345,5 @@
     \caption{
       \protect\label{fig:zad_Aimp_coeff}
-      The value of the partitioning coefficient ($Cf$) used to partition vertical velocities into parts to
+      The value of the partitioning coefficient ($C\kern-0.14em f$) used to partition vertical velocities into parts to
       be treated implicitly and explicitly for a range of typical Courant numbers (\forcode{ln_zad_Aimp=.true.})
     }
@@ -1359,22 +1357,38 @@
 \begin{align}
   \label{eq:Eqn_zad_Aimp_partition2}
-    w_{i_{ijk}} &= Cf_{ijk} w_{n_{ijk}}     \nonumber \\
-    w_{n_{ijk}} &= (1-Cf_{ijk}) w_{n_{ijk}}           
+    w_{i_{ijk}} &= C\kern-0.14em f_{ijk} w_{n_{ijk}}     \nonumber \\
+    w_{n_{ijk}} &= (1-C\kern-0.14em f_{ijk}) w_{n_{ijk}}           
 \end{align}
 
 \noindent Note that the coefficient is such that the treatment is never fully implicit;
 the three cases from \autoref{eq:Eqn_zad_Aimp_partition} can be considered as:
-fully-explicit; mixed explicit/implicit and mostly-implicit.  The $w_i$ component is added
-to the implicit solvers for the vertical mixing in \mdl{dynzdf} and \mdl{trazdf} in a
-similar way to \citep{shchepetkin_OM15}.  This is sufficient for the flux-limited
-advection scheme (\forcode{ln_traadv_mus}) but further intervention is required when using
-the flux-corrected scheme (\forcode{ln_traadv_fct}). For these schemes the implicit
-upstream fluxes must be added to both the monotonic guess and to the higher order solution
-when calculating the antidiffusive fluxes. The implicit vertical fluxes are then removed
-since they are added by the implicit solver later on.
+fully-explicit; mixed explicit/implicit and mostly-implicit.  With the settings shown the
+coefficient ($C\kern-0.14em f$) varies as shown in \autoref{fig:zad_Aimp_coeff}. Note with these values
+the $Cu_{cut}$ boundary between the mixed implicit-explicit treatment and 'mostly
+implicit' is 0.45 which is just below the stability limited given in
+\autoref{tab:zad_Aimp_CFLcrit}  for a 3rd order scheme.
+
+The $w_i$ component is added to the implicit solvers for the vertical mixing in
+\mdl{dynzdf} and \mdl{trazdf} in a similar way to \citep{shchepetkin_OM15}.  This is
+sufficient for the flux-limited advection scheme (\forcode{ln_traadv_mus}) but further
+intervention is required when using the flux-corrected scheme (\forcode{ln_traadv_fct}).
+For these schemes the implicit upstream fluxes must be added to both the monotonic guess
+and to the higher order solution when calculating the antidiffusive fluxes. The implicit
+vertical fluxes are then removed since they are added by the implicit solver later on.
 
 The adaptive-implicit vertical advection option is new to NEMO at v4.0 and has yet to be 
 used in a wide range of simulations. The following test simulation, however, does illustrate
 the potential benefits and will hopefully encourage further testing and feedback from users:
+
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
+    \caption{
+      \protect\label{fig:zad_Aimp_overflow_frames}
+      A time-series of temperature vertical cross-sections for the OVERFLOW test case. These results are for the default
+      settings with \forcode{nn_rdt=10.0} and without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).
+    }
+  \end{center}
+\end{figure}
 
 \subsection{Adaptive-implicit vertical advection in the OVERFLOW test-case}
@@ -1389,6 +1403,6 @@
      ln_zdfnpc     = .true.
      ln_traadv_fct = .true.
-        nn_fct_h   =  4
-        nn_fct_v   =  4
+        nn_fct_h   =  2
+        nn_fct_v   =  2
 \end{verbatim}
 
@@ -1404,26 +1418,45 @@
 candidate, therefore, for use of the adaptive-implicit vertical advection scheme.
 
