Index: trunk/DOC/BETA/Biblio/Biblio.bib
===================================================================
--- trunk/DOC/BETA/Biblio/Biblio.bib	(revision 812)
+++ trunk/DOC/BETA/Biblio/Biblio.bib	(revision 817)
@@ -85,4 +85,15 @@
 
 @STRING{Tellus = {Tellus}}
+
+@ARTICLE{Adcroft_Campin_OM04,
+  author = {A. Adcroft and J.-M. Campin},
+  title = {Re-scaled height coordinates for accurate representation of free-surface
+	flows in ocean circulation models},
+  journal = {Ocean Modelling},
+  year = {2004},
+  volume = {7},
+  owner = {gm},
+  timestamp = {2008.01.27}
+}
 
 @ARTICLE{Arakawa1966,
@@ -154,12 +165,12 @@
 
 @ARTICLE{Arhan2006,
-  author = {M. Arhan and A.M. Treguier and  B. Bourles and  S. Michel},
-  year = { 2006 },
-  title = {Diagnosing the annual cycle of the Equatorial Undercurrent in the Atlantic Ocean from a general circulation model},
-  journal =  JPO,
-  volume = { 36}, 
+  author = {M. Arhan and A.M. Treguier and B. Bourles and S. Michel},
+  title = {Diagnosing the annual cycle of the Equatorial Undercurrent in the
+	Atlantic Ocean from a general circulation model},
+  journal = JPO,
+  year = {2006},
+  volume = { 36},
   pages = {1502-1522}
 }
-
 
 @ARTICLE{ASSELIN1972,
@@ -185,4 +196,41 @@
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Barnier_al_OD06,
+  author = {B. Barnier and G. Madec and T. Penduff and J.-M. Molines and A.-M.
+	Treguier and J. Le Sommer and A. Beckmann and A. Biastoch and C.
+	Boning and J. Dengg and C. Derval and E. Durand and S. Gulev and
+	E. Remy and C. Talandier and S. Theetten and M. Maltrud and J. McClean
+	and B. De Cuevas},
+  title = {Impact of partial steps and momentum advection schemes in a global
+	ocean circulation model at eddy-permitting resolution.},
+  journal = {Ocean Dyn.},
+  year = {2006},
+  pages = {doi: 10.1007/s10236-006-0082-1.},
+  owner = {gm},
+  timestamp = {2008.01.25}
+}
+
+@INCOLLECTION{Barnier1996,
+  author = {B. Barnier and P. Marchesiello and A.P. de Miranda},
+  title = {Modeling the ocean circulation in the South Atlantic: A strategy
+	for dealing with open boundaries},
+  booktitle = {The South Atlantic: Present and Past Circulation},
+  publisher = {Springer-Verlag, Berlin},
+  year = {1996},
+  editor = {G.Wefer and W.H. Berger and G Siedler and D. Webb},
+  pages = {289-304}
+}
+
+@ARTICLE{Barnier1998,
+  author = {B. Barnier and P. Marchesiello and A. P. de Miranda and J.M. Molines
+	and M. Coulibaly},
+  title = {A sigma-coordinate primitive equation model for studying the circulation
+	in the South Atlantic I, Model configuration with error estimates},
+  journal = {Deep Sea Res.},
+  year = {1998},
+  volume = {45},
+  pages = {543-572}
 }
 
@@ -336,4 +384,14 @@
 }
 
+@ARTICLE{Blayo2005,
+  author = {E. Blayo and L. Debreu},
+  title = {Revisiting open boundary conditions from the point of view of characteristic
+	variables},
+  journal = {Ocean Modelling},
+  year = {2005},
+  volume = {9},
+  pages = {231-252}
+}
+
 @ARTICLE{Bougeault1989,
   author = {P. Bougeault and P. Lacarrere},
@@ -429,4 +487,16 @@
 }
 
+@ARTICLE{Campin_Goosse_Tel99,
+  author = {J. M. Campin and H. Goosse},
+  title = {Parameterization of density-driven downsloping flow for a coarse-resolution
+	ocean model in z-coordinate},
+  journal = {Tellus},
+  year = {1999},
+  volume = {51},
+  pages = {412-430},
+  owner = {gm},
+  timestamp = {2008.01.20}
+}
+
 @ARTICLE{Cox1987,
   author = {M. Cox},
@@ -435,7 +505,43 @@
   year = {1987},
   volume = {74},
-  pages = {1-5},
+  pages = {1-9},
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Dorscher_Beckmann_JAOT00,
+  author = {R. D\"{o}scher and A. Beckmann},
+  title = {Effects of a Bottom Boundary Layer Parameterization in a Coarse-Resolution
+	Model of the North Atlantic Ocean},
+  journal = JAOT,
+  year = {2000},
+  volume = {17},
+  pages = {698-707},
+  owner = {gm},
+  timestamp = {2008.01.23}
+}
+
+@ARTICLE{Debreu_al_CG2008,
+  author = {L. Debreu and C. Vouland and E. Blayo},
+  title = {AGRIF: Adaptive Grid Refinement In Fortran},
+  journal = {Computers and Geosciences},
+  year = {2008},
+  volume = {34},
+  pages = {8-13},
+  owner = {gm},
+  timestamp = {2008.02.03}
+}
+
+@ARTICLE{Delecluse_Madec_Bk00,
+  author = {P. Delecluse and G. Madec},
+  title = {Ocean modelling and the role of the ocean in the climate system},
+  journal = {In \textit{Modeling the Earth's Climate and its Variability}, Les
+	Houches, Session, LXVII 1997,
+	
+	Eds. W. R. Holland, S. Joussaume and F. David, Elsevier Science,},
+  year = {2000},
+  pages = {237-313},
+  owner = {gm},
+  timestamp = {2008.02.03}
 }
 
@@ -449,4 +555,13 @@
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@INCOLLECTION{Durran2001,
+  author = {D.R. Durran },
+  title = {Open boundary conditions: fact and fiction},
+  booktitle = {Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics},
+  publisher = {Kluwer Academic Publishers},
+  year = {2001},
+  editor = {P.F. Hodnett}
 }
 
@@ -619,4 +734,13 @@
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@TECHREPORT{Gibson_TR86,
+  author = {J. K. Gibson},
+  title = {Standard software development and maintenance},
+  institution = {Operational Dep., ECMWF, Reading, UK.},
+  year = {1986},
+  owner = {gm},
+  timestamp = {2008.02.03}
 }
 
@@ -883,4 +1007,13 @@
 }
 
+@ARTICLE{Killworth1992,
+  author = {P. D. Killworth},
+  title = {An equivalent-barotropic mode in the fine resolution Antarctic model},
+  journal = JPO,
+  year = {1992},
+  volume = {22},
+  pages = {1379-1387}
+}
+
 @ARTICLE{Killworth1991,
   author = {Killworth, P. D. and Stainforth, D. and Webb, D. J. and Paterson,
@@ -909,14 +1042,4 @@
   timestamp = {2007.08.03}
 }
-
-@ARTICLE{Killworth1992,
-  author = {P. D. Killworth},
-   year = {1992}, 
-  title = {An equivalent-barotropic mode in the fine resolution Antarctic model},
-  journal = JPO,
-  volume = {22},
-  pages = {1379-1387}
-}.
-
 
 @ARTICLE{Kolmogorov1942,
@@ -1043,4 +1166,16 @@
 }
 
+@ARTICLE{Lengaigne_al_JGR03,
+  author = {M. Lengaigne and G. Madec and G. Alory and C. Menkes},
+  title = {Sensitivity of the tropical Pacific Ocean to isopycnal diffusion
+	on tracer and dynamics},
+  journal = JGR,
+  year = {2003},
+  volume = {108 (C11)},
+  pages = {3345, doi:10.1029/2002JC001704},
+  owner = {gm},
+  timestamp = {2008.01.26}
+}
+
 @ARTICLE{Leonard1991,
   author = {B. P. Leonard},
@@ -1121,5 +1256,5 @@
   title = {La formation d'eau profonde et son impact sur la circulation r\'{e}gionale
 	en M\'{e}diterran\'{e}e Occidentale - une approche num\'{e}rique},
-  school = {Universit\'{e}Pierre et Marie Curie, Paris,  France, 194pp.},
+  school = {Universit\'{e}Pierre et Marie Curie, Paris, France, 194pp.},
   year = {1990},
   owner = {gm},
@@ -1254,4 +1389,14 @@
 }
 
+@ARTICLE{Marchesiello2001,
+  author = { P. Marchesiello and J. Mc Williams and A. Shchepetkin },
+  title = {Open boundary conditions for long-term integrations of Regional Oceanic
+	Models},
+  journal = {Ocean Modelling},
+  year = {2001},
+  volume = {3},
+  pages = {1-20}
+}
+
 @PHDTHESIS{MartiTh1992,
   author = {O. Marti},
@@ -1338,4 +1483,13 @@
 }
 
+@BOOK{Mesinger_Arakawa_Bk76,
+  title = {Numerical methods used in Atmospheric models},
+  publisher = {GARP Publication Series No 17},
+  year = {1976},
+  author = {F. Mesinger and A. Arakawa},
+  owner = {gm},
+  timestamp = {2008.02.09}
+}
+
 @ARTICLE{Murray1996,
   author = {R. J. Murray},
@@ -1387,4 +1541,18 @@
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Pacanowski_Gnanadesikan_MWR98,
+  author = {R. C. Pacanowski and A. Gnanadesikan},
+  title = {Transient response in a z-level ocean model that resolves topography
+	
+	
+	with partial-cells},
+  journal = MWR,
+  year = {1998},
+  volume = {126},
+  pages = {3248-3270},
+  owner = {gm},
+  timestamp = {2008.01.26}
 }
 
@@ -1422,6 +1590,27 @@
 }
 
+@ARTICLE{Penduff2000,
+  author = {T. Penduff and B. Barnier and A. Colin de Verdi\`{e}re},
+  title = { Self-adapting open boundaries for a regional model of the eastern
+	North Atlantic},
+  journal = JGR,
+  year = {2000},
+  volume = {105},
+  pages = {11,279-11,297}
+}
+
+@ARTICLE{Penduff2007,
+  author = {T. Penduff and J. Le Sommer and B. Barnier and A.M. Treguier and
+	J. Molines and G. Madec},
+  title = {Influence of numerical schemes on current-topography interactions
+	in 1/4$^{\circ}$ global ocean simulations},
+  journal = {Ocean Science},
+  year = {2007},
+  volume = {?},
+  pages = {in revision}
+}
+
 @ARTICLE{Phillips1959,
-  author = {R. S.  Phillips},
+  author = {R. S. Phillips},
   title = {Dissipative Operators and Hyperbolic Systems of Partial Differential
 	Equations},
@@ -1433,4 +1622,15 @@
   owner = {gm},
   timestamp = {2007.08.10}
+}
+
+@ARTICLE{Redi_JPO82,
+  author = {M. H. Redi},
+  title = {Oceanic isopycnal mixing by coordinate rotation},
+  journal = JPO,
+  year = {1982},
+  volume = {13},
+  pages = {1154-1158},
+  owner = {gm},
+  timestamp = {2008.02.02}
 }
 
@@ -1466,4 +1666,13 @@
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@INCOLLECTION{Roed1986,
+  author = {L.P. Roed and C.K. Cooper},
+  title = {Open boundary conditions in numerical ocean models},
+  booktitle = {Advanced Physical Oceanography Numerical Modelling},
+  publisher = { NATO ASI Series, vol. 186.},
+  year = {1986},
+  editor = {J.J. O'Brien}
 }
 
@@ -1804,12 +2013,25 @@
   timestamp = {2007.08.05}
 }
- 
+
 @ARTICLE{Treguier1992,
-  author = {A.M. Treguier},
-  year = { 1992},
-  title = {Kinetic energy analysis of an eddy resolving, primitive equation North Atlantic model},
-  journal = JGR, 
-  volume = {97}, 
+  author = {A.M. Tr\'{e}guier},
+  title = {Kinetic energy analysis of an eddy resolving, primitive equation
+	North Atlantic model},
+  journal = JGR,
+  year = {1992},
+  volume = {97},
   pages = {687-701}
+}
+
+@ARTICLE{Treguier2001,
+  author = {A.M Tr\'{e}guier and B. Barnier and A.P. de Miranda and J.M. Molines
+	and N. Grima and M. Imbard and G. Madec and C. Messager and T. Reynaud
+	and S. Michel},
+  title = {An Eddy Permitting model of the Atlantic circulation: evaluating
+	open boundary conditions},
+  journal = JGR,
+  year = {2001},
+  volume = {106},
+  pages = {22115-22129}
 }
 
@@ -1970,99 +2192,4 @@
   month = jul,
   owner = {gm}
-}
-
-@ARTICLE{Marchesiello2001,
-  author = { P. Marchesiello  and J. Mc Williams  and A. Shchepetkin },
-  title = {Open boundary conditions for long-term integrations of Regional Oceanic Models},
-  journal = {Ocean Modelling},
-  year = {2001},
-  volume = {3},
-  pages = {1-20}
-}
-
-@INCOLLECTION{Durran2001,
-  author = {D.R.  Durran },
-  title = {Open boundary conditions: fact and fiction},
-  booktitle = {Advances in Mathematical Modelling of Atmosphere and Ocean Dynamics},
-  year = {2001},
-  editor = {P.F. Hodnett},
-  publisher = {Kluwer Academic Publishers}
-}
-
-@ARTICLE{Treguier2001,
-  author = {A.M Treguier and  B. Barnier and A.P. de Miranda and J.M. Molines and N. Grima and M. Imbard and G. Madec and C. Messager and  T. Reynaud and S. Michel},
-  title = {An Eddy Permitting model of the Atlantic circulation: evaluating open boundary conditions},
-  journal = JGR,
-  year = {2001},
-  volume = {106},
-  pages = {22115-22129}
-}
-
-@ARTICLE{Blayo2005,
-  author = {E. Blayo and L. Debreu},
-  title = {Revisiting open boundary conditions from the point of view of characteristic variables},
-  journal = {Ocean Modelling}, 
-  year = {2005},
-  volume = {9},
-  pages = {231-252}
-}
-
-@ARTICLE{Barnier1998,
-   author = {B. Barnier  and P. Marchesiello and A. P. de Miranda and  J.M. Molines and M. Coulibaly}, 
-   title = {A sigma-coordinate primitive equation model for studying the circulation in the South Atlantic I, Model configuration with error estimates}, 
-   journal = {Deep Sea Res.}, 
-   volume = {45}, 
-   pages = {543-572}, 
-   year =  {1998}
-}
-
-@ARTICLE{Penduff2000,
-   author = {T. Penduff and B. Barnier  and A. Colin de Verdi\`{e}re}, 
-   title = { Self-adapting open boundaries for a regional model of the eastern North Atlantic},
-   journal = JGR,
-   volume = {105}, 
-   pages = {11,279-11,297}, 
-   year =  {2000}
-}
-
-@ARTICLE{Penduff2007,
-   author = {T. Penduff and J. Le Sommer and B. Barnier and A.M. Treguier and J. Molines and G. Madec},
-   title = {Influence of numerical schemes on current-topography interactions in 1/4$^{\circ}$ global ocean simulations},  
-   journal = {Ocean Science},
-   volume = {?}, 
-   pages = {in revision}, 
-   year =  {2007}
-}
-
-  
-@INCOLLECTION{Barnier1996,
-  author = { B. Barnier  and P. Marchesiello  and  A.P. de Miranda },
-  title = {Modeling the ocean circulation in the South Atlantic : A strategy for dealing with open boundaries},
-  booktitle = {The South Atlantic : Present and Past Circulation},
-  editor = {G.Wefer and  W.H. Berger and G Siedler and D. Webb},
-  publisher ={Springer-Verlag, Berlin},
-  pages = {289-304},
-   year =  {1996}
-}
-
-@INCOLLECTION{Roed1986,
-  author = {L.P. Roed  and  C.K. Cooper},
-  year =  {1986},
-  title = {Open boundary conditions in numerical ocean models},
-  booktitle = {Advanced Physical Oceanography Numerical Modelling}, 
-  editor = {J.J. O'Brien},
-  publisher = { NATO ASI Series, vol. 186.}
-}
-
-
-@INCOLLECTION{Delecluse2000,
-  author = {P. Delecluse},
-  title = {Ocean modelling and the role of the ocean in the climate system},
-  pages = {237-313},
-  booktitle = {Modeling the Earth's Climate and its Variability, Les Houches, 1997},
-  year = {2000},
-  editor = {W. R. Holland and  S. Joussaume and F. David},
-  owner = {gm},
-  timestamp = {2007.08.17}
 }
 
Index: trunk/DOC/BETA/Chapters/Abstracts_Foreword.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Abstracts_Foreword.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Abstracts_Foreword.tex	(revision 817)
@@ -9,10 +9,10 @@
 
 \small{ 
-The ocean engine of NEMO is a primitive equation model adapted to regional and global ocean circulation. It is intended to be a flexible tool for studying the ocean and its interactions with the others components of the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. Prognostic variables are the three-dimensional velocity field, the sea surface height, the temperature and the salinity. In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction, a z full or partial step coordinate, or s-coordinate, or a mixture of the two. The distribution of variables is a three-dimensional Arakawa C-type grid. Various physical choices are available to describe ocean physics, including TKE and KPP turbulent closures for the vertical mixing. Within NEMO, the ocean is interfaced with a sea-ice model (LIM), passive tracer and biogeochemical models (TOP) and, via the OASIS coupler, with several atmospheric general circulation models. 
+The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive equation model adapted to regional and global ocean circulation problems. It is intended to be a flexible tool for studying the ocean and its interactions with the others components of the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. Prognostic variables are the three-dimensional velocity field, a linear or non-linear sea surface height, the temperature and the salinity. In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction, a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two. The distribution of variables is a three-dimensional Arakawa C-type grid. Various physical choices are available to describe ocean physics, including TKE and KPP vertical physics. Within NEMO, the ocean is interfaced with a sea-ice model (LIM), passive tracer and biogeochemical models (TOP) and, via the OASIS coupler, with several atmospheric general circulation models. 
 
