Index: branches/2015/dev_r5003_MERCATOR6_CRS/DOC/TexFiles/Chapters/Chap_DYN.tex
===================================================================
--- branches/2015/dev_r5003_MERCATOR6_CRS/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 5602)
+++ branches/2015/dev_r5003_MERCATOR6_CRS/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 7260)
@@ -1,10 +1,10 @@
-% ================================================================
-% Chapter � Ocean Dynamics (DYN)
+\documentclass[NEMO_book]{subfiles}
+\begin{document}
+% ================================================================
+% Chapter ——— Ocean Dynamics (DYN)
 % ================================================================
 \chapter{Ocean Dynamics (DYN)}
 \label{DYN}
 \minitoc
-
-% add a figure for  dynvor ens, ene latices
 
 %\vspace{2.cm}
@@ -165,11 +165,10 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The vector invariant form of the momentum equations is the one most 
-often used in applications of the \NEMO ocean model. The flux form option 
-(see next section) has been present since version $2$. Options are defined
-through the \ngn{namdyn\_adv} namelist variables
-Coriolis and momentum advection terms are evaluated using a leapfrog 
-scheme, $i.e.$ the velocity appearing in these expressions is centred in 
-time (\textit{now} velocity). 
+The vector invariant form of the momentum equations (\np{ln\_dynhpg\_vec}~=~true) is the one most 
+often used in applications of the \NEMO ocean model. The flux form option (\np{ln\_dynhpg\_vec}~=false)
+(see next section) has been present since version $2$. 
+Options are defined through the \ngn{namdyn\_adv} namelist variables.
+Coriolis and momentum advection terms are evaluated using a leapfrog scheme, 
+$i.e.$ the velocity appearing in these expressions is centred in time (\textit{now} velocity). 
 At the lateral boundaries either free slip, no slip or partial slip boundary 
 conditions are applied following Chap.\ref{LBC}.
@@ -296,5 +295,5 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!ht]    \begin{center}
-\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_DYN_een_triad.pdf}
+\includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
 \caption{ \label{Fig_DYN_een_triad}  
 Triads used in the energy and enstrophy conserving scheme (een) for 
@@ -303,10 +302,13 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and 
-\textbf{j}- directions uses the masked vertical scale factor but is always divided by 
-$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of 
-$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and 
-extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 
-the $z$-coordinate with partial steps.
+A key point in \eqref{Eq_een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 
+It uses the sum of masked t-point vertical scale factor divided either 
+by the sum of the four t-point masks (\np{ln\_dynvor\_een\_old}~=~false), 
+or  just by $4$ (\np{ln\_dynvor\_een\_old}~=~true).
+The latter case preserves the continuity of $e_{3f}$ when one or more of the neighbouring $e_{3t}$ 
+tends to zero and extends by continuity the value of $e_{3f}$ into the land areas. 
+This case introduces a sub-grid-scale topography at f-points (with a systematic reduction of $e_{3f}$ 
+when a model level intercept the bathymetry) that tends to reinforce the topostrophy of the flow 
+($i.e.$ the tendency of the flow to follow the isobaths) \citep{Penduff_al_OS07}. 
 
 Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as 
@@ -374,4 +376,11 @@
 \end{aligned}         \right.
 \end{equation} 
+When \np{ln\_dynzad\_zts}~=~\textit{true}, a split-explicit time stepping with 5 sub-timesteps is used 
+on the vertical advection term.
+This option can be useful when the value of the timestep is limited by vertical advection \citep{Lemarie_OM2015}. 
+Note that in this case, a similar split-explicit time stepping should be used on 
+vertical advection of tracer to ensure a better stability, 
+an option which is only available with a TVD scheme (see \np{ln\_traadv\_tvd\_zts} in \S\ref{TRA_adv_tvd}).
+
 
 % ================================================================
@@ -491,5 +500,6 @@
 those in the centred second order method. As the scheme already includes 
 a diffusion component, it can be used without explicit  lateral diffusion on momentum 
-($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
+($i.e.$ setting both \np{ln\_dynldf\_lap} and \np{ln\_dynldf\_bilap} to \textit{false}), 
+and it is recommended to do so.
 
 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
@@ -629,7 +639,4 @@
 ($e_{3w}$).
  
-$\bullet$ Traditional coding with adaptation for ice shelf cavities (\np{ln\_dynhpg\_isf}=true).
-This scheme need the activation of ice shelf cavities (\np{ln\_isfcav}=true).
-
 $\bullet$ Pressure Jacobian scheme (prj) (a research paper in preparation) (\np{ln\_dynhpg\_prj}=true)
 
@@ -646,4 +653,20 @@
 pressure Jacobian method is used to solve the horizontal pressure gradient. This method can provide
 a more accurate calculation of the horizontal pressure gradient than the standard scheme.
+
+\subsection{Ice shelf cavity}
+\label{DYN_hpg_isf}
+Beneath an ice shelf, the total pressure gradient is the sum of the pressure gradient due to the ice shelf load and
+ the pressure gradient due to the ocean load. If cavities are present (\np{ln\_isfcav}~=~true) these two terms can be
+ calculated by setting \np{ln\_dynhpg\_isf}~=~true. No other scheme is working with ice shelves.\\
+
+$\bullet$ The main hypothesis to compute the ice shelf load is that the ice shelf is in isostatic equilibrium.
+ The top pressure is computed integrating a reference density profile (prescribed as density of a water at 34.4 
+PSU and -1.9$^{\circ}C$) from the sea surface to the ice shelf base, which corresponds to the load of the water
+column in which the ice shelf is floatting. This top pressure is constant over time. A detailed description of 
+this method is described in \citet{Losch2008}.\\
+
+$\bullet$ The ocean load is computed using the expression \eqref{Eq_dynhpg_sco} described in \ref{DYN_hpg_sco}. 
+A treatment of the top and bottom partial cells similar to the one described in \ref{DYN_hpg_zps} is done 
+to reduce the residual circulation generated by the top partial cell. 
 