-The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps are shown
-in \autoref{fig:zad_Aimp_overflow_all_rdt} (together with the equivalent frames from the
-base run).  In this simple example the use of the adaptive-implicit vertcal advection
-scheme has enabled a 12x increase in the model timestep without significantly altering the
-solution (although at this extreme the plume is more diffuse and has not travelled so far).
-Notably, the solution with and without the scheme is slightly different even with
-\forcode{nn_rdt = 10.}; suggesting that the base run was close enough to instability to
-trigger the scheme despite completing successfully.  To assist in diagnosing how active
-the scheme is, in both location and time, the 3D implicit and explicit components of the
-vertical velocity are available via XIOS as \texttt{wimp} and \texttt{wexp} respectively.
-Likewise, the partitioning coefficient ($Cf$) is also available as \texttt{wi\_cff}.
-
-\begin{figure}[!t]
-  \begin{center}
-    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_overflow_frames}
-    \caption{
-      \protect\label{fig:zad_Aimp_overflow_frames}
-      A time-series of temperature vertical cross-sections for the OVERFLOW test case. These results are for the default
-      settings with \forcode{nn_rdt=10.0} and without adaptive implicit vertical advection (\forcode{ln_zad_Aimp=.false.}).
-    }
-  \end{center}
-\end{figure}
+The results with \forcode{ln_zad_Aimp=.true.} and a variety of model timesteps
+are shown in \autoref{fig:zad_Aimp_overflow_all_rdt} (together with the equivalent
+frames from the base run).  In this simple example the use of the adaptive-implicit
+vertcal advection scheme has enabled a 12x increase in the model timestep without
+significantly altering the solution (although at this extreme the plume is more diffuse
+and has not travelled so far).  Notably, the solution with and without the scheme is
+slightly different even with \forcode{nn_rdt = 10.}; suggesting that the base run was
+close enough to instability to trigger the scheme despite completing successfully.
+To assist in diagnosing how active the scheme is, in both location and time, the 3D
+implicit and explicit components of the vertical velocity are available via XIOS as
+\texttt{wimp} and \texttt{wexp} respectively.  Likewise, the partitioning coefficient
+($C\kern-0.14em f$) is also available as \texttt{wi\_cff}. For a quick oversight of
+the schemes activity the global maximum values of the absolute implicit component
+of the vertical velocity and the partitioning coefficient are written to the netCDF
+version of the run statistics file (\texttt{run.stat.nc}) if this is active (see
+\autoref{sec:MISC_opt} for activation details).
+
+\autoref{fig:zad_Aimp_maxCf} shows examples of the maximum partitioning coefficient for
+the various overflow tests.  Note that the adaptive-implicit vertical advection scheme is
+active even in the base run with \forcode{nn_rdt=10.0s} adding to the evidence that the
+test case is close to stability limits even with this value. At the larger timesteps, the
+vertical velocity is treated mostly implicitly at some location throughout the run. The
+oscillatory nature of this measure appears to be linked to the progress of the plume front
+as each cusp is associated with the location of the maximum shifting to the adjacent cell.
+This is illustrated in \autoref{fig:zad_Aimp_maxCf_loc} where the i- and k- locations of the
+maximum have been overlaid for the base run case.
+
+\medskip
+\noindent Only limited tests have been performed in more realistic configurations. In the
+ORCA2\_ICE\_PISCES reference configuration the scheme does activate and passes
+restartability and reproducibility tests but it is unable to improve the model's stability
+enough to allow an increase in the model time-step. A view of the time-series of maximum
+partitioning coefficient (not shown here)  suggests that the default time-step of 5400s is
+already pushing at stability limits, especially in the initial start-up phase. The
+time-series does not, however, exhibit any of the 'cuspiness' found with the overflow
+tests.
+
+\medskip
+\noindent A short test with an eORCA1 configuration promises more since a test using a
+time-step of 3600s remains stable with \forcode{ln_zad_Aimp=.true.} whereas the
+time-step is limited to 2700s without.
 
 \begin{figure}[!t]
@@ -1440,4 +1473,26 @@
 \end{figure}
 
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf}
+    \caption{
+      \protect\label{fig:zad_Aimp_maxCf}
+      The maximum partitioning coefficient during a series of test runs with increasing model timestep length.
+      At the larger timesteps, the vertical velocity is treated mostly implicitly at some location throughout 
+      the run.
+    }
+  \end{center}
+\end{figure}
+
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=\textwidth]{Fig_ZDF_zad_Aimp_maxCf_loc}
+    \caption{
+      \protect\label{fig:zad_Aimp_maxCf_loc}
+      The maximum partitioning coefficient for the  \forcode{nn_rdt=10.0s} case overlaid with  information on the gridcell i- and k-
+      locations of the maximum value. 
+    }
+  \end{center}
+\end{figure}
 
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