 % ================================================================
- \vspace{0.25cm}
+ \vspace{0.5cm}
 
-Le moteur oc\'{e}anique de NEMO est un mod\`{e}le aux \'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et globale. Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre spatiotemporel l'oc\'{e}an et ses interactions avec les autres composantes du syst\`{e}me climatique terrestre (atmosph\`{e}re, glace de mer, traceurs biog\'{e}ochimiques...). Les variables pronostiques sont le champ tridimensionnel de vitesse, la hauteur de la mer, la temperature et la salinit\'{e}. La distribution des variables se fait sur une grille C d'Arakawa tridimensionnelle utilisant une coordonn\'{e}e verticale z \`{a} niveaux entiers ou partiels, ou une coordonn\'{e}e s, ou encore une combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour d\'{e}crire la physique oc\'{e}anique, incluant notamment une fermeture turbulente TKE et KPP pour le m\'{e}lange vertical. Via l'infrastructure NEMO, l'oc\'{e}an est interfac\'{e} avec un mod\`{e}le de glace de mer, des mod\`{e}les biog\'{e}ochimiques et de traceur passif, et, via le coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation g\'{e}n\'{e}rale atmosph\'{e}rique. 
+Le moteur oc\'{e}anique de NEMO (Nucleus for European Modelling of the Ocean) est un mod\`{e}le aux \'{e}quations primitives de la circulation oc\'{e}anique r\'{e}gionale et globale. Il se veut un outil flexible pour \'{e}tudier sur un vaste spectre spatiotemporel l'oc\'{e}an et ses interactions avec les autres composantes du syst\`{e}me climatique terrestre (atmosph\`{e}re, glace de mer, traceurs biog\'{e}ochimiques...). Les variables pronostiques sont le champ tridimensionnel de vitesse, une hauteur de la mer lin\'{e}aire ou non, la temperature et la salinit\'{e}. La distribution des variables se fait sur une grille C d'Arakawa tridimensionnelle utilisant une coordonn\'{e}e verticale $z$ \`{a} niveaux entiers ou partiels, ou une coordonn\'{e}e s, ou encore une combinaison des deux. Diff\'{e}rents choix sont propos\'{e}s pour d\'{e}crire la physique oc\'{e}anique, incluant notamment des physiques verticales TKE et KPP . A travers l'infrastructure NEMO, l'oc\'{e}an est interfac\'{e} avec un mod\`{e}le de glace de mer, des mod\`{e}les biog\'{e}ochimiques et de traceur passif, et, via le coupleur OASIS, \`{a} plusieurs mod\`{e}les de circulation g\'{e}n\'{e}rale atmosph\'{e}rique. 
 } 
 
@@ -22,10 +22,14 @@
 \chapter*{Disclaimer}
 
-OPA (an acronym for Ocean PArall\'{e}lis\'{e}) is the ocean component of NEMO (Nucleus for European Modelling of the Ocean (www.locean-ipsl.upmc.fr/NEMO). Like all components of NEMO, it is developed under the CECILL license, which is a french adaptation of the GNU GPL (General Public license). Anyone may use OPA freely for research purposes, and is encouraged to communicate back to the NEMO team its own developments and improvements. The model and the present document have been made available as a service to the community. We cannot certify that the code and its manual are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase. Users are encouraged to bring them to our attention. The author assumes no responsibility for problems, errors, or incorrect usage of OPA.
+Like all components of NEMO, the ocean component is developed under the CECILL license, which is a French adaptation of the GNU GPL (General Public License). Anyone may use it freely for research purposes, and is encouraged to communicate back to the NEMO team its own developments and improvements. The model and the present document have been made available as a service to the community. We cannot certify that the code and its manual are free of errors. Bugs are inevitable and some have undoubtedly survived the testing phase. Users are encouraged to bring them to our attention. The author assumes no responsibility for problems, errors, or incorrect usage of NEMO.
 
  \vspace{1cm}
-The OPA OGCM reference in papers and other publications is as follows:
+NEMO reference in papers and other publications is as follows:
  \vspace{0.5cm}
 
-Madec, G., 2007: NEMO ocean engine. \textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, NXX, \colorbox{yellow}{YYpp}.\\
+Madec, G., 2008: NEMO ocean engine. \textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, \colorbox{yellow}{NXX, 193pp}.\\
 
+ \vspace{0.5cm}
+Additional information can be found on www.locean-ipsl.upmc.fr/NEMO
+ \vspace{0.5cm}
+
Index: trunk/DOC/BETA/Chapters/Annex_A.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Annex_A.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Annex_A.tex	(revision 817)
@@ -3,79 +3,87 @@
 % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
 % ================================================================
-\chapter{Appendix A : Curvilinear $s$-Coordinate Equations}
+\chapter{Curvilinear $s$-Coordinate Equations}
 \label{Apdx_A}
 \minitoc
 
-In order to establish the set of Primitive Equation in curvilinear 
-$s$-coordinates (i.e. orthogonal curvilinear coordinates in the horizontal and 
-$s$-coordinates in the vertical), we start from the set of equation established 
-in {\S}~I.3 for the special case $k = z$ and thus $e_3 = 1$, and we introduce an arbitrary 
-vertical coordinate $s = s(i,j,z)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z)$) and the horizontal slope of $s$-surfaces by :
-\begin{equation} \label{Apdx_A_A1}
+In order to establish the set of Primitive Equation in curvilinear $s$-coordinates ($i.e.$ 
+orthogonal curvilinear coordinate in the horizontal and $s$-coordinate in the vertical), we 
+start from the set of equation established in \S\ref{PE_zco_Eq} for the special case 
+$k = z$ and thus $e_3 = 1$, and we introduce an arbitrary vertical coordinate 
+$s = s(i,j,z,t)$. Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ 
+(which now depends on $(i,j,z,t)$) and the horizontal slope of $s$-surfaces by :
+\begin{equation} \label{Apdx_A_s_slope}
 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
-\quad \text{and}
+\quad \text{and} \quad 
 \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
 \end{equation}
 
-The chain rule to establish the model equations in the curvilinear 
-s-coordinate system is:
-\begin{equation} \label{Apdx_A_A2}
+The chain rule to establish the model equations in the curvilinear $s$-coordinate system 
+is:
+\begin{equation} \label{Apdx_A_s_chain_rule}
 \begin{aligned}
-&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =\left. 
-{\frac{\partial \bullet }{\partial i}} \right|_s +\frac{\partial \bullet 
-}{\partial s}\;\frac{\partial s}{\partial i}=\left. {\frac{\partial \bullet 
-}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet 
-}{\partial s} \\
-&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =\left. 
-{\frac{\partial \bullet }{\partial j}} \right|_s +\frac{\partial \bullet 
-}{\partial s}\;\frac{\partial s}{\partial j}=\left. {\frac{\partial \bullet 
-}{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet 
-}{\partial s} \\
-&\;\frac{\partial \bullet }{\partial z}  =\frac{1}{e_3 }\frac{\partial \bullet 
-}{\partial s} \\
+&\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
+\left. {\frac{\partial \bullet }{\partial t}} \right|_s 
+	 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
+&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
+  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
+  	  -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
+	  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
+	  -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
+&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
+	- \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
+\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
+	- \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
+&\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
 \end{aligned}
 \end{equation}
 
-Using (\ref{Apdx_A_A2}), the divergence of the velocity is transformed as follows:
-
-
-\begin{equation*}
-\nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 }\left[ {\left. 
-{\frac{\partial (e_2 \,u)}{\partial i}} \right|_z +\left. {\frac{\partial 
-(e_1 \,v)}{\partial j}} \right|_z } \right]+\frac{\partial w}{\partial z} \\
-\end{equation*}
-
-%\begin{equation} \label{   }
-\begin{multline*}
-=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}} 
-\right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,u)}{\partial s}} 
-\right. \\ 
-\shoveright { \left. { +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s -\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 v)}{\partial s}} \right]+\frac{\partial w}{\partial s}\frac{\partial s}{\partial z} }\\ 
-\end{multline*}
-%\end{equation}
-
-\begin{equation*}
-%\begin{multline}
-=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,u)}{\partial i}} 
-\right|_s +\left. {\frac{\partial (e_1 \,v)}{\partial j}} \right|_s } 
-\right]+\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1 
-\frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} 
-\right]
-%\end{multline}
-\end{equation*}
-
-%\begin{equation} \label{   }
-\begin{multline*}
- =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 
-\,u)}{\partial i}} \right|_s -\left. {e_2 \,u\frac{\partial e_3 }{\partial 
-i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} 
-\right|_s -\left. {e_1 v\frac{\partial e_3 }{\partial j}} \right|_s } 
-\right] \\ 
-\shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-\sigma _1 
-\frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} 
-\right]} \\ 
-\end{multline*}
-%\end{equation}
-
+In particular applying the time derivative chain rule to $z$ provide the expression of $w_s$,  the vertical velocity of the $s-$surfaces:
+\begin{equation} \label{Apdx_A_w_in_s}
+w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s 
+				= \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} 
+			    = e_3 \, \frac{\partial s}{\partial t} 
+\end{equation}
+
+% ================================================================
+% continuity equation
+% ================================================================
+\section{Continuity Equation}
+\label{Apdx_B_continuity}
+
+Using (\ref{Apdx_A_s_chain_rule}) and the fact that the horizontal scale factors $e_1$ and $e_2$ do not depend on the vertical coordinate, the divergence of the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows:
+
+\begin{align*}
+\nabla \cdot {\rm {\bf U}} 
+&= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z 
+						+\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right]
++ \frac{\partial w}{\partial z} 		\\
+\\
+&     = \frac{1}{e_1 \,e_2 }  \left[ 
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
+		  - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
+		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       
+		  - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}	\right]
+	+ \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\
+\\
+&     = \frac{1}{e_1 \,e_2 }   \left[ 
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
+		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       	\right]
+	+ \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
+						-  \sigma _1 \frac{\partial u}{\partial s}
+						-  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\
+\\
+&     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[ 
+		  \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s  
+		  -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s      
+		+ \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
+		  - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\
+& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+	+ \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
+						-  \sigma _1 \frac{\partial u}{\partial s}
+						-  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\
+\\
+\end{align*}
 
 Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}} 
@@ -86,90 +94,126 @@
 \right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes:
 
-\begin{multline*}
- =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 
-\,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 
-\,v)}{\partial j}} \right|_s } \right] \\ 
-\shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right]} \\ 
- \end{multline*}
+\begin{align*}
+\nabla \cdot {\rm {\bf U}} 
+& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[   
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]        \\ 
+& \qquad \qquad \qquad \qquad \qquad \quad
+ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\ 
+\\
+& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[   
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
+   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right]
+ \end{align*} 
  
-\begin{multline*}
- =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 
-\,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 
-\,v)}{\partial j}} \right|_s } \right] \\ 
-\shoveright{ +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial 
-s}-\frac{\partial (u\;\sigma _1 )}{\partial s}-\frac{\partial (v\;\sigma _2 
-)}{\partial s}} \right]} \\ 
- \end{multline*}
- 
-Introducing a "vertical" velocity $\omega $ as the velocity normal to $s$-surfaces:
-
-\begin{equation} \label{Apdx_A_A3}
-\omega =w-\sigma _1 \,u-\sigma _2 \,v
-\end{equation}
-
-the divergence of the velocity is given in curvilinear $s$-coordinates by:
-\begin{equation} \label{Apdx_A_A4}
-\nabla \cdot {\rm {\bf U}}=\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. 
-{\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s +\left. 
-{\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } 
-\right]+\frac{1}{e_3 }\frac{\partial \omega }{\partial s}
-\end{equation}
-
-
-As a result, the continuity equation (I.1.3) in $s$-coordinates becomes:
+Here, $w$ is the vertical velocity relative to the $z-$coordinate system. Introducing the dia-surface velocity component, $\omega $, defined as the velocity relative to the moving $s$-surfaces and normal to them:
+\begin{equation} \label{Apdx_A_w_s}
+\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
+\end{equation}
+with $w_s$ given by \eqref{Apdx_A_w_in_s}, we obtain the expression of the divergence of the velocity in the curvilinear $s$-coordinate system:
+\begin{align*} \label{Apdx_A_A4}
+\nabla \cdot {\rm {\bf U}} 
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
++ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}     \\
+\\
+&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
+		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
++ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
++ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\
+\end{align*}
+
+As a result, the continuity equation \eqref{Eq_PE_continuity} in $s$-coordinates becomes:
 \begin{equation} \label{Apdx_A_A5}
-\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\frac{\partial (e_2 \,e_3 
-\,u)}{\partial i}} \right|_s +\left. {\frac{\partial (e_1 \,e_3 
-\,v)}{\partial j}} \right|_s } \right]+\frac{1}{e_3 }\frac{\partial \omega 
-}{\partial s}=0
-\end{equation}
-
-
-\textbf{Momentum equation:}
-
-As an example let us consider (I.3.10), the first component of the momentum 
-equation. Its non linear term can be transformed as follows:
-
-\begin{equation*}
-\begin{aligned}
-&+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial 
-(u^2+v^2)}{\partial i}} \right|_z -w\frac{\partial u}{\partial z} \\
-&=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
+\frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
++ \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
+			{\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 
+		    +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
+ +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0   
+\end{equation}
+
+% ================================================================
+% momentum equation
+% ================================================================
+\section{Momentum Equation}
+\label{Apdx_B_momentum}
+
+Let us consider \eqref{Eq_PE_dyn_vect}, the first component of the 
+momentum equation in the vector invariant form (similar manipulations can be performed on the second one). Its non linear term can be transformed 
+as follows:
+
+\begin{align*}
+&+\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 
+- w \;\frac{\partial u}{\partial z} \\
+\\
+&\qquad=\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
 \right|_z -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_z } 
 \right]\;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} 
-\right|_z -w\frac{\partial u}{\partial z}
-\end{aligned}
-\end{equation*}
-\begin{multline*}
- =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
-\right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s } 
-\right. \\ 
+\right|_z -w\frac{\partial u}{\partial z}      \\
+\\
+&\qquad =\frac{1}{e_1 \,e_2 }\left[ {\left. {\frac{\partial (e_2 \,v)}{\partial i}} 
+\right|_s -\left. {\frac{\partial (e_1 \,u)}{\partial j}} \right|_s }     \right. 
  \left. {-\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (e_2 \,v)}{\partial s}+\frac{e_2 }{e_3 }\sigma _2 \frac{\partial (e_1 \,u)}{\partial s}} \right]\;v \\ 
-\shoveright{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right)-\frac{w}{e_3 }\frac{\partial u}{\partial s} }\\
- \end{multline*}
-
-\begin{equation*}
- =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial 
-(u^2+v^2)}{\partial i}} \right|_s -\frac{w}{e_3 }\frac{\partial u}{\partial 
-s}-\left[ {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}-\frac{\sigma 
-_2 }{e_3 }\frac{\partial u}{\partial s}} \right]\;v \\ 
- +\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s} 
-\end{equation*}
-
-
-\begin{multline*}
- =\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
-\shoveright{ -\frac{1}{e_3 }\left[ {w\frac{\partial u}{\partial s}+\sigma _1 v\frac{\partial v}{\partial s}-\sigma _2 v\frac{\partial u}{\partial s}-\sigma _1 u\frac{\partial u}{\partial s}-\sigma _1 v\frac{\partial v}{\partial s}} \right] }\\ 
- \end{multline*}
-
-\begin{equation} \label{Apdx_A_A6}
-=\left. \zeta \right|_s \;v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 
-\end{equation}
+&\qquad \qquad \qquad \qquad \qquad
+{ -\frac{1}{2e_1 }\left( {\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial (u^2+v^2)}{\partial s}} \right)
+-\frac{w}{e_3 }\frac{\partial u}{\partial s} }    \\
+\end{align*}
+\begin{align*}
+\qquad  &= \left. \zeta \right|_s \;v
+	- \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s 
+	- \frac{w}{e_3 }\frac{\partial u}{\partial s}
+	- \left[   {\frac{\sigma _1 }{e_3 }\frac{\partial v}{\partial s}
+	           - \frac{\sigma_2 }{e_3 }\frac{\partial u}{\partial s}}   \right]\;v      \\
+\qquad&\qquad \qquad \qquad \qquad \qquad \qquad
+\qquad  \qquad \qquad \qquad \quad
+	+\frac{\sigma _1 }{2e_3 }\frac{\partial (u^2+v^2)}{\partial s}      \\
+%\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s \\ 
+\qquad&\qquad \qquad \qquad
+ -\frac{1}{e_3} \left[    {w\frac{\partial u}{\partial s}
+	+\sigma _1 v\frac{\partial v}{\partial s} - \sigma _2 v\frac{\partial u}{\partial s}
+	-\sigma _1 u\frac{\partial u}{\partial s} - \sigma _1 v\frac{\partial v}{\partial s}} \right] \\
+\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. \frac{\partial (u^2+v^2)}{\partial i} \right|_s    
+		  - \frac{1}{e_3} \left[  w - \sigma _2 v - \sigma _1 u  \right] 
+		  			 \; \frac{\partial u}{\partial s}   \\
+\\
+\qquad &= \left. \zeta \right|_s \;v
+		- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
+		  - \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 
+		  - \frac{\partial s}{\partial t}  \frac{\partial u}{\partial s} 
+\end{align*}
 
 Therefore, the non-linear terms of the momentum equation have the same form 
-in $z- $and $s-$coordinates
+in $z-$ and $s-$coordinates but with the addition of the time derivative of the velocity: 
+\begin{multline}  \label{Apdx_A_momentum_NL}
++\left. \zeta \right|_z v-\frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_z 
+- w \;\frac{\partial u}{\partial z}    \\
+= - \frac{\partial u}{\partial t} + \left. \zeta \right|_s \;v
+	- \frac{1}{2e_1 }\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s   
+	- \frac{1}{e_3 }\omega \frac{\partial u}{\partial s} 
+\end{multline}
 
 The pressure gradient term can be transformed as follows:
-\begin{equation} \label{Apdx_A_A7}
+\begin{equation} \label{Apdx_A_grad_p}
 \begin{split}
  -\frac{1}{\rho _o e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z& =-\frac{1}{\rho _o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial p}{\partial s}} \right] \\
@@ -179,21 +223,50 @@
 \end{equation}
 
-An additional term appears in (\ref{Apdx_A_A7}) which accounts for the tilt of model 
+An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the tilt of model 
 levels.
 
