Index: NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_time_domain.tex
===================================================================
--- NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_time_domain.tex	(revision 11003)
+++ NEMO/branches/2019/dev_r10984_HPC-13_IRRMANN_BDY_optimization/doc/latex/NEMO/subfiles/chap_time_domain.tex	(revision 11263)
@@ -11,5 +11,5 @@
 
 % Missing things:
-%	- daymod: definition of the time domain (nit000, nitend andd the calendar)
+%	- daymod: definition of the time domain (nit000, nitend and the calendar)
 
 \gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here,
@@ -22,5 +22,5 @@
 a key feature of an ocean model as it exerts a strong influence on the structure of the computer code
 (\ie on its flowchart).
-In the present chapter, we provide a general description of the \NEMO time stepping strategy and
+In the present chapter, we provide a general description of the \NEMO  time stepping strategy and
 the consequences for the order in which the equations are solved.
 
@@ -40,7 +40,7 @@
 $\rdt$ is the time step;
 and the superscripts indicate the time at which a quantity is evaluated.
-Each term of the RHS is evaluated at a specific time step depending on the physics with which it is associated.
-
-The choice of the time step used for this evaluation is discussed below as well as
+Each term of the RHS is evaluated at a specific time stepping depending on the physics with which it is associated.
+
+The choice of the time stepping used for this evaluation is discussed below as well as
 the implications for starting or restarting a model simulation.
 Note that the time stepping calculation is generally performed in a single operation.
@@ -53,7 +53,5 @@
 is usually not the variable at the after time step;
 but rather it is used to store the time derivative (RHS in \autoref{eq:STP}) prior to time-stepping the equation.
-Generally, the time stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} modules,
-except when using implicit vertical diffusion or calculating sea surface height in which
-case time-splitting options are used.
+The time stepping itself is performed once at each time step where implicit vertical diffusion is computed, \ie in the \mdl{trazdf} and \mdl{dynzdf} modules.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -64,5 +62,5 @@
 
 The time stepping used for processes other than diffusion is the well-known leapfrog scheme
-\citep{Mesinger_Arakawa_Bk76}.
+\citep{mesinger.arakawa_bk76}.
 This scheme is widely used for advection processes in low-viscosity fluids.
 It is a time centred scheme, \ie the RHS in \autoref{eq:STP} is evaluated at time step $t$, the now time step.
@@ -80,5 +78,5 @@
 To prevent it, the leapfrog scheme is often used in association with a Robert-Asselin time filter
 (hereafter the LF-RA scheme).
-This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied by \citet{Asselin_MWR72},
+This filter, first designed by \citet{robert_JMSJ66} and more comprehensively studied by \citet{asselin_MWR72},
 is a kind of laplacian diffusion in time that mixes odd and even time steps:
 \begin{equation}
@@ -88,11 +86,11 @@
 where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin coefficient.
 $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter).
-Its default value is \np{rn\_atfp}~\forcode{= 10.e-3} (see \autoref{sec:STP_mLF}),
-causing only a weak dissipation of high frequency motions (\citep{Farge1987}).
+Its default value is \np{rn\_atfp}\forcode{ = 10.e-3} (see \autoref{sec:STP_mLF}),
+causing only a weak dissipation of high frequency motions (\citep{farge-coulombier_phd87}).
 The addition of a time filter degrades the accuracy of the calculation from second to first order.
 However, the second order truncation error is proportional to $\gamma$, which is small compared to 1.
 Therefore, the LF-RA is a quasi second order accurate scheme.
 The LF-RA scheme is preferred to other time differencing schemes such as predictor corrector or trapezoidal schemes,
-because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme.
+because the user has an explicit and simple control of the magnitude of the time diffusion of the scheme. 
 When used with the 2nd order space centred discretisation of the advection terms in
 the momentum and tracer equations, LF-RA avoids implicit numerical diffusion:
@@ -107,5 +105,5 @@
 
 The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes.
-For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology
+For a tendency $D_x$, representing a diffusion term or a restoring term to a tracer climatology
 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used :
 \[
@@ -115,5 +113,5 @@
 
 This is diffusive in time and conditionally stable.
-The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
+The conditions for stability of second and fourth order horizontal diffusion schemes are \citep{griffies_bk04}:
 \begin{equation}
   \label{eq:STP_euler_stability}
@@ -130,22 +128,6 @@
 