 %--------------------------------------------------------------------------------------------------------------
@@ -718,9 +741,17 @@
 $\ $\newline      %force an empty line
 
-%%%
 Options are defined through the \ngn{namdyn\_spg} namelist variables.
-The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface) and the variable volume case (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case (\S\ref{PE_free_surface}). With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
-
-%%%
+The surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). 
+The main distinction is between the fixed volume case (linear free surface) and the variable volume case 
+(nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) 
+the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case 
+(\S\ref{PE_free_surface}). 
+With both linear and nonlinear free surface, external gravity waves are allowed in the equations, 
+which imposes a very small time step when an explicit time stepping is used. 
+Two methods are proposed to allow a longer time step for the three-dimensional equations: 
+the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), 
+and the split-explicit free surface described below. 
+The extra term introduced in the filtered method is calculated implicitly, 
+so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
 
 
@@ -736,5 +767,4 @@
 implicitly, so that a solver is used to compute it. As a consequence the update of the $next$ 
 velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
-
 
 
@@ -779,5 +809,6 @@
 $\rdt_e = \rdt / nn\_baro$. This parameter can be optionally defined automatically (\np{ln\_bt\_nn\_auto}=true) 
 considering that the stability of the barotropic system is essentially controled by external waves propagation. 
-Maximum allowed Courant number is in that case time independent, and easily computed online from the input bathymetry.
+Maximum Courant number is in that case time independent, and easily computed online from the input bathymetry.
+Therefore, $\rdt_e$ is adjusted so that the Maximum allowed Courant number is smaller than \np{rn\_bt\_cmax}.
 
 %%%
@@ -798,9 +829,9 @@
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 \begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_DYN_dynspg_ts.pdf}
+\includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
 \caption{  \label{Fig_DYN_dynspg_ts}
 Schematic of the split-explicit time stepping scheme for the external 
 and internal modes. Time increases to the right. In this particular exemple, 
-a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_filt=1$) and $nn\_baro=5$.
+a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
 Internal mode time steps (which are also the model time steps) are denoted 
 by $t-\rdt$, $t$ and $t+\rdt$. Variables with $k$ superscript refer to instantaneous barotropic variables, 
@@ -808,7 +839,7 @@
 The former are used to obtain time filtered quantities at $t+\rdt$ while the latter are used to obtain time averaged 
 transports to advect tracers.
-a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=true. 
-b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_ave}=true. 
-c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_ave}=false. }
+a) Forward time integration: \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=true. 
+b) Centred time integration: \np{ln\_bt\_fw}=false, \np{ln\_bt\_av}=true. 
+c) Forward time integration with no time filtering (POM-like scheme): \np{ln\_bt\_fw}=true, \np{ln\_bt\_av}=false. }
 \end{center}    \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -816,5 +847,5 @@
 In the default case (\np{ln\_bt\_fw}=true), the external mode is integrated 
 between \textit{now} and  \textit{after} baroclinic time-steps (Fig.~\ref{Fig_DYN_dynspg_ts}a). To avoid aliasing of fast barotropic motions into three dimensional equations, time filtering is eventually applied on barotropic 
-quantities (\np{ln\_bt\_ave}=true). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities. 
+quantities (\np{ln\_bt\_av}=true). In that case, the integration is extended slightly beyond  \textit{after} time step to provide time filtered quantities. 
 These are used for the subsequent initialization of the barotropic mode in the following baroclinic step. 
 Since external mode equations written at baroclinic time steps finally follow a forward time stepping scheme, 
@@ -837,5 +868,5 @@
 %%%
 
-One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_ave}=false). 
+One can eventually choose to feedback instantaneous values by not using any time filter (\np{ln\_bt\_av}=false). 
 In that case, external mode equations are continuous in time, ie they are not re-initialized when starting a new 
 sub-stepping sequence. This is the method used so far in the POM model, the stability being maintained by refreshing at (almost) 
@@ -1158,12 +1189,19 @@
 
 Besides the surface and bottom stresses (see the above section) which are 
-introduced as boundary conditions on the vertical mixing, two other forcings 
-enter the dynamical equations. 
-
-One is the effect of atmospheric pressure on the ocean dynamics.
-Another forcing term is the tidal potential.
-Both of which will be introduced into the reference version soon. 
-
-\gmcomment{atmospheric pressure is there!!!!    include its description }
+introduced as boundary conditions on the vertical mixing, three other forcings 
+may enter the dynamical equations by affecting the surface pressure gradient. 
+
+(1) When \np{ln\_apr\_dyn}~=~true (see \S\ref{SBC_apr}), the atmospheric pressure is taken 
+into account when computing the surface pressure gradient.
+
+(2) When \np{ln\_tide\_pot}~=~true and \key{tide} is defined (see \S\ref{SBC_tide}), 
+the tidal potential is taken into account when computing the surface pressure gradient.
+
+(3) When \np{nn\_ice\_embd}~=~2 and LIM or CICE is used ($i.e.$ when the sea-ice is embedded in the ocean), 
+the snow-ice mass is taken into account when computing the surface pressure gradient.
+
+
+\gmcomment{ missing : the lateral boundary condition !!!   another external forcing
+ }
 
 % ================================================================
@@ -1296,2 +1334,3 @@
 
 % ================================================================
+\end{document}