-\textbf{Tracer equation:}
+Introducing \eqref{Apdx_A_momentum_NL} and \eqref{Apdx_A_grad_p} in \eqref{Eq_PE_dyn_vect} and regrouping the time derivative terms in the left hand side, and performing the same manipulation on the second component, we obtain the vector invariant form of momentum equation in $s-$coordinate :
+\begin{subequations} \label{Apdx_A_dyn_vect}
+\begin{multline} \label{Apdx_A_PE_dyn_vect_u}
+ \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
+	+  g\frac{\rho }{\rho _o}\sigma _1 
+	+   D_u^{\vect{U}}  +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Apdx_A_dyn_vect_v}
+ \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	
+	-   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
+	 +  g\frac{\rho }{\rho _o }\sigma _2   
+	+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+
+It has the same form as in $z-$coordinate but the vertical scale factor that has appeared inside the time derivative. The form of the vertical physics and forcing terms remain unchanged. The form of the lateral physics is discussed in appendix~\ref{Apdx_B}.  
+
+% ================================================================
+% Tracer equation
+% ================================================================
+\section{Tracer Equation}
+\label{Apdx_B_tracer}
 
 The tracer equation is obtained using the same calculation as for the 
-continuity equation:
-
-%\begin{equation} \label{Eq_   }
-\begin{multline} \label{Apdx_A_A8}
- \frac{\partial T}{\partial t} = -\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) + \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \;\right .\\
- \shoveright{\left . +\frac{\partial }{\partial k} \left( {e_1 \,e_2 \;T\omega } \right) \right] +D^{lT} +D^{vT} }\\
-\end{multline}
-%\end{equation}
+continuity equation and then regrouping the time derivative terms in the left hand side :
+
+\begin{multline} \label{Apdx_A_tracer}
+ \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
+ 	= -\frac{1}{e_1 \,e_2 \,e_3 } 
+		\left[ {\frac{\partial }{\partial i}} \left( {e_2 \,e_3 \;Tu} \right) \right .
+		    +         \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right)                \\
+		    + \left. \frac{\partial }{\partial j} \left( {e_1 \,e_3 \;Tv} \right) \right] +D^{T} +F^{T} \; \;
+\end{multline}
 
 
 The expression of the advection term is a straight consequence of (A.4), the 
-expression of the 3D divergence in $s$-coordinates established above.
-
+expression of the 3D divergence in $s$-coordinates established above. 
+
Index: trunk/DOC/BETA/Chapters/Annex_B.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Annex_B.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Annex_B.tex	(revision 817)
@@ -13,29 +13,19 @@
 
 
-In $z$-coordinates, the horizontal/vertical second order tracer diffusive 
-operator is given by:
-
-
-
-\begin{equation} \label{Apdx_B1}
-\begin{split}
- D^T&=D^{lT}+D^{vT} \\ 
- &=\frac{1}{e_1 e_2 }\left[ {\left. {\frac{\partial }{\partial i}\left( 
-{\frac{e_2 }{e_1 }A^{lT} \;\left. {\frac{\partial T}{\partial i}} \right|_z 
-} \right)} \right|_z +\left. {\frac{\partial }{\partial j}\left( {\frac{e_1 
-}{e_2 }A^{lT} \;\left. {\frac{\partial T}{\partial j}} \right|_z } \right)} 
-\right|_z } \right] +\frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial 
-z}} \right) \\ 
-\end{split}
-\end{equation}
-
-
-In $s$-coordinates, we defined the slopes of $s-$surfaces, $\sigma_1$ and $\sigma_2$ by (A.1), the vertical/horizontal ratio of diffusive coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusive operator is given by:
+In $z$-coordinate, the horizontal/vertical second order tracer diffusive operator is given by:
+\begin{multline} \label{Apdx_B1}
+ D^T = \frac{1}{e_1 \, e_2}      \left[ 
+  \left. \frac{\partial}{\partial i} \left( 	\frac{e_2}{e_1}A^{lT} \;\left. \frac{\partial T}{\partial i} \right|_z   \right)   \right|_z      \right.          \\
+						     \left.
++ \left. \frac{\partial}{\partial j} \left(  \frac{e_1}{e_2}A^{lT} \;\left. \frac{\partial T}{\partial j} \right|_z   \right)   \right|_z      \right]         
++ \frac{\partial }{\partial z}\left( {A^{vT} \;\frac{\partial T}{\partial z}} \right)
+\end{multline}
+
+In $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by (A.1) and the vertical/horizontal ratio of diffusive coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusive operator is given by:
 
 \begin{equation} \label{Apdx_B2}
-\begin{aligned} 
-&D^T=D^{lT}+D^{vT} =\left. \nabla \right|_s \cdot \left[ {A^{lT} \;\Re \cdot \left. \nabla 
-\right|_s \left( T \right)} \right] \\
-&\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
+D^T = \left. \nabla \right|_s \cdot 
+           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
+\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
  1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
  0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
@@ -43,38 +33,27 @@
 ^2+\sigma _2 ^2} \hfill \\
 \end{array} }} \right)
-\end{aligned}
 \end{equation}
-
 or in expended form:
-
 \begin{multline} \label{Apdx_B3}
-D^T=\frac{1}{e_1 e_2 e_3 }\;\left[ {\;\,e_2 e_3 A^{lT} \;\left. 
+D^T=\frac{1}{e_1\,e_2\,e_3 }\;\left[ {\quad \; \; e_2\,e_3\,A^{lT} \;\left. 
 {\frac{\partial }{\partial i}\left( {\frac{1}{e_1 }\;\left. {\frac{\partial 
 T}{\partial i}} \right|_s -\frac{\sigma _1 }{e_3 }\;\frac{\partial 
 T}{\partial s}} \right)} \right|_s } \right.  \\
-\;\;+e_1 e_3 A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
- \;\;+e_1 e_2 A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. \\
-\shoveright{\;\;\left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;\,} \right]} \\ 
++e_1\,e_3\,A^{lT} \;\left. {\frac{\partial }{\partial j}\left( {\frac{1}{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s -\frac{\sigma _2 }{e_3 }\;\frac{\partial T}{\partial s}} \right)} \right|_s \\
+ \;\;+e_1\,e_2\,A^{lT} \;\frac{\partial }{\partial s}\left( {-\frac{\sigma _1 }{e_1 }\;\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{\sigma _2 }{e_2 }\;\left. {\frac{\partial T}{\partial j}} \right|_s } \right. \; \\
+\shoveright{\;\;\left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;\,} \right]} 
 \end{multline}
 
 
-Equation (\ref{Apdx_B2}) (or equivalently (\ref{Apdx_B3})~) is obtained from (\ref{Apdx_B1}) without any additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, 
-we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix A 
-\newline
-\newline
-and use (A.1) and (A.2). Since no cross horizontal derivate $\partial _i \,\partial _j $ appears neither in (B.1) nor in (A.2), there is a decoupling between $(i,z)$ and $(j,z)$ plans as well as 
-$(i,s)$ and $(j,s)$ plans. The demonstration can then be done for the $(i,s)\;\to \;(j,s)$ transformation without loss of generality:
-
-\begin{equation*}
-D^T=\frac{1}{e_1 \,e_2 }\left. {\frac{\partial }{\partial i}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)
-\end{equation*}
-
-\begin{multline*}
- =\frac{1}{e_1 \,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
- \left. { -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right] \\ 
+Equation \eqref{Apdx_B2} (or equivalently \eqref{Apdx_B3}) is obtained from \eqref{Apdx_B1} without any additional assumption. Indeed, for the special case $k=z$ and thus $e_3 =1$, we introduce an arbitrary vertical coordinate $s = s (i,j,z)$ as in Appendix~\ref{Apdx_A} and use \eqref{Apdx_A_s_slope} and \eqref{Apdx_A_s_chain_rule}. Since no cross horizontal derivate $\partial _i \partial _j $ appears in \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. The demonstration can then be done for the ($i$,$z$)~$\to$~($j$,$s$) transformation without any loss of generality:
+
+\begin{equation*}
+D^T=\frac{1}{e_1\,e_2 }\left. {\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_z } \right)} \right|_z +\frac{\partial }{\partial z}\left( {A^{vT}\;\frac{\partial T}{\partial z}} \right)        \qquad  \qquad  \qquad  \qquad \qquad  \qquad \qquad     \\
+\end{equation*}
+\begin{multline*}
+ =\frac{1}{e_1\,e_2 }\left[ {\left. {\;\frac{\partial }{\partial i}\left( {\frac{e_2}{e_1}A^{lT}\;\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right)} \right|_s } \right. \\ 
+ \left. { -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\left( {\left. {\frac{\partial T}{\partial i}} \right|_s -\frac{e_1 \,\sigma _1 }{e_3 }\frac{\partial T}{\partial s}} \right)} \right|_s } \right)\;} \right]   
 \shoveright{ +\frac{1}{e_3 }\frac{\partial }{\partial s}\left[ {\frac{A^{vT}}{e_3 }\;\frac{\partial T}{\partial s}} \right]} \\ 
 \end{multline*}
-
-
 \begin{multline*}
  =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s -\left. {\frac{e_2 }{e_1 
@@ -85,14 +64,12 @@
 \end{multline*}
 
-
 Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:
 
 \begin{multline*}
- =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-e_3 \frac{\partial }{\partial 
-i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
- -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
+ =\frac{1}{e_1\,e_2\,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2\,e_3 }{e_1}\,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. 
+-\, {e_3 \frac{\partial }{\partial i}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
+\qquad -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s -e_1 \,\sigma_1 \frac{\partial }{\partial s}\left( {\frac{e_2 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
 \shoveright{ \left. { +e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial z}} \right)\;\;\;} \right] }\\ 
 \end{multline*}
-
 \begin{multline*}
  =\frac{1}{e_1 \,e_2 \,e_3 } \left[ {\left. {\;\;\;\frac{\partial }{\partial i} \left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)} \right|_s \left. {-\frac{\partial }{\partial 
@@ -110,5 +87,6 @@
 \end{multline*}
 
-Since the horizontal scale factor do not depend on the vertical coordinates, the last term of the first line and the first term of the last line cancel, while the second line reduces to a single vertical derivative, so it becomes:
+Since the horizontal scale factor do not depend on the vertical coordinate, the last term of the first line and the first term of the last line cancel, while the second line reduces to a single vertical derivative, so it becomes:
+
 \begin{multline*}
  =\frac{1}{e_1 \,e_2 \,e_3 }\left[ {\left. {\;\;\;\frac{\partial }{\partial i}\left( {\frac{e_2 \,e_3 }{e_1 }A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s -e_2 \,\sigma _1 \,A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
@@ -116,19 +94,19 @@
 \end{multline*}
 
-in other words:
+in other words, the horizontal Laplacian operator in the ($i$,$s$)plane takes the following expression :
 
 \begin{equation*}
 D^T = {\frac{1}{e_1\,e_2\,e_3}}
-\left( {{\begin{array}{*{20}c}
+\left( {{\begin{array}{*{30}c}
 {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
 {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
 \end{array}}}\right)
 \cdot \left[ {A^{lT}
-\left( {{\begin{array}{*{20}c}
+\left( {{\begin{array}{*{30}c}
  {1} \hfill & {-\sigma_1 } \hfill \\
  {-\sigma_1} \hfill & {\varepsilon_1^2} \hfill \\
 \end{array} }} \right)
 \cdot 
-\left( {{\begin{array}{*{20}c}
+\left( {{\begin{array}{*{30}c}
 {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
 {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
@@ -136,14 +114,14 @@
 \right) \left( T \right)} \right]
 \end{equation*}
-
-
-% ================================================================
-% Isopycnal/Vertical Second Order Tracer Diffusive Operators
-% ================================================================
-\section{Isopycnal/Vertical Second Order Tracer Diffusive Operators}
+ 
+
+% ================================================================
+% Isopycnal/Vertical 2nd Order Tracer Diffusive Operators
+% ================================================================
+\section{Iso/diapycnal 2nd Order Tracer Diffusive Operators}
 \label{Apdx_B_2}
 
 
-The isopycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the curvilinear coordinate system $(i,j,k)$, in which the equations of the ocean circulation model are formulated, takes the following expression [Redi 1982]:
+The iso/diapycnal diffusive tensor $\textbf {A}_{\textbf I}$ expressed in the  ($i$,$j$,$k$) curvilinear coordinate system in which the equations of the ocean circulation model are formulated, takes the following expression \citep{Redi_JPO82}:
 
 \begin{equation*}
@@ -155,15 +133,15 @@
 \end{array} }} \right]
 \end{equation*}
-where$(a_1$, $a_2)$ are the isopycnal slopes in $(\textbf{i}$, $\textbf{j})$ directions:
+where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions:
 \begin{equation*}
 a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
-\end{equation*}
-\begin{equation*}
+\qquad , \qquad
 a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}} 
 \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
 \end{equation*}
-In practice, the isopycnal slopes are generally less than $10${\small $^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably (Cox, 1987) :
-\begin{equation*}
-{\textbf{A}_{\textbf{I}}} =A^{lT}
+
+In practice, the isopycnal slopes are generally less than $10^{-2}$ in the ocean, so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
+\begin{equation*}
+{\textbf{A}_{\textbf{I}}} \approx A^{lT}
 \left[ {{\begin{array}{*{20}c}
  1 \hfill & 0 \hfill & {-a_1 } \hfill \\
@@ -172,29 +150,26 @@
 \end{array} }} \right]
 \end{equation*}
-The resulting isopycnal operator conserves the quantity it diffuses, and dissipates the square of this quantity. The demonstration of the first property is trivial. Let us demonstrate the second one:
+
+The resulting isopycnal operator conserves the quantity and dissipates its square. The demonstration of the first property is trivial as \eqref{Apdx_B2} is the divergence of fluxes. Let us demonstrate the second one:
 \begin{equation*}
 \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
 \end{equation*}
 since
-\begin{multline*}
+\begin{align*}
  \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
-\right)=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
+\right)&=A^{lT}\left[ {\left( {\frac{\partial T}{\partial i}} \right)^2-2a_1 
 \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
 {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
-\shoveright{ \left. {-2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial 
+&\qquad \qquad \qquad \quad
+{ \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial 
 k}+\left( {a_1 ^2+a_2 ^2} \right)\left( {\frac{\partial T}{\partial k}} 
 \right)^2} \right]} \\
-\end{multline*}
-
-\begin{equation*}
-=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]\quad \geq 0
-\end{equation*}
-the property becomes obvious.
-\\
-Note that the resulting tensor is similar to those obtained for geopotential diffusion in $s$-coordinates. The simplification leads to a decoupling between $(i,z)$ and $(j,z)$ plans.
+&=A_h \left[ {\left( {\frac{\partial T}{\partial i}-a_1 \frac{\partial T}{\partial k}} \right)^2+\left( {\frac{\partial T}{\partial j}-a_2 \frac{\partial T}{\partial k}} \right)^2} \right]\quad \geq 0
+\end{align*}
+the property becomes obvious. 
 
 The resulting diffusive operator in $z$-coordinates has the following 
 expression :
-\begin{multline*}
+\begin{multline*} \label{Apdx_B_ldfiso}
  D^T=\frac{1}{e_1 e_2 }\left\{ {\;\frac{\partial }{\partial i}\left[ {A_h \left( {\frac{e_2 }{e_1 }\frac{\partial T}{\partial i}-a_1 \frac{e_2}{e_3}\frac{\partial T}{\partial k}} \right)} \right]\;} \right.\;\; \\ 
  \;\left. {\;\;\;+\frac{\partial}{\partial j}\left[ {A_h \left( {\frac{e_1 }{e_2 }\frac{\partial T}{\partial j}-a_2 \frac{e_1 }{e_3 }\frac{\partial T}{\partial k}} \right)} \right]\;} \right\} \\
@@ -202,4 +177,5 @@
 \end{multline*}
 
+It has to be emphasised that the simplification introduced leads to a decoupling between ($i$,$z$) and ($j$,$z$) planes. The operator has therefore the same expression as \eqref{Apdx_B3}, the diffusive operator obtained for geopotential diffusion in $s$-coordinate. 
 
 % ================================================================
@@ -209,17 +185,11 @@
 \label{Apdx_B_3}
 
-
-*{\footnotesize }lateral/vertical{\footnotesize }2{\small nd}{\footnotesize 
-}order{\footnotesize }momentum diffusive operator
-
-Following (I.3.6), the Laplacian of the horizontal velocity can be expressed 
-in $z$-coordinates:
-\begin{equation*}
-\Delta {\textbf{U}}_h =\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
-\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)
-\end{equation*}
-
-\begin{equation*}
-=\left( {{\begin{array}{*{20}c}
+The second order momentum diffusive operator (Laplacian) in $z$-coordinate is found by applying \eqref{Eq_PE_lap_vector}, the expression of the Laplacian of a vector,  to the horizontal velocity vector : 
+\begin{align*}
+\Delta {\textbf{U}}_h 
+&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
+\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
+\\
+&=\left( {{\begin{array}{*{20}c}
  {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
  {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
@@ -237,7 +207,7 @@
 \hfill \\
 \end{array} }} \right)
-\end{equation*}
-\begin{equation*}
-=\left( {{\begin{array}{*{20}c}
+\\
+\\
+&=\left( {{\begin{array}{*{20}c}
 {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
 {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
@@ -250,42 +220,32 @@
 {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
 \end{array} }} \right)
-\end{equation*}
-Using (I.3.8), the definition of the horizontal divergence, the third componant of the second vector is obviously zero and thus :
+\end{align*}
+Using \eqref{Eq_PE_div}, the definition of the horizontal divergence, the third componant of the second vector is obviously zero and thus :
 \begin{equation*}
 \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
 \end{equation*}
 
-
-The lateral/vertical second order (Laplacian type) operator used to diffuse 
-horizontal momentum in $z$-coordinates therefore takes the following expression :
-\begin{equation*}
-\begin{split}
- {\textbf{D}}^{\textbf{U}}&={\textbf{D}}^{lm}+{\textbf{D}}^{vm} \\ 
-&=\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)+\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
-\end{split}
-\end{equation*}
-
-\begin{equation*}
-{\textbf{D}}^{\textbf{U}}=
-\left( {{\begin{array}{*{20}c}
-{\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial i}-\frac{1}{e_2 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}} \\
-{\frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi } \right)}{\partial j}+\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}} \\
-\end{array} }} \right)
-+\frac{1}{e_3 }\frac{\partial }{\partial k}
-\left( {{\begin{array}{*{20}c}
-{\frac{A^{vm}}{e_3 }\frac{\partial u}{\partial k}} \\
-{\frac{A^{vm}}{e_3 }\frac{\partial v}{\partial k}} \\
-\end{array} }} \right)
-\end{equation*}
-
-% ================================================================
-% References
-% ================================================================
-\section{References}
-
-
-{\small Cox, M. D., 1987 : Isopycnal diffusion in a z-coordinate ocean 
-model. }{\small \textit{Ocean Modelling}}{\small , 74, 1-9.}
-
-{\small Redi, M. H., 1982 : oceanic isopycnal mixing by coordinate rotation. 
-}{\small \textit{J. Phys. Oceanogr., 13}}{\small , 1154-1158.}
+Note that this operator ensures a full separation between the vorticity and horizontal divergence fields (see Appendix~\ref{Apdx_C}). It is only equal to a Laplacian applied on each component in Cartesian coordinate, not on the sphere.
+
+The horizontal/vertical second order (Laplacian type) operator used to diffuse 
+horizontal momentum in $z$-coordinate takes therefore the following expression :
+\begin{equation} \label{Apdx_B_Lap_U}
+ {\textbf{D}}^{\textbf{U}} = 
+ 	  \nabla _h \left( {A^{lm}\;\chi } \right)
+	- \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
+	+ \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
+				\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\ 
+\end{equation}
+that is in expanded form:
+\begin{align*}
+D^{\textbf{U}}_u 
+& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
+     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
+     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
+D^{\textbf{U}}_v   
+& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
+     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
+     +\frac{1}{e_3} \frac{\partial v}{\partial k}
+\end{align*}
+
+Note Bene: introducing a rotation in \eqref{Apdx_B_Lap_U} does not lead to any useful expression for the iso/diapycnal Laplacian operator in $z$-coordinate. Similarly, we did not found an expression of practical use for the geopotential horizontal/vertical Laplacian operator in $s$-coordinate. Generally, \eqref{Apdx_B_Lap_U} is used in both $z$- and $s$-coordinate system, that is a Laplacian diffusion is applied on momentum along the coordinate directions.
Index: trunk/DOC/BETA/Chapters/Annex_C.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Annex_C.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Annex_C.tex	(revision 817)
@@ -2,7 +2,12 @@
 % Chapter Ñ Appendix C : Discrete Invariants of the Equations
 % ================================================================
-\chapter{Appendix C : Discrete Invariants of the Equations}
+\chapter{Discrete Invariants of the Equations}
 \label{Apdx_C}
 \minitoc
+
+%%%  Appendix put in gmcomment as it has not been updated for z* and s coordinate
+I'm writting this appendix. It will be available in a forthcoming release of the documentation
+
+\gmcomment{
 