 For the vertical diffusion terms, a forward time differencing scheme can be used,
-but usually the numerical stability condition imposes a strong constraint on the time step.
-Two solutions are available in \NEMO to overcome the stability constraint:
-$(a)$ a forward time differencing scheme using a time splitting technique (\np{ln\_zdfexp}~\forcode{= .true.}) or
-$(b)$ a backward (or implicit) time differencing scheme                   (\np{ln\_zdfexp}~\forcode{= .false.}).
-In $(a)$, the master time step $\Delta$t is cut into $N$ fractional time steps so that
-the stability criterion is reduced by a factor of $N$.
-The computation is performed as follows:
-\begin{alignat*}{2}
-  % \label{eq:STP_ts}
-  &x_\ast^{t - \rdt}                      &= &x^{t - \rdt} \\
-  &x_\ast^{t - \rdt + L \frac{2 \rdt}{N}} &=   &x_\ast ^{t - \rdt + (L - 1) \frac{2 \rdt}{N}}
-                                             + \frac{2 \rdt}{N} \; DF^{t - \rdt + (L - 1) \frac{2 \rdt}{N}}
-  \quad \text{for $L = 1$ to $N$} \\
-  &x^{t + \rdt}                           &= &x_\ast^{t + \rdt}
-\end{alignat*}
-with DF a vertical diffusion term.
-The number of fractional time steps, $N$, is given by setting \np{nn\_zdfexp}, (namelist parameter).
-The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows:
+but usually the numerical stability condition imposes a strong constraint on the time step. To overcome the stability constraint, a 
+backward (or implicit) time differencing scheme is used. This scheme is unconditionally stable but diffusive and can be written as follows:
 \begin{equation}
   \label{eq:STP_imp}
@@ -157,7 +139,5 @@
 %%gm
 
-This scheme is rather time consuming since it requires a matrix inversion,
-but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme.
-For example, the finite difference approximation of the temperature equation is:
+This scheme is rather time consuming since it requires a matrix inversion. For example, the finite difference approximation of the temperature equation is:
 \[
   % \label{eq:STP_imp_zdf}
@@ -183,5 +163,5 @@
 $c(k)$ and $d(k)$ are positive and the diagonal term is greater than the sum of the two extra-diagonal terms,
 therefore a special adaptation of the Gauss elimination procedure is used to find the solution
-(see for example \citet{Richtmyer1967}).
+(see for example \citet{richtmyer.morton_bk67}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -191,5 +171,15 @@
 \label{sec:STP_spg_ts}
 
-===>>>>  TO BE written....  :-)
+The leapfrog environment supports a centred in time computation of the surface pressure, \ie evaluated 
+at \textit{now} time step. This refers to as the explicit free surface case in the code (\np{ln\_dynspg\_exp}\forcode{ = .true.}). 
+This choice however imposes a strong constraint on the time step which should be small enough to resolve the propagation 
+of external gravity waves. As a matter of fact, one rather use in a realistic setup, a split-explicit free surface 
+(\np{ln\_dynspg\_ts}\forcode{ = .true.}) in which barotropic and baroclinic dynamical equations are solved separately with ad-hoc 
+time steps. The use of the time-splitting (in combination with non-linear free surface) imposes some constraints on the design of 
+the overall flowchart, in particular to ensure exact tracer conservation (see \autoref{fig:TimeStep_flowchart}).
+
+Compared to the former use of the filtered free surface in \NEMO v3.6 (\citet{roullet.madec_JGR00}), the use of a split-explicit free surface is advantageous 
+on massively parallel computers. Indeed, no global computations are anymore required by the elliptic solver which saves a substantial amount of communication 
+time. Fast barotropic motions (such as tides) are also simulated with a better accuracy. 
 
 %\gmcomment{ 
@@ -197,14 +187,12 @@
 \begin{figure}[!t]
   \begin{center}
-    \includegraphics[]{Fig_TimeStepping_flowchart}
+    \includegraphics[width=\textwidth]{Fig_TimeStepping_flowchart_v4}
     \caption{
       \protect\label{fig:TimeStep_flowchart}
-      Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}.
-      The use of a semi -implicit computation of the hydrostatic pressure gradient requires the tracer equation to
-      be stepped forward prior to the momentum equation.
-      The need for knowledge of the vertical scale factor (here denoted as $h$) requires the sea surface height and
-      the continuity equation to be stepped forward prior to the computation of the tracer equation.
-      Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here
-      (see \autoref{sec:DYN_spg}).
+      Sketch of the leapfrog time stepping sequence in \NEMO with split-explicit free surface. The latter combined
+       with non-linear free surface requires the dynamical tendency being updated prior tracers tendency to ensure 
+       conservation. Note the use of time integrated fluxes issued from the barotropic loop  in subsequent calculations 
+       of tracer advection and in the continuity equation. Details about the time-splitting scheme can be found 
+       in \autoref{subsec:DYN_spg_ts}.
     }
   \end{center}
@@ -219,5 +207,5 @@
 \label{sec:STP_mLF}
 
-Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the LF-RA scheme in order to
+Significant changes have been introduced by \cite{leclair.madec_OM09} in the LF-RA scheme in order to
 ensure tracer conservation and to allow the use of a much smaller value of the Asselin filter parameter.
 The modifications affect both the forcing and filtering treatments in the LF-RA scheme.
@@ -237,17 +225,17 @@
 The change in the forcing formulation given by \autoref{eq:STP_forcing} (see \autoref{fig:MLF_forcing})
 has a significant effect:
-the forcing term no longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}.
+the forcing term no longer excites the divergence of odd and even time steps \citep{leclair.madec_OM09}.
 % forcing seen by the model....
-This property improves the LF-RA scheme in two respects.
+This property improves the LF-RA scheme in two aspects.
 First, the LF-RA can now ensure the local and global conservation of tracers.
 Indeed, time filtering is no longer required on the forcing part.
-The influence of the Asselin filter on the forcing is be removed by adding a new term in the filter
+The influence of the Asselin filter on the forcing is explicitly removed by adding a new term in the filter
 (last term in \autoref{eq:STP_RA} compared to \autoref{eq:STP_asselin}).
 Since the filtering of the forcing was the source of non-conservation in the classical LF-RA scheme,
-the modified formulation becomes conservative \citep{Leclair_Madec_OM09}.
+the modified formulation becomes conservative \citep{leclair.madec_OM09}.
 Second, the LF-RA becomes a truly quasi -second order scheme.
 Indeed, \autoref{eq:STP_forcing} used in combination with a careful treatment of static instability
-(\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}),
-the two other main sources of time step divergence,
+(\autoref{subsec:ZDF_evd}) and of the TKE physics (\autoref{subsec:ZDF_tke_ene}) 
+(the two other main sources of time step divergence),
 allows a reduction by two orders of magnitude of the Asselin filter parameter.
 
@@ -255,5 +243,5 @@
 $Q^{t + \rdt / 2}$ is the forcing applied over the $[t,t + \rdt]$ time interval.
 This and the change in the time filter, \autoref{eq:STP_RA},
-allows an exact evaluation of the contribution due to the forcing term between any two time steps,
+allows for an exact evaluation of the contribution due to the forcing term between any two time steps,
 even if separated by only $\rdt$ since the time filter is no longer applied to the forcing term.
 
@@ -261,5 +249,5 @@
 \begin{figure}[!t]
   \begin{center}
-    \includegraphics[]{Fig_MLF_forcing}
+    \includegraphics[width=\textwidth]{Fig_MLF_forcing}
     \caption{
       \protect\label{fig:MLF_forcing}
@@ -294,5 +282,5 @@
 This is done simply by keeping the leapfrog environment (\ie the \autoref{eq:STP} three level time stepping) but
 setting all $x^0$ (\textit{before}) and $x^1$ (\textit{now}) fields equal at the first time step and
-using half the value of $\rdt$.
+using half the value of a leapfrog time step ($2 \rdt$). 
 
 It is also possible to restart from a previous computation, by using a restart file.
@@ -303,14 +291,9 @@
 This requires saving two time levels and many auxiliary data in the restart files in machine precision.
 
-Note that when a semi -implicit scheme is used to evaluate the hydrostatic pressure gradient
-(see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to
-be added to the restart file to ensure an exact restartability.
-This is done optionally via the  \np{nn\_dynhpg\_rst} namelist parameter,
-so that the size of the restart file can be reduced when restartability is not a key issue
-(operational oceanography or in ensemble simulations for seasonal forecasting).
-
-Note the size of the time step used, $\rdt$, is also saved in the restart file.
-When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed.
-Options are defined through the  \ngn{namrun} namelist variables.
+Note that the time step $\rdt$, is also saved in the restart file.
+When restarting, if the time step has been changed, or one of the prognostic variables at \textit{before} time step 
+is missing, an Euler time stepping scheme is imposed. A forward initial step can still be enforced by the user by setting 
+the namelist variable \np{nn\_euler}\forcode{=0}. Other options to control the time integration of the model 
+are defined through the  \ngn{namrun} namelist variables.
 %%%
 \gmcomment{