 % ================================================================
@@ -14,5 +19,8 @@
 
 First, the boundary condition on the vertical velocity (no flux through the surface and the bottom) is established for the discrete set of momentum equations. Then, it is shown that the non linear terms of the momentum equation are written such that the potential enstrophy of a horizontally non divergent flow is preserved while all the other non-diffusive terms preserve the kinetic energy: the energy is also preserved in practice. In addition, an option is also offer for the vorticity term discretization which provides 
-a total kinetic energy conserving discretization for that term. Note that although these properties are established in the curvilinear $s$-coordinate system, they still hold in the curvilinear $z$-coordinate system.
+a total kinetic energy conserving discretization for that term. 
+
+Nota Bene: these properties are established here in the rigid-lid case and for the 2nd order centered scheme. A forthcoming update will be their generalisation to the free surface case
+and higher order scheme.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -25,18 +33,12 @@
 The discrete set of momentum equations used in rigid lid approximation 
 automatically satisfies the surface and bottom boundary conditions 
-($w_{surface} =w_{bottom} =~0$, no flux through the surface and the bottom). 
+(no flux through the surface and the bottom: $w_{surface} =w_{bottom} =~0$). 
 Indeed, taking the discrete horizontal divergence of the vertical sum of the 
 horizontal momentum equations (Eqs. (II.2.1) and (II.2.2)~) wheighted by the 
 vertical scale factors, it becomes:
 \begin{flalign*}
-\frac{\partial } {\partial t}
-	\left( 
-	\sum\limits_k \chi 
-	\right)
-\equiv \frac{\partial } {\partial t}
-	\left( 
-	w_{surface} -w_{bottom}  
-	\right)
-	&&&\\
+\frac{\partial } {\partial t}  \left( \sum\limits_k    \chi    \right)
+\equiv 
+\frac{\partial } {\partial t}  \left( 	w_{surface} -w_{bottom}     \right)&&&\\
 \end{flalign*}
 \begin{flalign*}
@@ -91,5 +93,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection{Coriolis and advection terms: vector invariant form}
-\label{Apdx_C.1.2}
+\label{Apdx_C_vor_zad}
 
 % -------------------------------------------------------------------------------------------------------------
@@ -97,111 +99,73 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsubsection{Vorticity Term}
-\label{Apdx_C.1.2.1} 
-
-Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity. It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs (II.1.10) and (II.1.11)), it can be shown that:
+\label{Apdx_C_vor} 
+
+Potential vorticity is located at $f$-points and defined as: $\zeta / e_{3f}$. The standard discrete formulation of the relative vorticity term obviously conserves potential vorticity (ENS scheme). It also conserves the potential enstrophy for a horizontally non-divergent flow (i.e. $\chi $=0) but not the total kinetic energy. Indeed, using the symmetry or skew symmetry properties of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can be shown that:
 \begin{equation} \label{Apdx_C_1.1}
-\int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3 }\nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
+\int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
 \end{equation}
-
-where $dv=e_1 \,e_2 \,e_3 \;di\,dj\,dk$ is the volume element. Indeed, using 
-(II.2.11), the discrete form of the right hand side of (C.1.1) can be 
-transformed as follow:
+where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using 
+\eqref{Eq_dynvor_ens}, the discrete form of the right hand side of \eqref{Apdx_C_1.1} 
+can be transformed as follow:
 \begin{flalign*} 
-\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
+&\int_D \zeta / e_3\,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
 	\left( 
 	\zeta \; \textbf{k} \times  \textbf{U}_h  
 	\right)\;
 	dv
-	&&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+	&&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k} 
 \frac{\zeta / e_{3f}} {e_{1f}\,e_{2f}\,e_{3f}} 
 	\biggl\{ \quad
 	\delta_{i+1/2} 
-		&\left[ 
+		\left[ 
 		   - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\;
 		      \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 
 		 \right]  
-	&&
-	\biggr. \\ 
-	\biggl. 
+	&&  \\ & \qquad \qquad \qquad \;\;
 	- \delta_{j+1/2} 
-		&\left[	\;\;\;
+		\left[	\;\;\;
 		     \overline {\left( \zeta / e_{3f} \right)}^{\,j}\;
 		     \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
 		\right] 
-	\biggr\} \;
-	e_{1f}\,e_{2f}\,e_{3f}
-	&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	\biggl\{	\quad
-	\delta_i 
-		&\left[ \zeta / e_{3f} \right] \;
-		\overline {\left( \zeta / e_{3f} \right)}^{\,i}\; 
-		\overline{\overline {\left( e_{1u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
-		&& 
-	\biggr. \\ 
-	\biggl. 
-	+ \delta_j 
-		&\left[ \zeta / e_{3f} \right] \;
-		\overline {\left( \zeta / e_{3f} \right)}^{\,j} \; 
-		\overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}	\;
+	\;\;\biggr\} \;  e_{1f}\,e_{2f}\,e_{3f}       && \displaybreak[0] \\ 
+%
+\equiv& \sum\limits_{i,j,k} 
+	\biggl\{   \delta_i     \left[   \frac{\zeta} {e_{3f}}   \right] \;
+		     \overline{  \left(   \frac{\zeta} {e_{3f}}   \right)  }^{\,i}\; 
+		     \overline{  \overline{   \left( e_{1u}\,e_{3u}\,u \right)  }  }^{\,i,j+1/2} 
+			+ \delta_j   \left[   \frac{\zeta} {e_{3f}}   \right] \;
+        		\overline{   \left(  \frac{\zeta} {e_{3f}}    \right)  }^{\,j} \; 
+		\overline{\overline {\left( e_{2v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}         \biggr\} 
+		&&&& \displaybreak[0] \\ 
+%
+\equiv& \frac{1} {2} \sum\limits_{i,j,k}  
+	\biggl\{	\delta_i    \Bigl[    \left(  \frac{\zeta} {e_{3f}} \right)^2   \Bigr]\;
+			\overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
+		      + \delta_j 	\Bigl[    \left( \zeta / e_{3f} \right)^2     \Bigr]\; 
+			\overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
 	\biggr\} 
-		&&  \\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \frac{1} {2} \sum\limits_{i,j,k}  
-	\biggl\{	\quad
-	\delta_i 
-		&\Bigl[ 
-			\left( \zeta / e_{3f} \right)^2 
-		\Bigr]\;
-	\overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
-	&&
-	\biggr. \\ 
-	\biggl. 
-	+ \delta_j 
-		&\Bigl[ 
-			\left( \zeta / e_{3f} \right)^2 
-		\Bigr]\; 
-		\overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
-	\biggr\} 
-	&&  \\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv - \frac{1} {2} \sum\limits_{i,j,k} 	\left( \zeta / e_{3f} \right)^2\;
-	\biggl\{		\quad
-	\delta_{i+1/2} 
-		&\left[ 
-		\overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2} 
-		\right]  
-		&&
-	\biggr. \\ 
-	\biggl. 
-	+ \delta_{j+1/2}
-		&\left[ 
-		\overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j} 
-		\right]  
-	\biggr\} 
-	&& \\ 
-\end{flalign*}
-
-
+	&& \displaybreak[0] \\ 
+%
+\equiv& - \frac{1} {2} \sum\limits_{i,j,k} 	\left(  \frac{\zeta} {e_{3f}} \right)^2\;
+	\biggl\{    \delta_{i+1/2} 
+	      \left[   \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/2}    \right]  
+	            + \delta_{j+1/2}
+		\left[   \overline{\overline {\left( e_{1v}\,e_{3v}\,v \right)}}^{\,i+1/2,j}     \right]  
+	\biggr\}    && \\ 
+\end{flalign*}
 Since $\overline {\;\cdot \;} $ and $\delta $ operators commute: $\delta_{i+1/2} 
 \left[ {\overline a^{\,i}} \right] = \overline {\delta_i \left[ a \right]}^{\,i+1/2}$, 
 and introducing the horizontal divergence $\chi $, it becomes:
-\begin{equation*}
-\equiv \sum\limits_{i,j,k} - \frac{1} {2} \left( \zeta / e_{3f} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \equiv 0
-\end{equation*}
+\begin{align*}
+\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left(  \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 
+\qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\
+\end{align*}
 
 Note that the demonstration is done here for the relative potential 
-vorticity but it still hold for the planetary ($f/e_3$ ) and the total 
+vorticity but it still hold for the planetary ($f/e_3$) and the total 
 potential vorticity $((\zeta +f) /e_3 )$. Another formulation of 
-the two components of the vorticity term is optionally offered :
+the two components of the vorticity term is optionally offered (ENE scheme) :
 \begin{equation*}
 \frac{1} {e_3 }\nabla \times 
@@ -228,43 +192,21 @@
 enstrophy on the relative vorticity term and energy on the Coriolis term.
 \begin{flalign*}
-\int\limits_D \textbf{U}_h \times 
-	\left( 
-	\zeta \;\textbf{k} \times \textbf{U}_h 
-	\right)\;
-	dv
-	&&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	\biggl\{ 	\quad
-	&   \overline {\left( \zeta / e_{3f}      \right) 
-	     \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} \; e_{2u}\,e_{3u}\,u 
-	&&
-	\biggr. \\
-	\biggl. 
-	-& \overline {\left( \zeta / e_{3f}       \right) 
-	    \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} \; e_{1v}\,e_{3v}\,v \;
-	\biggr\} 
-	&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}
-\Bigl( 
-\zeta / e_{3f} 
-\Bigr)\;
-	\biggl\{	\quad
-	&  \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\;\;
-	    \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2} 
-	&&
-	\biggr. \\ 
-	\biggl. 
-	-& \overline {\left( e_{2u}\,e_{3u} \,u \right)}^{\,j+1/2}\;\;
-	     \overline {\left( e_{1v}\,e_{3v} \,v \right)}^{\,i+1/2}\; 
-	\biggr\} \; 
+&\int\limits_D \textbf{U}_h \times   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
+\equiv& \sum\limits_{i,j,k} 	\biggl\{ 	
+	   \overline {\left(  \frac{\zeta} {e_{3f}}      \right) 
+	     \overline {\left( e_{1v}e_{3v}v \right)}^{\,i+1/2}} ^{\,j} \, e_{2u}e_{3u}u 
+	- \overline {\left(  \frac{\zeta} {e_{3f}}       \right) 
+	    \overline {\left( e_{2u}e_{3u}u \right)}^{\,j+1/2}} ^{\,i} \, e_{1v}e_{3v}v \;
+	                                \biggr\}     
+\\
+\equiv& \sum\limits_{i,j,k}  \frac{\zeta} {e_{3f}}
+	\biggl\{  \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}\;
+	    	    \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}        
+		  - \overline {\left( e_{2u}e_{3u} u \right)}^{\,j+1/2}\;
+	            \overline {\left( e_{1v}e_{3v} v \right)}^{\,i+1/2}
+	\biggr\}
 	\equiv 0
-	&& \\ 
-\end{flalign*}
+\end{flalign*}
+
 
 % -------------------------------------------------------------------------------------------------------------
@@ -272,56 +214,30 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsubsection{Gradient of Kinetic Energy / Vertical Advection}
-\label{Apdx_C.1.2.2} 
+\label{Apdx_C_zad} 
 
 The change of Kinetic Energy (KE) due to the vertical advection is exactly 
 balanced by the change of KE due to the horizontal gradient of KE~:
 \begin{equation*}
-      \int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\;\textbf{U}_h^2 \right)\;dv
+      \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv
  = - \int_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv
 \end{equation*}
-Indeed, using successively (II.1.10) and the continuity of mass (II.2.5), 
-(II.1.10), and the commutativity of operators $\overline {\,\cdot \,} $ and 
-$\delta$, and finally (II.1.11) successively in the horizontal and in the vertical 
-direction, it becomes:
-\begin{flalign*}
-\int_D \textbf{U}_h \cdot \nabla_h \left( 1/2\; \textbf{U}_h^2 \right)\;dv	&&&\\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \frac{1} {2} \sum\limits_{i,j,k} 
-	\biggl\{	\quad
-	&\frac{1} {e_{1u}}  \delta_{i+1/2} 
-		\left[ 
-		   \overline {u^2}^{\,i} 
-		+ \overline {v^2}^{\,j} 
-		\right]
-		\;u\;e_{1u}\,e_{2u}\,e_{3u}  
-		&&
-	\biggr. \\ 
-	\biggl. 
-	& + \frac{1} {e_{2v}}  \delta_{j+1/2} 
-		\left[ 
-		   \overline {u^2}^{\,i} 
-		+ \overline {v^2}^{\,j} 
-		\right]
-		\;v\;e_{1v}\,e_{2v}\,e_{3v}  \;
+Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) and the incompressibility, then again \eqref{DOM_di_adj}, then 
+the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} ($i.e.$ the  symmetry property of the $\overline {\,\cdot \,}$ operator) applied in the horizontal and vertical direction, it becomes:
+\begin{flalign*}
+&\int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv	&&&\\
+\equiv& \frac{1}{2} \sum\limits_{i,j,k} 
+	\biggl\{	
+	\frac{1} {e_{1u}}  \delta_{i+1/2} 
+	\left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}    \right]  u\,e_{1u}e_{2u}e_{3u}  
+     + \frac{1} {e_{2v}}  \delta_{j+1/2} 
+	\left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  v\,e_{1v}e_{2v}e_{3v} 
 	\biggr\} 
-	&& \\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv  &\frac{1} {2}\quad \sum\limits_{i,j,k} 
-	\left( 
-	   \overline {u^2}^{\,i}
-	+ \overline {v^2}^{\,j} 
-	\right)\;
-\delta_k 
-	\left[
-	 e_{1T}\,e_{2T} \,w 
-	 \right]
-	 && \\
-\end{flalign*}
-\begin{flalign*}
-\equiv &\frac{1} {2} - \sum\limits_{i,j,k}  \delta_{k+1/2} 
+	&&& \displaybreak[0] \\ 
+%
+\equiv&  \frac{1}{2} \sum\limits_{i,j,k} 
+	\left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)\;
+	\delta_k \left[  e_{1T}\,e_{2T} \,w   \right]
+%
+\;\; \equiv -\frac{1}{2} \sum\limits_{i,j,k}  \delta_{k+1/2} 
 	\left[ 
 	   \overline{ u^2}^{\,i} 
@@ -329,38 +245,26 @@
 	\right] \;
 	e_{1v}\,e_{2v}\,w 
-	&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv &\frac{1} {2} \quad \sum\limits_{i,j,k} 
-	\left( 
-	   \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
-	+ \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j} 
-	\right)\;
-	e_{1T}\,e_{2T} \,w 
-	&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv \frac{1} {2} \quad \sum\limits_{i,j,k} 
-	\biggl\{	\quad
-	& \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 	\overline {u}^{\,k+1/2}\; \delta_{k+1/2} 
-		\left[ u \right] 
-	&&
-	\biggr. \\ 
-	\biggl.
-	& + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 	\overline {v}^{\,k+1/2}\; \delta_{k+1/2} 			\left[ v \right]  \;
+	&&& \displaybreak[0]\\
+%
+\equiv &\frac{1} {2} \sum\limits_{i,j,k} 
+	\left(    \overline {\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
+		+ \overline {\delta_{k+1/2} \left[ v^2 \right]}^{\,j}    \right) \; e_{1T}\,e_{2T} \,w 
+	&& \displaybreak[0] \\
+%	
+\equiv &\frac{1} {2} \sum\limits_{i,j,k} 
+	\biggl\{  \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\;2 	
+			\overline {u}^{\,k+1/2}\; \delta_{k+1/2}         \left[ u \right]     %&&&  \\
+	 + \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2}\;2 	\overline {v}^{\,k+1/2}\; \delta_{k+1/2} 			\left[ v \right]  \;
 	\biggr\} 
-	&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \quad -\sum\limits_{i,j,k}  
-	\biggl\{	\quad
-	&\frac{1} {b_u } \;
+	&&\displaybreak[0] \\ 
+%
+\equiv& -\sum\limits_{i,j,k}  
+	\biggl\{
+	\quad \frac{1} {b_u } \;
 	\overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,i+1/2}\,\delta_{k+1/2}
 		\left[ u \right] 
 		       \Bigr\} }^{\,k} \;u\;e_{1u}\,e_{2u}\,e_{3u}  
-	&&
-	\biggr. \\ 
-	\biggl.  
-	& + \frac{1} {b_v } \;
+	&& \\
+	&\qquad \quad\; + \frac{1} {b_v } \;
 	\overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2}
 		 \left[ v \right] 
@@ -368,10 +272,8 @@
 	\biggr\} 
 	&& \\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv	&&&\\
-\end{flalign*}
-
-The main point here is that the respect of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be defined as $1/2(\overline u^{\,i} + \overline v^{\,j})$, but this leads to the following expression for the vertical advection~:
+\equiv& -\int\limits_D \textbf{U}_h \cdot w \frac{\partial \textbf{U}_h} {\partial k}\;dv	&&&\\
+\end{flalign*}
+
+The main point here is that the satisfaction of this property links the choice of the discrete formulation of vertical advection and of horizontal gradient of KE. Choosing one imposes the other. For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following expression for the vertical advection:
 \begin{equation*}
 \frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k}
@@ -385,6 +287,6 @@
 \end{array}} } \right)
 \end{equation*}
-
-This formulation requires one more horizontal mean, and thus the use of 9 velocity points instead of 3. This is the reason why it has not been retained in the model.
+a formulation that requires a additional horizontal mean compare to the one used in NEMO. Nine velocity points have to be used instead of 3. This is the reason why it has not been choosen.
+
 % -------------------------------------------------------------------------------------------------------------
 %       Coriolis and advection terms: flux form
@@ -399,5 +301,5 @@
 \label{Apdx_C.1.3.1} 
 
-In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is thus discretised at F-point. It is given by: 
+In flux from the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the ``metric'' term. This altered Coriolis parameter is discretised at F-point. It is given by: 
 \begin{equation*}
 f+\frac{1} {e_1 e_2 } 
@@ -409,5 +311,5 @@
 \end{equation*}
 
-Then any of the scheme presented above for the vorticity term in the vector invariant formulation can be used, except of course the mixed scheme. However, the energy conserving scheme has exclusively been used to date.
+The ENE scheme is then applied to obtain the vorticity term in flux form. It therefore conserves the total KE. The demonstration is same as for the vorticity term in vector invariant form (\S\ref{Apdx_C_vor}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -417,5 +319,5 @@
 \label{Apdx_C.1.3.2} 
 
-The flux form operator of the momentum advection is evaluated using a second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Moreover, it conserves the horizontal kinetic energy, that is :
+The flux form operator of the momentum advection is evaluated using a centered second order finite difference scheme. Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is :
 
 \begin{equation} \label{Apdx_C_I.3.10}
@@ -429,100 +331,49 @@
 Let us demonstrate this property for the first term of the scalar product (i.e. considering just the the terms associated with the i-component of the advection):
 \begin{flalign*}
-\int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right)\;dv	&&&\\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+&\int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv    &&&\\
+%
+\equiv& \sum\limits_{i,j,k} 
+\biggl\{    \frac{1} {e_{1u}\, e_{2u}\,e_{3u}}    \biggl(   
+      \delta_{i+1/2}  \left[   \overline {e_{2u}\,e_{3u}\,u}^{\,i}      \;\overline u^{\,i}          \right]   
+   + \delta_j           \left[   \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right] 
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad
+   + \delta_k          \left[   \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} \right]
+			\biggr)   \;   \biggr\} \, e_{1u}\,e_{2u}\,e_{3u} \;u 
+		&&& \displaybreak[0] \\ 
+%
+\equiv& \sum\limits_{i,j,k} 
 	\biggl\{ 
-	\frac{1} {e_{1u}\, e_{2u}\,e_{3u}} 
-		\biggl(	\quad	\biggr.
-		&    \delta_{i+1/2} 
-			\left[ 
-			\overline {e_{2u}\,e_{3u}\,u}^{\,i}\;\overline u^{\,i} 
-			\right]
-		&& \\
-		& + \delta_j 
-			\left[ 
-			\overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 
-			\right] 
-		&&
-	\biggr.\\ 
-	\biggl. 
-		& + \delta_k  		\biggl. 
-			\left[ 
-			\overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;\overline u^{\,k+1/2} 
-			\right]\; 
-		\biggr)\; 
-	\biggr\} \, 
-	e_{1u}\,e_{2u}\,e_{3u} \;u 
-	&& \\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	\biggl\{ 
-	\delta_{i+1/2} 
-		\left[ 
-		\overline {e_{2u}\,e_{3u}\,u}^{\,i}\;	  \overline u^{\,i} 
-		\right]
-	& + \delta_j 
-		\left[ 
-		\overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2} 
-		\right]
-	&&
-	\biggr.\\
-	\biggl.
-	& + \delta_k 
-		\left[ 
-		\overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2} 
-		\right]\; 
-	\biggr\}
-	&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}
+      \delta_{i+1/2} \left[   \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;  \overline u^{\,i}  \right]
+   + \delta_j          \left[   \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right]
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+   + \delta_k         \left[   \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2}  \right]
+   	\; \biggr\}    &&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
 	\biggl\{ 
 	   \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;			\overline u^{\,i} 		\delta_i 
 		\left[ u \right] 
-	& + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;	\overline u^{\,j+1/2} 	\delta_{j+1/2} 
+        + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2}\;	\overline u^{\,j+1/2} 	\delta_{j+1/2} 
 		\left[ u \right] 
-	&&
-	\biggr. \\
-	\biggl.
-	& + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;	\overline u^{\,k+1/2}	\delta_{k+1/2} 
-		\left[ u \right] 
-	\biggr\} 
-	&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}
+		&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+       + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\;	\overline u^{\,k+1/2}	\delta_{k+1/2} 	\left[ u \right]     \biggr\}     && \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
 	\biggl\{ 
 	     \overline {e_{2u}\,e_{3u}\,u}^{\,i} 			\delta_i 		\left[ u^2 \right] 
-	& + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} 		\delta_{j+/2} 	\left[ u^2 \right]
-	&& 
-	\biggr. \\
-	\biggl.
-	& + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} 	\delta_{k+1/2} 	\left[ u^2 \right] 
+	 + \overline {e_{1u}\,e_{3u}\,v}^{\,i+1/2} 		\delta_{j+/2} 	\left[ u^2 \right]
+	 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} 	\delta_{k+1/2} 	\left[ u^2 \right] 
 	\biggr\} 
-	&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}
+	&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
 	\bigg\{ 
 	   e_{2u}\,e_{3u}\,u\;		\delta_{i+1/2} 	   \left[ \overline {u^2}^{\,i} \right]
-	& + e_{1u}\,e_{3u}\,v\;	\delta_{j+1/2} 	\; \left[ \overline {u^2}^{\,i} \right]
-	&&
-	\biggr.\\
-	\biggl.
-	& + e_{1w}\,e_{2w}\,w\;	\delta_{k+1/2}  	   \left[ \overline {u^2}^{\,i} \right] 
+        + e_{1u}\,e_{3u}\,v\;	\delta_{j+1/2} 	\; \left[ \overline {u^2}^{\,i} \right]
+	 + e_{1w}\,e_{2w}\,w\;	\delta_{k+1/2}  	   \left[ \overline {u^2}^{\,i} \right] 
 	\biggr\} 
-	&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}
+	&& \displaybreak[0] \\
+%
+\equiv& \sum\limits_{i,j,k}
 \overline {u^2}^{\,i} 
 	\biggl\{ 
@@ -530,11 +381,9 @@
 	+ \delta_{j+1/2} 	\left[ e_{1u}\,e_{3u}\,v  \right]
 	+ \delta_{k+1/2} 	\left[ e_{1w}\,e_{2w}\,w \right] 
-	\biggr\} 
+	\biggr\}  \;\;  \equiv 0
 	&&& \\
 \end{flalign*}
 
-\begin{flalign*}
-\equiv 0 	&&&\\
-\end{flalign*}
+When the UBS scheme is used to evaluate the flux form momentum advection, the discrete operator does not contribute to the global budget of linear momentum (flux form). The horizontal kinetic energy is not conserved, but forced to decrease (the scheme is diffusive). 
 
 % -------------------------------------------------------------------------------------------------------------
@@ -545,5 +394,5 @@
 
 
-A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z-$coordinates, this properties is satisfied locally with the choice of discretization we have made (property (II.1.9)~). When the equation of state is linear (i.e. when a advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces. 
+A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. In $z$-coordinate, this properties is satisfied locally on a C-grid with 2nd order finite differences (property \eqref{Eq_DOM_curl_grad}). When the equation of state is linear ($i.e.$ when an advective-diffusive equation for density can be derived from those of temperature and salinity) the change of KE due to the work of pressure forces is balanced by the change of potential energy due to buoyancy forces: 
 \begin{equation*}
 \int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv 
@@ -551,80 +400,49 @@
 \end{equation*}
 
-This property is satisfied in both $z- $and $s-$coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, it can be written as:
-\begin{flalign*}
-\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv 	&&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	\biggl\{ \;		
-	& - \frac{1} {\rho_o e_{1u}} 
-		\Bigl( 
-		\delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} 
-			\left[ z_T \right] 
-		\Bigr)\;
-		u\;e_{1u}\,e_{2u}\,e_{3u}  
-	&&
-	\biggr. \\ 
-	\biggl.
-	& - \frac{1} {\rho_o e_{2v}} 	 
-		\Bigl( 
-		\delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} 
-			\left[ z_T \right] 
-		\Bigr)\;
-		v\;e_{1v}\,e_{2v}\,e_{3v} \;
-	\biggr\} 
-	&& \\ 
-\end{flalign*}
-
-Using (II.1.10), the continuity equation (II.2.5), and the hydrostatic 
-equation (II.2.4), it turns out to be: 
-\begin{multline*} 
-\equiv \frac{1} {\rho_o} \sum\limits_{i,j,k} 	
-	\biggl\{ 
+This property can be satisfied in discrete sense for both $z$- and $s$-coordinates. Indeed, defining the depth of a $T$-point, $z_T$ defined as the sum of the vertical scale factors at $w$-points starting from the surface, the workof pressure forces can be written as:
+\begin{flalign*}
+&\int_D - \frac{1} {\rho_o} \left. \nabla p^h \right|_z \cdot \textbf{U}_h \;dv   &&& \\
+\equiv& \sum\limits_{i,j,k} \biggl\{ \;  - \frac{1} {\rho_o e_{1u}}   \Bigl( 
+\delta_{i+1/2} \left[ p^h \right] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right] 
+		  			\Bigr)  \; u\;e_{1u}\,e_{2u}\,e_{3u}  
+	&&  \\ & \qquad \qquad
+						      	      - \frac{1} {\rho_o e_{2v}}    \Bigl( 
+\delta_{j+1/2} \left[ p^h \right] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2}  \left[ z_T \right] 
+					\Bigr)  \; v\;e_{1v}\,e_{2v}\,e_{3v} \;
+	\biggr\}   && \\ 
+\end{flalign*}
+
+Using  \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation), and \eqref{Eq_dynhpg_sco}, the hydrostatic 
+equation in $s$-coordinate, it turns out to be: 
+\begin{flalign*} 
+\equiv& \frac{1} {\rho_o} \sum\limits_{i,j,k}    \biggl\{ 
 		e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2}[ z_T]    
-	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T]  
-	 \biggr. \\  
-\shoveright {
-	+\biggl. 
-		\Bigl(
-		 \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v]
-		 \Bigr)\;p^h
-	\biggr\} } \\
-\end{multline*}
-
-\begin{multline*}
-\equiv \frac{1} {\rho_o } \sum\limits_{i,j,k}
+	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2}[ z_T]     
+&& \\  & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	+\Bigl(  \delta_i[ e_{2u}\,e_{3u}\,u] + \delta_j [ e_{1v}\,e_{3v}\,v]  \Bigr)\;p^h \biggr\}  &&\\
+%
+\equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k}
 	\biggl\{ 
 		 e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2} \delta_{i+1/2} \left[ z_T \right]
 	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2} \delta_{j+1/2} \left[ z_T \right] 
-	\biggr. \\ 
-\shoveright {
-	\biggl. 
-	 - \delta_k 
-		\left[ e_{1w} e_{2w}\,w \right]\;p^h 
-	\biggr\} } \\ 
-\end{multline*}
-
-\begin{multline*}
-\equiv \frac{1} {\rho_o } \sum\limits_{i,j,k}
+	&&&\\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
+	 - \delta_k \left[ e_{1w} e_{2w}\,w \right]\;p^h   \biggr\}   &&&\\ 
+%
+\equiv& \frac{1} {\rho_o } \sum\limits_{i,j,k}
 	\biggl\{ 
 		e_{2u}\,e_{3u} \;u\;g\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
-	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] 	\biggr. \\ 
-\shoveright{
-	\biggl. 
+	+ 	e_{1v}\,e_{3v} \;v\;g\;	\overline \rho^{\,j+1/2} \;\delta_{j+1/2} \left[ z_T \right] 
+	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
 	+	 e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 
-	\biggr\} }  \\ 
-\end{multline*}
-
-\begin{multline*}
-\equiv \frac{g} {\rho_o}  \sum\limits_{i,j,k}
+	\biggr\}  &&&\\ 
+%
+\equiv& \frac{g} {\rho_o}  \sum\limits_{i,j,k}
 	\biggl\{ 
 		e_{2u}\,e_{3u} \;u\;	\overline \rho^{\,i+1/2}\;\delta_{i+1/2} \left[ z_T \right]
-	+	e_{1v}\,e_{3v} \;v\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] 		\biggr. \\
-\shoveright{ 
-	\biggl. 
+	+	e_{1v}\,e_{3v} \;v\;	\overline \rho^{\,j+1/2}\;\delta_{j+1/2} \left[ z_T \right] 	
+	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
 	- 	e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 
-	\biggr\} }  \\ 
-\end{multline*}
+	\biggr\}   &&&\\ 
+\end{flalign*}
 noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus:
 \begin{multline*}
@@ -639,7 +457,7 @@
 	\biggr\} } \\ 
 \end{multline*}
-Using (II.1.10), it becomes~:
-\begin{flalign*}
-\equiv - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 
+Using \eqref{DOM_di_adj}, it becomes:
+\begin{flalign*}
+\equiv& - \frac{g} {\rho_o} \sum\limits_{i,j,k} z_T 
 	\biggl\{ 
 		\delta_i 	\left[ e_{2u}\,e_{3u}\,u\;	\overline \rho^{\,i+1/2} 	\right]
@@ -648,12 +466,12 @@
 	\biggr\} 
 	&&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv 	&&& \\
+%
+\equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv 	&&& \\
 \end{flalign*}
 
 Note that this property strongly constraints the discrete expression of both 
 the depth of $T-$points and of the term added to the pressure gradient in 
-$s-$coordinates. 
+$s$-coordinate. Nevertheless, it is almost never satisfied as a linear equation of state 
+is rarely used.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -669,62 +487,34 @@
 With the rigid-lid approximation, the change of KE due to the work of surface pressure forces is exactly zero. This is satisfied in discrete form, at the precision required on the elliptic solver used to solve this equation. This can be demonstrated as follows:
 \begin{flalign*}
-\int\limits_D  - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv	&&& \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	\biggl\{ 	\quad
-		& \left( 
-		 - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j 	\left[ \partial_t \psi  \right] 
-		  \right)\;
-		  u\;e_{1u}\,e_{2u}\,e_{3u} 
-	&&
-	\biggr. \\
-	\biggl. 
-		+ & \left( 
-		   - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
-		    \right)\;
-		    v\;e_{1v}\,e_{2v}\,e_{3v} 	\;
-	\biggr\} 
-	&&  \\
-\end{flalign*}
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j} 
-	\Biggl\{ 	\quad
-		&\biggl( 
-		- M_u - \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi  \right] 
-		\biggr)
-		\biggl( 
-		\sum\limits_k  u\;e_{3u}  
-		\biggr)\;
-		e_{1u}\,e_{2u} 
-	&&
-	\Biggr.  \\
-	\Biggl. 
-	+ & \biggl( 
-		- M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi  \right]
-	      \biggr)
-	      \biggl( 
-	      \sum\limits_k   v\;e_{3v}
-	       \biggr)\;
-	       e_{1v}\,e_{2v} \;
-	\Biggr\} 
+\int\limits_D  - \frac{1} {\rho_o} \nabla_h \left( p_s \right) \cdot \textbf{U}_h \;dv%	&&& \\
+%
+&\equiv \sum\limits_{i,j,k}   \biggl\{ 	\;
+	 \left(  - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j 	\left[ \partial_t \psi  \right]   \right)\;
+	 u\;e_{1u}\,e_{2u}\,e_{3u}  
+&&&\\&  \qquad \;\;\,
+      + \left(  - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right]     \right)\;
+	  v\;e_{1v}\,e_{2v}\,e_{3v} 	\; \biggr\}      
+&&&\\
+\\
+%
+&\equiv \sum\limits_{i,j}  \Biggl\{   \;
+	\biggl( - M_u - \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi  \right]   \biggr)
+	\biggl( \sum\limits_k  u\;e_{3u}   \biggr)\;  e_{1u}\,e_{2u} 
+&&&\\&  \qquad \;\;\,
+	+ \biggl( - M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi  \right]   \biggr)
+	   \biggl(   \sum\limits_k   v\;e_{3v}   \biggr)\;   e_{1v}\,e_{2v} \;   \Biggr\} 
 	&& \\ 
-\end{flalign*}
-using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes~:
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j} 
-	\biggl\{ 	\quad
-		&\left( 
+%
+\intertext{using the relation between \textit{$\psi $} and the vertically sum of the velocity, it becomes:}
+%
+&\equiv \sum\limits_{i,j} 
+	\biggl\{  \;	
+		\left( \;\;\,
 		M_u + \frac{1} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
 		\right)\;
 		e_{1u} \,\delta_j 
 			\left[ \partial_t \psi  \right] 
-	&&
-	\biggr. \\ 
-	\biggl. 
-		+& \left( 
+	&& \\ &  \qquad \;\;\,
+		+ \left( 
 		- M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
 		\right)\;
@@ -732,16 +522,13 @@
 	\biggr\} 
 	&& \\ 
-\end{flalign*}
-applying the adjoint of the $\delta$ operator, it is now:
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j}  - \partial_t \psi \;
-	\biggl\{ 	\quad
-	&  \delta_{j+1/2} \left[ e_{1u} M_u \right] 
+%
+\intertext{applying the adjoint of the $\delta$ operator, it is now:}
+%
+&\equiv \sum\limits_{i,j}  - \partial_t \psi \;
+	\biggl\{ 	\;
+	  \delta_{j+1/2} \left[ e_{1u} M_u \right] 
 	  - \delta_{i+1/2} \left[ e_{1v} M_v \right] 
-	&&
-	\biggr.  \\ 
-	\biggl. 
-	+& \delta_{i+1/2} 
+	&& \\ &  \qquad \;\;\,
+	+ \delta_{i+1/2} 
 		\left[ \frac{e_{2v}} {H_v \,e_{2v}}  \delta_i \left[ \partial_t \psi \right] 
 		\right] 
@@ -749,10 +536,9 @@
 	 	 \left[ \frac{e_{1u}} {H_u \,e_{2u}}  \delta_j \left[ \partial_t \psi \right] 
 		 \right]  
-	\biggr\} 
-	\equiv 0 
-	&& \\ 
-\end{flalign*}
-
-The last equality is obtained using (II.2.3), the discrete barotropic streamfunction time evolution equation. By the way, this shows that (II.2.3) is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver.
+	\biggr\}   &&&\\
+	&\equiv 0                   && \\ 
+\end{flalign*}
+
+The last equality is obtained using \eqref{Eq_dynspg_rl}, the discrete barotropic streamfunction time evolution equation. By the way, this shows that \eqref{Eq_dynspg_rl} is the only way do compute the streamfunction, otherwise the surface pressure forces will work. Nevertheless, since the elliptic equation verified by $\psi $ is solved numerically by an iterative solver, the property is only satisfied at the precision required on the solver.
 
 % ================================================================
@@ -763,6 +549,5 @@
 
 
-The numerical schemes are written such that the heat and salt contents are conserved by the internal dynamics (equations in flux form, second order centered finite differences). As a form flux is used to compute the temperature and salinity, the quadratic form of these quantities (i.e. their variance) is globally conserved, too. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 
-
+All the numerical schemes used in NEMO are written such that the tracer content is conserved by the internal dynamics and physics (equations in flux form). For advection, only the CEN2 scheme ($i.e.$ 2nd order finite different scheme) conserves the global variance of tracer. Nevertheless the other schemes ensure that the global variance decreases ($i.e.$ they are at least slightly diffusive). For diffusion, all the schemes ensure the decrease of the total tracer variance, but the iso-neutral operator. There is generally no strict conservation of mass, as the equation of state is non linear with respect to $T$ and $S$. In practice, the mass is conserved with a very good accuracy. 
 % -------------------------------------------------------------------------------------------------------------
 %       Advection Term
@@ -771,40 +556,25 @@
 \label{Apdx_C.2.1}
 
-Conservation of the tracer
-
-The flux form 
-\begin{flalign*}
-\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv &&&
-\end{flalign*}
-\begin{flalign*}
- \equiv  \sum\limits_{i,j,k}     \left\{
+Whatever the advection scheme considered it conserves of the tracer content as all the scheme are written in flux form. Let $\tau$ be the tracer interpolated at velocity point (whatever the interpolation is). The conservation of the tracer content is obtained as follows: 
+\begin{flalign*}
+&\int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\
+&\equiv  \sum\limits_{i,j,k}    \biggl\{
 	 \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
-	 \left( 
-	  \delta_i 
-	  	\left[ 
-		e_{2u}\,e_{3u} \,\overline T^{\,i+1/2}\,u 
-		\right]
-	 \right. \right.
-& + \left. \delta_j \left[ e_{1v}\,e_{3v} \,\overline T^{\,j+1/2}\,v \right] \right) && \\ 
- & + \left. \frac{1} {e_{3T}} \delta_k \left[ \overline T^{\,k+1/2}\,w \right] \ \ \right\} 
- 	\ \ e_{1T}\,e_{2T}\,e_{3T} &&
-\end{flalign*}
-
-\begin{flalign*}
- \equiv&  \sum\limits_{i,j,k}     \left\{
+	 \left(  \delta_i    \left[   e_{2u}\,e_{3u}\; u \;\tau_u   \right]
+	        + \delta_j    \left[   e_{1v}\,e_{3v}\; v  \;\tau_v   \right] \right) 
+&&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+	+ \frac{1} {e_{3T}} \delta_k \left[ w\;\tau \right]    \biggl\}  e_{1T}\,e_{2T}\,e_{3T} &&&\\
+%
+&\equiv  \sum\limits_{i,j,k}     \left\{
 	   \delta_i  \left[ e_{2u}\,e_{3u}  \,\overline T^{\,i+1/2}\,u \right]
        	+ \delta_j  \left[ e_{1v}\,e_{3v}  \,\overline T^{\,j+1/2}\,v \right] 
  	+ \delta_k \left[ e_{1T}\,e_{2T} \,\overline T^{\,k+1/2}\,w \right] \right\} 
- 	 && 
-\end{flalign*}
-\begin{flalign*}
- \equiv 0 &&&
-\end{flalign*}
-
-Conservation of the variance of tracer
-\begin{flalign*}
-\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
-\end{flalign*}
-\begin{flalign*}
+ 	 && \\
+&\equiv 0 &&&
+\end{flalign*}
+
+The conservation of the variance of tracer can be achieved only with the CEN2 scheme. It can be demonstarted as follows:
+\begin{flalign*}
+&\int\limits_D T\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
 \equiv& \sum\limits_{i,j,k} T\;
 	\left\{
@@ -814,16 +584,12 @@
 	\right\} 
 	&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+\equiv& \sum\limits_{i,j,k} 
 	\left\{
 	-           e_{2u}\,e_{3u} \overline T^{\,i+1/2}\,u\,\delta_{i+1/2}  \left[ T \right] \right.
-	-&        e_{1v}\,e_{3v}  \overline T^{\,j+1/2}\,v\;\delta_{j+1/2}  \left[ T \right]
-	 &&\\
-	-& \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right]
+	-           e_{1v}\,e_{3v}  \overline T^{\,j+1/2}\,v\;\delta_{j+1/2}  \left[ T \right]
+&&&\\& \qquad \qquad \qquad \qquad \qquad \qquad \quad \;
+	- \left. e_{1T}\,e_{2T} \overline T^{\,k+1/2}w\;\delta_{k+1/2} \left[ T \right]
 	\right\} 
 	&&\\
-\end{flalign*}
-\begin{flalign*}
 \equiv&  -\frac{1} {2}  \sum\limits_{i,j,k}
 	\Bigl\{
@@ -833,6 +599,4 @@
 	\Bigr\} 
 	&& \\ 
-\end{flalign*}
-\begin{flalign*}
 \equiv& \frac{1} {2}  \sum\limits_{i,j,k} T^2
 	\Bigl\{
@@ -841,8 +605,5 @@
 	+ \delta_k \left[ e_{1T}\,e_{2T}\,w \right]
 	\Bigr\} 
-	&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv 0 &&&
+\quad \equiv 0 &&&
 \end{flalign*}
 
@@ -866,7 +627,5 @@
 
 These properties of the horizontal diffusive operator are a direct 
-consequence of properties (II.1.8) and (II.1.9). When the vertical curl of 
-the horizontal diffusion of momentum (discrete sense) is taken, the term 
-associated to the horizontal gradient of the divergence is zero locally. 
+consequence of properties \eqref{Eq_DOM_curl_grad} and \eqref{Eq_DOM_div_curl}. When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, the term associated to the horizontal gradient of the divergence is zero locally. 
 
 % -------------------------------------------------------------------------------------------------------------
@@ -878,11 +637,12 @@
 The lateral momentum diffusion term conserves the potential vorticity :
 \begin{flalign*}
-\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
+&\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
 	\Bigl[ \nabla_h 
 		\left( A^{\,lm}\;\chi  \right)
 	- \nabla_h \times
 		\left( A^{\,lm}\;\zeta \; \textbf{k} \right)
-	\Bigr]\;dv &&& \\
-\end{flalign*}
+	\Bigr]\;dv  = 0
+\end{flalign*}
+%%%%%%%%%%  recheck here....  (gm)
 \begin{flalign*}
 = \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
@@ -891,5 +651,4 @@
 	\Bigr]\;dv &&& \\ 
 \end{flalign*}
-
 \begin{flalign*}
 \equiv& \sum\limits_{i,j}
@@ -907,8 +666,7 @@
 	\right\} 
 	&& \\ 
-\end{flalign*}
-Using (II.1.10), it follows:
-
-\begin{flalign*}
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
 \equiv& \sum\limits_{i,j,k} 
 	-\,\left\{
@@ -928,76 +686,49 @@
 
 
-The lateral momentum diffusion term dissipates the horizontal kinetic 
-energy:
-\begin{flalign*}
+The lateral momentum diffusion term dissipates the horizontal kinetic energy:
+%\begin{flalign*}
+\begin{equation*}
+\begin{split}
 \int_D \textbf{U}_h \cdot 
-	\left[ \nabla_h 
-		\left( A^{\,lm}\;\chi \right)
-	- \nabla_h \times 
-		\left( A^{\,lm}\;\zeta \;\textbf{k} \right) 
-	\right]\;dv &&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} \quad
-	&\left\{
-	\frac{1} {e_{1u}}            \delta_{i+1/2}
-		\left[ A_T^{\,lm} \chi  \right]
-	- \frac{1} {e_{2u}\,e_{3u}}  \delta_j
-		\left[ A_f^{\,lm} e_{3f} \zeta   \right]
-	\right\}\; 
-	e_{1u}\,e_{2u}\,e_{3u} \;u 
-	 &&\\
-	& + \left\{
-	\frac{1} {e_{2u}}             \delta_{j+1/2}
-		\left[ A_T^{\,lm} \chi  \right] 
-	+ \frac{1} {e_{1v}\,e_{3v}} \delta_i
-	\left[ A_f^{\,lm} e_{3f} \zeta  \right] 
-	\right\}\; 
-	e_{1v}\,e_{2u}\,e_{3v} \;v  
-	&& \\ 
-\end{flalign*}
-\begin{flalign*} 
-\equiv \sum\limits_{i,j,k} \quad
- 	&\left\{
-	  e_{2u}\,e_{3u} \;u\;\delta_{i+1/2} 
-		\left[ A_T^{\,lm} \chi  \right]
-	- e_{1u} \;u\;\delta_j 
-		\left[ A_f^{\,lm} e_{3f} \zeta  \right]
-	 \right\} 
-	 &&\\ 
-	& + \left\{
-	   e_{1v}\,e_{3v} \;v\;\delta_{j+1/2} 
-		\left[ A_T^{\,lm} \chi  \right]
-	+ e_{2v} \;v\;\delta_i 
-		\left[ A_f^{\,lm} e_{3f} \zeta  \right]
-	\right\} 
-	&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} \quad
-	-& \Bigl(
-	  \delta_i 
-		\left[ e_{2u}\,e_{3u} \;u \right]
-	+ \delta_j 
-		\left[ e_{1v}\,e_{3v} \;v \right] 
-	     \Bigr)\;
-	A_T^{\,lm} \chi 
-	&& \\ 
-	-& \Bigl(
-	  \delta_{i+1/2} 
-		\left[ e_{2v} \;v \right]
-	- \delta_{j+1/2} 
-		\left[ e_{1u} \;u \right] 
-	     \Bigr)\;
-	A_f^{\,lm} e_{3f} \zeta 
-	&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
-	- A_T^{\,lm} \,\chi^2   \;e_{1T}\,e_{2T}\,e_{3T}
-	- A_f^{\,lm}  \,\zeta^2 \;e_{1f}\,e_{2f}\,e_{3f}  
-	\quad \leq 0
-	&&&\\
-\end{flalign*}
+	\left[ \nabla_h 		\right.   &     \left.       \left( A^{\,lm}\;\chi \right)      
+	- \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	\left\{
+	  \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right]
+	- \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right]
+	\right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\
+&\;\; + 	\left\{
+	   \frac{1} {e_{2u}}             \delta_{j+1/2}	\left[ A_T^{\,lm}          \chi    \right] 
+	+ \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
+	\right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+ 	\Bigl\{
+	  e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right]
+	- e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	 \Bigl\} 
+	 \\ 
+&\;\; + \Bigl\{
+	   e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right]
+	+ e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	\Bigl\}      \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	- \Bigl(
+	  \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right]
+	+ \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right] 
+	     \Bigr) \;  A_T^{\,lm} \chi   \\ 
+&\;\; - \Bigl(
+	  \delta_{i+1/2}  \left[  e_{2v}  \;v  \right]
+	- \delta_{j+1/2}  \left[  e_{1u} \;u  \right] 
+	     \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\ 
+\\  %%%
+\equiv& \sum\limits_{i,j,k} 
+	- A_T^{\,lm}  \,\chi^2   \;e_{1T}\,e_{2T}\,e_{3T}
+	- A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}  
+\quad \leq 0       \\
+\end{split}
+\end{equation*}
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1011,5 +742,5 @@
 coefficients are horizontally uniform:
 \begin{flalign*}
-\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
+&\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
 	\left[ 
 	  \nabla_h 
@@ -1018,14 +749,10 @@
 		\left( A^{\,lm}\;\zeta \; \textbf{k} \right)
 	\right]\;dv &&&\\
-\end{flalign*}
-\begin{flalign*}
- = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
+&= A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
  	\left[ 
 	\nabla_h \times 
 		\left( \zeta \; \textbf{k} \right)
-	\right]\;dv &&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
+	\right]\;dv &&&\displaybreak[0]\\
+&\equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
 	\left\{
 	  \delta_{i+1/2} 
@@ -1041,9 +768,8 @@
 	\right\} 
 	&&&\\ 
-\end{flalign*}
-Using (II.1.10), it becomes~:
-
-\begin{flalign*}
-\equiv  - A^{\,lm} \sum\limits_{i,j,k} 
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv  - A^{\,lm} \sum\limits_{i,j,k} 
 	\left\{
 	  \left( 
@@ -1055,7 +781,6 @@
 	  	\left[ e_{3f} \zeta  \right]
 	  \right)^2   e_{1u}\,e_{2u}\,e_{3u}
-	  \right\} 
-	  \; \leq \;0 
-	  &&&\\ 
+	  \right\}      &&&\\
+& \leq \;0       &&&\\ 
 \end{flalign*}
 
@@ -1072,5 +797,5 @@
 horizontally uniform.
 \begin{flalign*}
-  \int\limits_D  \nabla_h \cdot 
+& \int\limits_D  \nabla_h \cdot 
 	\Bigl[ 
 	  \nabla_h 
@@ -1084,7 +809,5 @@
 	dv 
 &&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+&\equiv \sum\limits_{i,j,k} 
 	\left\{ 
 	  \delta_i
@@ -1100,9 +823,8 @@
 	\right\}
 	&&&\\ 
-\end{flalign*}
-Using (II.1.10), it follows:
-
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+%
+\intertext{Using \eqref{DOM_di_adj}, it follows:}
+%
+&\equiv \sum\limits_{i,j,k} 
 	- \left\{
 	\frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} 
@@ -1126,53 +848,23 @@
 
 \begin{flalign*}
-		\int\limits_D \chi \;\nabla_h \cdot 
-	\left[ 
-	  \nabla_h 
-		\left( A^{\,lm}\;\chi  \right)
-	- \nabla_h \times 
-		\left( A^{\,lm}\;\zeta \;\textbf{k} \right)
-	\right]\;
-	dv
- = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h 
- 	\left( \chi \right)\;
-	dv
-&&&\\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1T}\,e_{2T}\,e_{3T}}  \chi 
+&\int\limits_D \chi \;\nabla_h \cdot 
+	\left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right)
+	        - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv
+ = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    &&&\\ 
+%
+&\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1T}\,e_{2T}\,e_{3T}}  \chi 
 	\left\{
-	\delta_i 
-	\left[ 
-	\frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} 
-		\left[ \chi \right] 
-	\right]
-	+ \delta_j 
-	\left[ 
-	\frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} 
-		\left[ \chi \right]
-	\right]
-	\right\} \;
-	e_{1T}\,e_{2T}\,e_{3T} 
-	&&&\\ 
-\end{flalign*}
-Using (II.1.10), it turns out to be:
-
-\begin{flalign*}
-\equiv - A^{\,lm} \sum\limits_{i,j,k}
+	   \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right]
+	+ \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right]
+	\right\} \;   e_{1T}\,e_{2T}\,e_{3T}    &&&\\ 
+%
+\intertext{Using \eqref{DOM_di_adj}, it turns out to be:}
+%
+&\equiv - A^{\,lm} \sum\limits_{i,j,k}
 	\left\{ 
-	\left( 
-	\frac{1} {e_{1u}}  \delta_{i+1/2} 
-		\left[ \chi \right]
-	\right)^2   
-	e_{1u}\,e_{2u}\,e_{3u}
-	+ \left(
-	\frac{1} {e_{2v}}  \delta_{j+1/2} 
-		\left[ \chi \right]
-	\right)^2
-	e_{1v}\,e_{2v}\,e_{3v}
-	\right\} \; 
-	\leq 0 
-	&&&\\
+   \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  e_{1u}\,e_{2u}\,e_{3u}
++ \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  e_{1v}\,e_{2v}\,e_{3v}
+	\right\} \;    &&&\\
+&\leq 0              &&&\\
 \end{flalign*}
 
@@ -1185,5 +877,5 @@
 
 As for the lateral momentum physics, the continuous form of the vertical diffusion of momentum satisfies the several integral constraints. The first two are associated to the conservation of momentum and the dissipation of horizontal kinetic energy:
-\begin{flalign*}
+\begin{align*}
  \int\limits_D  
 	 \frac{1} {e_3 }\; \frac{\partial } {\partial k}
@@ -1191,9 +883,9 @@
 	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
 	\right)\;
-	dv \quad  = \vec{\textbf{0}}
-	&&&\\
-\end{flalign*}
-and
-\begin{flalign*}
+	dv \qquad \quad &= \vec{\textbf{0}}
+	\\
+%
+\intertext{and}
+%
 \int\limits_D 
 	\textbf{U}_h \cdot 
@@ -1202,7 +894,7 @@
 	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
 	\right)\;
-	dv \quad \leq 0
-	&&&\\
-\end{flalign*}
+	dv \quad &\leq 0
+	\\
+\end{align*}
 The first property is obvious. The second results from:
 
@@ -1217,5 +909,5 @@
 \end{flalign*}
 \begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+&\equiv \sum\limits_{i,j,k} 
 	\left( 
 	  u\; \delta_k 
@@ -1233,9 +925,8 @@
 	\right)
 	&&&\\ 
-\end{flalign*}
-as the horizontal scale factor do not depend on $k$, it follows:
-
-\begin{flalign*}
-\equiv - \sum\limits_{i,j,k} 
+%
+\intertext{as the horizontal scale factor do not depend on $k$, it follows:}
+%
+&\equiv - \sum\limits_{i,j,k} 
 	\left( 
 	   \frac{A_u^{\,vm}} {e_{3uw}}
@@ -1254,4 +945,5 @@
 	 &&&\\
 \end{flalign*}
+
 The vorticity is also conserved. Indeed:
 \begin{flalign*}
@@ -1532,8 +1224,7 @@
 constraint of conservation of tracers:
 \begin{flalign*}
-\int\limits_D  T\;\nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} 
+&\int\limits_D  T\;\nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
+\\
+&\equiv \sum\limits_{i,j,k} 
 	\biggl\{ 	\biggr.
 	\delta_i 
@@ -1543,20 +1234,16 @@
 		\right]
 	+ \delta_j 
-		&\left[ 
+		\left[ 
 		A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
 			\left[ T \right] 
 		\right]
-	&&\\ 
-	\biggl. 
+	&&\\  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
 	+ \delta_k 
-		&\left[ 
+		\left[ 
 		A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
 			\left[ T \right] 
 		\right]
-	\biggr\} 
+	\biggr\}   \quad  \equiv 0
 	&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv 0	&&&\\
 \end{flalign*}
 
@@ -1572,5 +1259,4 @@
 \int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv	&&&\\ 
 \end{flalign*}
-
 \begin{flalign*}
 \equiv \sum\limits_{i,j,k} T
@@ -1596,5 +1282,4 @@
 	&&\\ 
 \end{flalign*}
-
 \begin{flalign*}
 \equiv - \sum\limits_{i,j,k} 
@@ -1626,2 +1311,5 @@
 \end{flalign*}
 
+
+%%%%  end of appendix in gm comment
+}
Index: trunk/DOC/BETA/Chapters/Annex_D.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Annex_D.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Annex_D.tex	(revision 817)
@@ -1,12 +1,12 @@
 % ================================================================
-% Chapter Ñ Appendix D : Coding Rules
+% Appendix D Ñ Coding Rules
 % ================================================================
-\chapter{Appendix D : Coding Rules}
+\chapter{Coding Rules}
 \label{Apdx_D}
 \minitoc
 
 
-The "model life" is about ten years and its software, composed by about one 
-hundred programs, is used by many people who are scientists or students and 
+A "model life" is more than ten years. Its software, composed by a few 
+hundred modules, is used by many people who are scientists or students and 
 do not necessary know very well all computer aspects. Moreover, a well 
 thought-out programme is easy to read and understand, less difficult to 
@@ -26,17 +26,18 @@
 - flexible.
 
-To satisfy part of these aims, OPA is written with a coding standard which 
-is close to the ECMWF rule, named DOCTOR [Gibson,1986]. These rules present 
-some advantages like :
+To satisfy part of these aims, \NEMO is written with a coding standard which 
+is close to the ECMWF rules, named DOCTOR \citep{Gibson_TR86}. These rules present some advantages like :
 
 - to provide a well presented program
 
-- to enable the extraction of several levels of on-line documentation
-
 - to use rules for variable names which allow recognition of their type 
-(integer, real, parameter, common variables, etc. ) so that debugging is 
+(integer, real, parameter, local or shared variables, etc. ) so that debugging is 
 facilitated.
 
-\textbf{The program structure}
+% ================================================================
+% The program structure
+% ================================================================
+\section{The program structure}
+\label{Apdx_D_structure}
 
 Each program begins with a set of headline comments containing :
@@ -61,5 +62,9 @@
 - A program has not more than 200 to 300 lines.
 
-\textbf{Coding conventions}
+% ================================================================
+% Coding conventions
+% ================================================================
+\section{Coding conventions}
+\label{Apdx_D_coding}
 
 - Use of the universal language \textsc{Fortran} 5 ANSI 77, with non 
@@ -106,5 +111,9 @@
 - STOP must be well documented with the name of the subroutine or a number.
 
-\textbf{Naming Conventions.}
+% ================================================================
+% Naming Conventions
+% ================================================================
+\section{Naming Conventions}
+\label{Apdx_D_naming}
 
 The purpose of the naming conventions is to use prefix letters to classify 
@@ -117,14 +126,7 @@
 \begin{center}
 \begin{tabular}{|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|p{50pt}|}
+\hline  Type \par / Status &   integer&   real&   logical &   character&   double \par precision&   complex \\  
 \hline
-Type \par / \par Status & 
-integer& 
-real& 
-logical & 
-character& 
-double \par precision& 
-complex \\
-\hline
-global \par or \par common& 
+public & 
 \textbf{m n} \par \textit{but not } \par \textbf{nam}& 
 \textbf{a b e f g h o} \textbf{q} \textit{to} \textbf{x} \par but not \par \textbf{sf}& 
@@ -179,7 +181,2 @@
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
-
-\textbf{References}
-
-{\small Gibson, J. K., 1986: Standard software development and maintenance. }{\small \textit{Tech. memorandum}}{\small , Operational Dep., ECMWF, Reading, UK.}
-
Index: trunk/DOC/BETA/Chapters/Chap_DOM.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_DOM.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_DOM.tex	(revision 817)
@@ -8,14 +8,25 @@
 
 % Missing things:
-% 	- istate: description of the initial state
+% 	- istate: description of the initial state   ==> this has to be put elsewhere..
+%                  perhaps in MISC ?  By the way the initialisation of T S and dynamics 
+%                  should be put outside of DOM routine (better with TRC staff and off-line
+%                  tracers)
 %	- daymod: definition of the time domain (nit000, nitend andd the calendar)
 % 	-geo2ocean:  how to switch from geographic to mesh coordinate
 %  - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
 
-
-
-
-
-Having defined the continuous equations in Chap.~\ref{PE}, we need to choose a discretization on a grid, and numerical algorithms. In the present chapter, we provide a general description of the staggered grid used in OPA, and other information relevant to the main directory routines (time stepping, main program) as well as the DOM (DOMain) directory. 
+\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction which could be referred to here, would help  ==> to be added}
+%%%%
+
+
+\newpage
+$\ $\newline    % force a new ligne
+
+
+Having defined the continuous equations in Chap.~\ref{PE}, we need to choose a 
+discretization on a grid, and numerical algorithms. In the present chapter, we 
+provide a general description of the staggered grid used in \NEMO, and other 
+information relevant to the main directory routines (time stepping, main program) 
+as well as the DOM (DOMain) directory. 
 
 % ================================================================
@@ -34,13 +45,40 @@
 \begin{figure}[!tb] \label{Fig_cell}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_cell.pdf}
-\caption{Arrangement of variables. $T$ indicates scalar points where temperature, salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) indicates vector points, and $f$ indicates vorticity points where both relative and planetary vorticities are defined}
+\caption{Arrangement of variables. $T$ indicates scalar points where temperature, 
+salinity, density, pressure and horizontal divergence are defined. ($u$,$v$,$w$) 
+indicates vector points, and $f$ indicates vorticity points where both relative and 
+planetary vorticities are defined}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-The numerical techniques used to solve the Primitive Equations in this model are based on the traditional, centred second-order finite difference approximation. Special attention has been given to the homogeneity of the solution in the three space directions. The arrangement of variables is the 
-same in all directions. It consists in cells centred on scalar points ($T$, $S$, $p$, $\rho$) with vector points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}). This is the generalisation to three dimensions of the well-known ``C'' grid in Arakawa's classification. The relative and planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge and the barotropic stream function $\psi$ is defined at horizontal points overlying the $\zeta$ and $f$-points.
-
-The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. The grid-points are located at integer or integer and a half value of $(i,j,k)$ as indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$, $f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale factors are defined. Each scale factor is defined as the local analytical value provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and $\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size unity. Discrete partial derivatives are formulated by the traditional, centred second order finite difference approximation while the scale factors are chosen equal to their local analytical value. An important point here is that the partial derivative of the scale factors must be evaluated by centred finite difference approximation, not from their 
-analytical expression. This preserves the symmetry of the discrete set of equations and therefore allows satisfying many of the continuous properties (see { \colorbox{yellow}{Annexe C}). A similar, related remark can be made about the domain size: when needed, an area, volume, or the total ocean depth must be evaluated as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section). 
+The numerical techniques used to solve the Primitive Equations in this model are 
+based on the traditional, centred second-order finite difference approximation. 
+Special attention has been given to the homogeneity of the solution in the three 
+space directions. The arrangement of variables is the same in all directions. 
+It consists of cells centred on scalar points ($T$, $S$, $p$, $\rho$) with vector 
+points $(u, v, w)$ defined in the centre of each face of the cells (Fig. \ref{Fig_cell}). 
+This is the generalisation to three dimensions of the well-known ``C'' grid in 
+Arakawa's classification \citep{Mesinger_Arakawa_Bk76}. The relative and 
+planetary vorticity, $\zeta$ and $f$, are defined in the centre of each vertical edge 
+and the barotropic stream function $\psi$ is defined at horizontal points overlying 
+the $\zeta$ and $f$-points.
+
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
+by the transformation that gives ($\lambda$ ,$\varphi$ ,$z$) as a function of $(i,j,k)$. 
+The grid-points are located at integer or integer and a half value of $(i,j,k)$ as 
+indicated on Table \ref{Tab_cell}. In all the following, subscripts $u$, $v$, $w$, 
+$f$, $uw$, $vw$ or $fw$ indicate the position of the grid-point where the scale 
+factors are defined. Each scale factor is defined as the local analytical value 
+provided by \eqref{Eq_scale_factors}. As a result, the mesh on which partial 
+derivatives $\frac{\partial}{\partial \lambda}, \frac{\partial}{\partial \varphi}$, and 
+$\frac{\partial}{\partial z} $ are evaluated is a uniform mesh with a grid size of unity. Discrete partial derivatives are formulated by the traditional, centred second order 
+finite difference approximation while the scale factors are chosen equal to their 
+local analytical value. An important point here is that the partial derivative of the 
+scale factors must be evaluated by centred finite difference approximation, not 
+from their analytical expression. This preserves the symmetry of the discrete set 
+of equations and therefore satisfies many of the continuous properties (see 
+Appendix~\ref{Apdx_C}). A similar, related remark can be made about the domain 
+size: when needed, an area, volume, or the total ocean depth must be evaluated 
+as the sum of the relevant scale factors (see \eqref{DOM_bar}) in the next section). 
 
 \begin{table}[!tb] \label{Tab_cell}
@@ -56,5 +94,8 @@
 fw	& $i+1/2$ 	& $j+1/2$	& $k+1/2$ 	\\ \hline
 \end{tabular}
-\caption{Location of grid-points as a function of integer or integer and a half value of the column, line or level. This indexation is only used for the writing of semi-discrete equation. In the code, the indexation use integer value only and has a reverse direction in the vertical (see \S\ref{DOM_Num_Index})}
+\caption{Location of grid-points as a function of integer or integer and a half value 
+of the column, line or level. This indexing is only used for the writing of the semi-
+discrete equation. In the code, the indexing uses integer values only and has a 
+reverse direction in the vertical (see \S\ref{DOM_Num_Index})}
 \end{center}
 \end{table}
@@ -66,14 +107,18 @@
 \label{DOM_operators}
 
-Given the values of a variable $q$ at adjacent points, the derivation and averaging operators at the midpoint between them are:
+Given the values of a variable $q$ at adjacent points, the differencing and 
+averaging operators at the midpoint between them are:
 \begin{subequations} \label{Eq_di_mi}
 \begin{align}
- \delta _i [q]  &=  \  \    q(i+1/2)  - q(i-1/2)		\\
- \overline q^i &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
+ \delta _i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)		\\
+ \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
 \end{align}
 \end{subequations}
 
-Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and $k+1/2$. Following 
-\eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a variable $q$ defined at $T$-point has its three components defined at $(u,v,w)$ while its Laplacien is defined at $T$-point. These operators have the following discrete forms in the curvilinear $s$-coordinate system:
+Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and 
+$k+1/2$. Following \eqref{Eq_PE_grad} and \eqref{Eq_PE_lap}, the gradient of a 
+variable $q$ defined at a $T$-point has its three components defined at $u$-, $v$- 
+and $w$-points while its Laplacien is defined at $T$-point. These operators have 
+the following discrete forms in the curvilinear $s$-coordinate system:
 \begin{equation} \label{Eq_DOM_grad}
 \nabla q\equiv 	\frac{1}{e_{1u} }\delta _{i+1/2} \left[ q \right]\;\,{\rm {\bf i}}
@@ -91,5 +136,7 @@
 \end{multline}
 
-Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ defined at vector points $(u,v,w)$ has its three curl components defined at $(vw,uw,f)$ and its divergence defined at $T$-points:
+Following \eqref{Eq_PE_curl} and \eqref{Eq_PE_div}, a vector ${\rm {\bf A}}=\left( a_1,a_2,a_3\right)$ defined at vector points $(u,v,w)$ has its three curl 
+components defined at $vw$-, $uw$, and $f$-points, and its divergence defined 
+at $T$-points:
 \begin{equation} \label{Eq_DOM_curl}
 \begin{split}
@@ -110,14 +157,26 @@
 \end{equation}
 
-In the special case of pure $z$-coordinates system, \eqref{Eq_DOM_lap} and \eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor becomes a function of the single variable $k$ and thus does not depend on the horizontal location of a grid point. It can be simplified from outside and inside the $\delta _i$ and $\delta_j$ operators. 
-
-The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
+In the special case of a pure $z$-coordinate system, \eqref{Eq_DOM_lap} and 
+\eqref{Eq_DOM_div} can be simplified. In this case, the vertical scale factor 
+becomes a function of the single variable $k$ and thus does not depend on the 
+horizontal location of a grid point. For example \eqref{Eq_DOM_div} reduces to: 
+\begin{equation*}
+\nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} }\left( {\delta 
+_i \left[ {e_{2u} a_1 } \right]+\delta _j \left[ {e_{1v}  a_2 } 
+\right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 } \right]
+\end{equation*}
+
+The vertical average over the whole water column denoted by an overbar becomes 
+for a quantity $q$ which is a masked field (i.e. equal to zero inside solid area):
 \begin{equation} \label{DOM_bar}
 \bar q 	= \frac{1}{H}\int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
 		\equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
 \end{equation}
-where $H_q$  the ocean depth, is the masked sum of the vertical scale factors at q points, $k^b$ and $k^o$ are the bottom and surface $k$-index, and the symbol $k^o$ referring to a summation over all grid points of the same species in the direction indicated by the subscript (here $k$). 
-
-In continuous, the following properties are satisfied:
+where $H_q$  is the ocean depth, which is the masked sum of the vertical scale 
+factors at $q$ points, $k^b$ and $k^o$ are the bottom and surface $k$-indices, 
+and the symbol $k^o$ refers to a summation over all grid points of the same type 
+in the direction indicated by the subscript (here $k$). 
+
+In continuous form, the following properties are satisfied:
 \begin{equation} \label{Eq_DOM_curl_grad}
 \nabla \times \nabla q ={\rm {\bf {0}}}
@@ -127,22 +186,33 @@
 \end{equation}
 
-It is straight forward to demonstrate that these properties are verified locally in discrete form as soon as the scalar $q$ is taken at $T$-points and the vector \textbf{A} has its components defined at vector points $(u,v,w)$.
-
-Let $a$ and $b$ be two fields defined on the ocean mesh, extended to zero inside continental area. By integration by part it can be shown that the derivation operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear operators, and further that the averaging operators $\overline{\cdot}^i$, $\overline{\cdot}^j$ and $\overline{\cdot}^k$) are symmetric linear operators, i.e.,
-\begin{equation} \label{DOM_di_adj}
-\sum\limits_i {a_i \;\delta _i \left[ b \right]} \equiv -\sum\limits_i 
-{\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }
-\end{equation}
-\begin{equation} \label{DOM_mi_adj}
-\sum\limits_i {a_i \;\overline b ^i} \equiv \sum\limits_i {\overline a ^{i+1/2}\;b_{i+1/2} } 
-\end{equation}
-
-In other words, the adjoint of the derivation and averaging operators are $\delta_i^*=\delta_{i+1/2}$ and $\overline{\cdot}^{i\,*}= \overline{\cdot}^{i+1/2}$, respectively. These two properties will be used extensively in the \colorbox{yellow} {Appendix C} to 
+It is straightforward to demonstrate that these properties are verified locally in 
+discrete form as soon as the scalar $q$ is taken at $T$-points and the vector 
+\textbf{A} has its components defined at vector points $(u,v,w)$.
+
+Let $a$ and $b$ be two fields defined on the mesh, with value zero inside 
+continental area. Using integration by parts it can be shown that the differencing 
+operators ($\delta_i$, $\delta_j$ and $\delta_k$) are anti-symmetric linear 
+operators, and further that the averaging operators $\overline{\,\cdot\,}^{\,i}$, 
+$\overline{\,\cdot\,}^{\,k}$ and $\overline{\,\cdot\,}^{\,k}$) are symmetric linear 
+operators, $i.e.$
+\begin{align} 
+\label{DOM_di_adj}
+\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
+	&\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
+\label{DOM_mi_adj}
+\sum\limits_i { a_i \;\overline b^{\,i}} 
+	& \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
+\end{align}
+
+In other words, the adjoint of the differencing and averaging operators are 
+$\delta_i^*=\delta_{i+1/2}$ and 
+${(\overline{\,\cdot \,}^{\,i})}^*= \overline{\,\cdot\,}^{\,i+1/2}$, respectively. 
+These two properties will be used extensively in the Appendix~\ref{Apdx_C} to 
 demonstrate integral conservative properties of the discrete formulation chosen.
 
 % -------------------------------------------------------------------------------------------------------------
-%        Numerical Indexation 
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Numerical Indexation}
+%        Numerical Indexing 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Numerical Indexing}
 \label{DOM_Num_Index}
 
@@ -150,47 +220,95 @@
 \begin{figure}[!tb] \label{Fig_index_hor}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_index_hor.pdf}
-\caption{Horizontal integer indexation used in the \textsc{Fortran} code. The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices}
+\caption{Horizontal integer indexing used in the \textsc{Fortran} code. The dashed 
+area indicates the cell in which variables contained in arrays have the same 
+$i$- and $j$-indices}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-The array representation used in the \textsc{Fortran} code requires an integer indexation while the analytical definition of the mesh (see \S\ref{DOM_cell}) is associated with the use of integer values of $(i,j,k)$ for $T$-points whereas all the other points use both integer and integer and a half values of $(i,j,k)$. Therefore a specific integer indexation must be defined for the latter grid-points 
-(i.e. velocity and vorticity grid-points). Furthermore, it has been chosen to change the direction of the vertical indexation so that the surface level is at $k=1$.
+The array representation used in the \textsc{Fortran} code requires an integer 
+indexing while the analytical definition of the mesh (see \S\ref{DOM_cell}) is 
+associated with the use of integer values for $T$-points and both integer and 
+integer and a half values for all the other points. Therefore a specific integer 
+indexing must be defined for points other than $T$-points ($i.e.$ velocity and 
+vorticity grid-points). Furthermore, the direction of the vertical indexing has 
+been changed so that the surface level is at $k=1$.
 
 % -----------------------------------
-%        Horizontal Indexation 
+%        Horizontal Indexing 
 % -----------------------------------
-\subsubsection{Horizontal Indexation}
+\subsubsection{Horizontal Indexing}
 \label{DOM_Num_Index_hor}
 
-The indexation in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. For an increasing $i$ index ($j$ index), the $T$-point and the eastward $u$-point (northward $v$-point) have the same index (see the dashed area in Fig.\ref{Fig_index_hor}). A $T$-point and its nearby northeast $f$-point have the same $i$-and $j$-indices.
+The indexing in the horizontal plane has been chosen as shown in Fig.\ref{Fig_index_hor}. For an increasing $i$ index ($j$ index), the $T$-point 
+and the eastward $u$-point (northward $v$-point) have the same index 
+(see the dashed area in Fig.\ref{Fig_index_hor}). A $T$-point and its 
+nearest northeast $f$-point have the same $i$-and $j$-indices.
 
 % -----------------------------------
-%        Vertical Indexation 
+%        Vertical indexing 
 % -----------------------------------
-\subsubsection{Vertical Indexation}
+\subsubsection{Vertical Indexing}
 \label{DOM_Num_Index_vertical}
 
-In the vertical plane, the chosen indexation requires special attention since the $k$-axis is re-oriented downward in the \textsc{Fortran} code compared to the indexation used for the semi-discrete equations and given in \S\ref{DOM_cell}. The sea surface corresponds to the $w$-level $k=1$ like the $T-$level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$) is either the ocean bottom or inside the ocean floor while the last $T-$level is always inside the floor (Fig.\ref{Fig_index_vert}). Note that for an increasing $k$ index, a $w$-point and the $T$-point just below have the same $k$ index, in opposition to what is done in the horizontal plane where 
-it is the $T-$point and the nearby velocity points in the direction of the horizontal axis that have the same $i$ or $j$ index (compare the dashed area in Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). As the scale factors are chosen to be strictly positive, \emph{a minus sign appears in the \textsc{Fortran} code before all the vertical derivatives of the discrete equations given in this documentation}.
+In the vertical, the chosen indexing requires special attention since the 
+$k$-axis is re-orientated downward in the \textsc{Fortran} code compared 
+to the indexing used in the semi-discrete equations and given in \S\ref{DOM_cell}. 
+The sea surface corresponds to the $w$-level $k=1$ which is the same index 
+as $T$-level just below (Fig.\ref{Fig_index_vert}). The last $w$-level ($k=jpk$) 
+either corresponds to the ocean floor or is inside the bathymetry while the last 
+$T$-level is always inside the bathymetry (Fig.\ref{Fig_index_vert}). Note that 
+for an increasing $k$ index, a $w$-point and the $T$-point just below have the 
+same $k$ index, in opposition to what is done in the horizontal plane where 
+it is the $T$-point and the nearest velocity points in the direction of the horizontal 
+axis that have the same $i$ or $j$ index (compare the dashed area in Fig.\ref{Fig_index_hor} and \ref{Fig_index_vert}). Since the scale factors are chosen 
+to be strictly positive, a \emph{minus sign} appears in the \textsc{Fortran} code 
+\emph{before all the vertical derivatives} of the discrete equations given in this 
+documentation.
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!pt] \label{Fig_index_vert}  \begin{center}
 \includegraphics[width=.90\textwidth]{./Figures/Fig_index_vert.pdf}
-\caption{Vertical integer indexation used in the \textsc{Fortran } code. Note that the $k$-axis is oriented downward. The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.}
+\caption{Vertical integer indexing used in the \textsc{Fortran } code. Note that 
+the $k$-axis is orientated downward. The dashed area indicates the cell in 
+which variables contained in arrays have the same $k$-index.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 % -----------------------------------
-%        Vertical Indexation 
+%        Domain Size
 % -----------------------------------
-\subsubsection{Domain size}
+\subsubsection{Domain Size}
 \label{DOM_size}
 
-The total size of the computational domain is set by the parameters \jp{jpiglo}, \jp{jpjglo} and \jp{jpk} in the $i$, $j$ and $k$ directions respectively. They are given as parameters in the \mdl{par\_oce} module (or additional files included in this module such as \textit{par\_ORCA\_R2.h90}, specific to a given configuration). The use of parameters rather than variables (together with dynamic allocation of arrays) was made because it ensured that the compiler would optimize the executable code efficiently, especially on vector machines (optimization may be less efficient when the problem size is unknown at the time of compilation). Nevertheless, it is possible to set up the code with full dynamical allocation by using the AGRIF packaged \colorbox{yellow}{(ref agrif!+ ref part of the doc)}. Note that are other parameters in \mdl{par\_oce} that refer to the domain size. The two parameters $jpidta$ and $jpjdta$, may be larger than $jpiglo$, $jpjglo$ when the user wants to use only a sub-region of a given configuration. This is the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters $jpi$ and $jpj$ refer to the size of each processor subdomain when the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined, see \S\ref{LBC_mpp}).
+The total size of the computational domain is set by the parameters \jp{jpiglo}, 
+\jp{jpjglo} and \jp{jpk} in the $i$, $j$ and $k$ directions respectively. They are 
+given as parameters in the \mdl{par\_oce} module\footnote{When a specific 
+configuration is used (ORCA2 global ocean, etc...) the parameter are actually 
+defined in additional files introduced by \mdl{par\_oce} module via CPP 
+\textit{include} command. For example, ORCA2 parameters are set in 
+\textit{par\_ORCA\_R2.h90} file}. The use of parameters rather than variables 
+(together with dynamic allocation of arrays) was chosen because it ensured that 
+the compiler would optimize the executable code efficiently, especially on vector 
+machines (optimization may be less efficient when the problem size is unknown 
+at the time of compilation). Nevertheless, it is possible to set up the code with full 
+dynamical allocation by using the AGRIF packaged \citep{Debreu_al_CG2008}. 
+%
+\gmcomment{  add the following ref 
+\colorbox{yellow}{(ref part of the doc)} } 
+%
+Note that are other parameters in \mdl{par\_oce} that refer to the domain size. 
+The two parameters $jpidta$ and $jpjdta$ may be larger than $jpiglo$, $jpjglo$ 
+when the user wants to use only a sub-region of a given configuration. This is 
+the "zoom" capability described in \S\ref{MISC_zoom}. In most applications of 
+the model, $jpidta=jpiglo$, $jpjdta=jpjglo$, and $jpizoom=jpjzoom=1$. Parameters 
+$jpi$ and $jpj$ refer to the size of each processor subdomain when the code is 
+run in parallel using domain decomposition (\key{mpp\_mpi} defined, see 
+\S\ref{LBC_mpp}).
 
 % ================================================================
 % Domain: Horizontal Grid (mesh) 
 % ================================================================
-\section{Domain: Horizontal Grid (mesh) (\mdl{domhgr} module)}
+\section  [Domain: Horizontal Grid (mesh) (\textit{domhgr})]               
+		{Domain: Horizontal Grid (mesh) \small{(\mdl{domhgr} module)} }
 \label{DOM_hgr}
 
@@ -201,16 +319,31 @@
 \label{DOM_hgr_coord_e}
 
-The ocean mesh (i.e. the position of all the scalar and vector points) is defined by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. The grid-points are located at integer or integer and a half values of as indicated in Table~\ref{Tab_cell}. The associated scale factors are defined using the analytical first derivative of the transformation \eqref{Eq_scale_factors}. These definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which provide the horizontal and vertical meshes, respectively. This section deals with the horizontal mesh parameters.
-
-In a horizontal plane, the location of all the model grid points is defined from the analytical expressions of the latitude $\varphi$ and the longitude $\lambda$ as a function of  $(i,j)$. The horizontal scale factors are calculated using \eqref{Eq_scale_factors}. For example, when the latitude and longitude are function of a single value ($j$ and $i$, respectively) (geographical configuration of the mesh), the horizontal mesh definition reduces to define the wanted $\varphi(j)$, $\varphi'(j)$, $\lambda(i)$, and $\lambda'(i)$ in the \mdl{domhgr} module. The model computes the grid-point positions and scale factors in the horizontal plane as follows:
+The ocean mesh ($i.e.$ the position of all the scalar and vector points) is defined 
+by the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$. 
+The grid-points are located at integer or integer and a half values of as indicated 
+in Table~\ref{Tab_cell}. The associated scale factors are defined using the 
+analytical first derivative of the transformation \eqref{Eq_scale_factors}. These 
+definitions are done in two modules, \mdl{domhgr} and \mdl{domzgr}, which 
+provide the horizontal and vertical meshes, respectively. This section deals with 
+the horizontal mesh parameters.
+
+In a horizontal plane, the location of all the model grid points is defined from the 
+analytical expressions of the longitude $\lambda$ and  latitude $\varphi$ as a 
+function of  $(i,j)$. The horizontal scale factors are calculated using 
+\eqref{Eq_scale_factors}. For example, when the longitude and latitude are 
+function of a single value ($i$ and $j$, respectively) (geographical configuration 
+of the mesh), the horizontal mesh definition reduces to define the wanted 
+$\lambda(i)$, $\varphi(j)$, and their derivatives $\lambda'(i)$ $\varphi'(j)$ in the 
+\mdl{domhgr} module. The model computes the grid-point positions and scale 
+factors in the horizontal plane as follows:
 \begin{flalign*}
-\lambda_T &\equiv \text{glamt} = \lambda(i)	&		\varphi_T &\equiv \text{gphit} = \varphi(j)\\
-\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)&		\varphi_u &\equiv \text{gphiu}= \varphi(j)\\
-\lambda_v &\equiv \text{glamv}= \lambda(i)		&		\varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
-\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)&		\varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
+\lambda_T &\equiv \text{glamt}= \lambda(i)	  & \varphi_T &\equiv \text{gphit} = \varphi(j)\\
+\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
+\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
+\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
 \end{flalign*}
 \begin{flalign*}
 e_{1T} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j)  |&
-e_{2T} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
+e_{2T} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
 e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)	\; \cos\varphi(j)  |&
 e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
@@ -220,33 +353,91 @@
 e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
 \end{flalign*}
-where the last letter of each computational name indicates the grid point considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with all universal constants). Note that the horizontal position and scale factors of $w$-points are exactly equal to those of $T-$points, thus no specific arrays are defined at those grid-points. 
-
-Note that the definition of the scale factors --- as the analytical first derivative of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$ --- is specific to the OPA model \citep{Marti1992}. As an example, $e_{1T}$ is defined locally at a $T$-point, whereas many other models on a C grid choose to define such a scale factor as the distance between the $U$-points on each side of the $T$-point. Relying on an analytical transformation has two advantages: firstly, there is no ambiguity in the scale factors appearing in the discrete equations, since they are first introduced in the continuous equations; secondly, analytical transformations encourage good practice by the definition of smooth grids \citep{Treguier1996}. An example of the effect of such a choice is shown in Fig.~\ref{Fig_zgr_e3}.
+where the last letter of each computational name indicates the grid point 
+considered and $r_a$ is the earth radius (defined in \mdl{phycst} along with 
+all universal constants). Note that the horizontal position of and scale factors 
+at $w$-points are exactly equal to those of $T$-points, thus no specific arrays 
+are defined at $w$-points. 
+
+Note that the definition of the scale factors ($i.e.$ as the analytical first derivative 
+of the transformation that gives $(\lambda,\varphi,z)$ as a function of $(i,j,k)$) is 
+specific to the \NEMO model \citep{Marti1992}. As an example, $e_{1T}$ is defined 
+locally at a $T$-point, whereas many other models on a C grid choose to define 
+such a scale factor as the distance between the $U$-points on each side of the 
+$T$-point. Relying on an analytical transformation has two advantages: firstly, there 
+is no ambiguity in the scale factors appearing in the discrete equations, since they 
+are first introduced in the continuous equations; secondly, analytical transformations 
+encourage good practice by the definition of smoothly varying grids (rather than 
+allowing the user to set arbitrary jumps in thickness between adjacent layers) 
+\citep{Treguier1996}. An example of the effect of such a choice is shown in 
+Fig.~\ref{Fig_zgr_e3}.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \label{Fig_zgr_e3}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_zgr_e3.pdf}
-\caption{(a) Traditional definition of grid-point position and grid-size in the vertical versus (b) analytically derived grid-point position and scale factors. For both grid here,a same $w$-point depth has been chosen but in (a) the $T$-points are set at the middle of $w$-points while in (b) they are defined from an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$. Note the resulting difference between the value of the grid-size $\Delta_k$ and those of the scale factor $e_k$. }
+\caption{Comparison of (a) traditional definitions of grid-point position and grid-size 
+in the vertical, and (b) analytically derived grid-point position and scale factors. For 
+both grids here,  the same $w$-point depth has been chosen but in (a) the 
+$T$-points are set half way between $w$-points while in (b) they are defined from 
+an analytical function: $z(k)=5\,(i-1/2)^3 - 45\,(i-1/2)^2 + 140\,(i-1/2) - 150$. 
+Note the resulting difference between the value of the grid-size $\Delta_k$ and 
+those of the scale factor $e_k$. }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 % -------------------------------------------------------------------------------------------------------------
-%        Choice of horizontal grids 
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Choice of horizontal grids}
+%        Choice of horizontal grid
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Choice of horizontal grid}
 \label{DOM_hgr_msh_choice}
 
-The user has three options to define a horizontal grid, involving the parameter $jphgr\_mesh$ of the \mdl{par\_oce} module. 
-\begin{enumerate}
-\item For the most general curvilinear orthogonal grids, the coordinates and their first derivatives with respect to $i$ and $j$ are provided in a file, read in \rou{hgr\_read} subroutine of the domhgr module: \jp{jphgr\_mesh}=0.
-\item A few simple analytical grids are provided as examples, that can be selected by setting \jp{jphgr\_mesh}=1 to 5 (see below) 
-\item For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
-\end{enumerate}
-
-There are two simple cases of geographical grids on the sphere. With \jp{jphgr\_mesh}=1, the grid is regular in space, with grid sizes specified by parameters \pp{ppe1\_deg} and \pp{ppe2\_deg}, respectively. A geographical grid 
-can be very anisotropic at high latitudes, because of the convergence of meridians (the zonal scale factors $e_1$ become much smaller than the meridional scale factors $e_2$). The Mercator grid (\jp{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale factors in the same way as the zonal ones. In that case, meridional scale factors and latitudes are calculated analytically using the formulae appropriate for a Mercator projection, based on \pp{ppe1\_deg} which is a reference grid spacing at the equator (this applies even when the geographical equator is situated outside the model domain). In those two cases (\jp{jphgr\_mesh}=1 or 4), the grid position is defined by the longitude and latitude of the south-westhernmost point (\pp{ppglamt0} and \pp{ppgphi0}). Note that for the Mercator grid the user need only provide an approximate starting latitude: the real latitude will be recalculated analytically, so as to ensure that the equator corresponds to a $T$- and$ U$-point.  
-
-Rectangular grids ignoring the spherical geometry are defined with \jp{jphgr\_mesh} = 2, 3, 5. The domain is either a $f$-plane (\jp{jphgr\_mesh} = 2, Coriolis factor is constant) or a beta-plane (\jp{jphgr\_mesh} = 3, the Coriolis factor is linear in the $j$-direction). The grid size is uniform in each direction, and given in meters by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. The zonal grid coordinate (glam. arrays) is in kilometers, starting at zero with the first T point. The meridional coordinate (gphi. arrays) is in kilometers, and the second $T$-point corresponds to coordinate gphit=0. The input parameter \pp{ppglam0} is ignored. \pp{ppgphi0} is used to set the reference latitude for computation of the Coriolis parameter. In the case of the beta plane, \pp{ppgphi0} corresponds to the center of the domain. Finally, the special case \jp{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the GYRE configuration representing a classical mid-latitude double gyre system. The rotation allows to maximize the jet length relative to the gyre areas (and the number of grid points). 
-
-The choice of the grid must be consistent with the boundary conditions specified by the parameter \jp{jperio} (see {\S\ref{LBC}).
+The user has three options available in defining a horizontal grid, which involve 
+the parameter $jphgr\_mesh$ of the \mdl{par\_oce} module. 
+\begin{description}
+\item[\jp{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
+The coordinates and their first derivatives with respect to $i$ and $j$ are 
+provided in a file, read in \rou{hgr\_read} subroutine of the domhgr module.
+\item[\jp{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below). 
+For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
+\end{description}
+
+There are two simple cases of geographical grids on the sphere. With 
+\jp{jphgr\_mesh}=1, the grid (expressed in degrees) is regular in space, 
+with grid sizes specified by parameters \pp{ppe1\_deg} and \pp{ppe2\_deg}, 
+respectively. Such a geographical grid can be very anisotropic at high latitudes 
+because of the convergence of meridians (the zonal scale factors $e_1$ 
+become much smaller than the meridional scale factors $e_2$). The Mercator 
+grid (\jp{jphgr\_mesh}=4) avoids this anisotropy by refining the meridional scale 
+factors in the same way as the zonal ones. In this case, meridional scale factors 
+and latitudes are calculated analytically using the formulae appropriate for 
+a Mercator projection, based on \pp{ppe1\_deg} which is a reference grid spacing 
+at the equator (this applies even when the geographical equator is situated outside 
+the model domain). 
+%%%
+\gmcomment{ give here the analytical expression of the Mercator mesh}
+%%%
+In these two cases (\jp{jphgr\_mesh}=1 or 4), the grid position is defined by the 
+longitude and latitude of the south-westernmost point (\pp{ppglamt0} 
+and \pp{ppgphi0}). Note that for the Mercator grid the user need only provide 
+an approximate starting latitude: the real latitude will be recalculated analytically, 
+in order to ensure that the equator corresponds to line passing through $T$- 
+and $u$-points.  
+
+Rectangular grids ignoring the spherical geometry are defined with 
+\jp{jphgr\_mesh} = 2, 3, 5. The domain is either an $f$-plane (\jp{jphgr\_mesh} = 2, 
+Coriolis factor is constant) or a beta-plane (\jp{jphgr\_mesh} = 3, the Coriolis factor 
+is linear in the $j$-direction). The grid size is uniform in meter in each direction, 
+and given by the parameters \pp{ppe1\_m} and \pp{ppe2\_m} respectively. 
+The zonal grid coordinate (\textit{glam} arrays) is in kilometers, starting at zero 
+with the first $T$-point. The meridional coordinate (gphi. arrays) is in kilometers, 
+and the second $T$-point corresponds to coordinate $gphit=0$. The input 
+parameter \pp{ppglam0} is ignored. \pp{ppgphi0} is used to set the reference 
+latitude for computation of the Coriolis parameter. In the case of the beta plane, 
+\pp{ppgphi0} corresponds to the center of the domain. Finally, the special case 
+\jp{jphgr\_mesh}=5 corresponds to a beta plane in a rotated domain for the 
+GYRE configuration, representing a classical mid-latitude double gyre system. 
+The rotation allows us to maximize the jet length relative to the gyre areas 
+(and the number of grid points). 
+
+The choice of the grid must be consistent with the boundary conditions specified 
+by the parameter \jp{jperio} (see {\S\ref{LBC}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -256,10 +447,19 @@
 \label{DOM_hgr_files}
 
-All the arrays related to a particular ocean model configuration (grid-point position, scale factors, masks) can be saved in files if $\np{nmsh} \not= 0$ (namelist parameter). This can be particularly useful for plots and off-line diagnostics. In some cases, the user may choose to make a local modification of a scale factor in the code. This is the case in global configurations when restricting the width of a specific strait (usually a one-grid-point strait that happens to be too wide due to the insufficient model resolution). On example is Lombok Strait in the ORCA2 configuration. When such modifications are done, the output grid written when $\np{nmsh} \not=0$ is not exactly equal to the input grid.
+All the arrays relating to a particular ocean model configuration (grid-point 
+position, scale factors, masks) can be saved in files if $\np{nmsh} \not= 0$ 
+(namelist parameter). This can be particularly useful for plots and off-line 
+diagnostics. In some cases, the user may choose to make a local modification 
+of a scale factor in the code. This is the case in global configurations when 
+restricting the width of a specific strait (usually a one-grid-point strait that 
+happens to be too wide due to insufficient model resolution). An example 
+is Gibraltar Strait in the ORCA2 configuration. When such modifications are done, 
+the output grid written when $\np{nmsh} \not=0$ is no more equal to the input grid.
 
 % ================================================================
 % Domain: Vertical Grid (domzgr)
 % ================================================================
-\section{Domain: Vertical Grid (\mdl{domzgr} module)}
+\section  [Domain: Vertical Grid (\textit{domzgr})]
+		{Domain: Vertical Grid \small{(\mdl{domzgr} module)} }
 \label{DOM_zgr}
 %-----------------------------------------nam_zgr & namdom-------------------------------------------
@@ -268,30 +468,73 @@
 %-------------------------------------------------------------------------------------------------------------
 
-In the vertical, the model mesh is determined by four things: (1) the bathymetry given in meters ; (2) the number of levels of the model (\jp{jpk}) ; (3) the analytical transformation $z(i,j,k)$ and the vertical scale factors 
-(derivatives of the transformation) ; and (4) the masking system, i.e. the number of wet model levels at each $(i,j)$.
+In the vertical, the model mesh is determined by four things: 
+(1) the bathymetry given in meters ; 
+(2) the number of levels of the model (\jp{jpk}) ; 
+(3) the analytical transformation $z(i,j,k)$ and the vertical scale factors 
+(derivatives of the transformation) ; 
+and (4) the masking system, $i.e.$ the number of wet model levels at each 
+$(i,j)$ column of points.
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!tb] \label{Fig_z_zps_s_sps}  \begin{center}
 \includegraphics[width=1.0\textwidth]{./Figures/Fig_z_zps_s_sps.pdf}
-\caption{The ocean bottom as seen by the model: (a) $z$-coordinate with full step, (b) $z$-coordinate with partial step, (c) $s$-coordinate: terrain following representation, (d) hybrid $s-z$ coordinate, (e) hybrid $s-z$ coordinate with partial step, and (f) same as (e) but with variable volume level associated with the non-linear free surface. Note that the variable volume level (\key{vvl}) could be used with any of the 5 coordinates (a) to (e).}
+\caption{The ocean bottom as seen by the model: 
+(a) $z$-coordinate with full step, 
+(b) $z$-coordinate with partial step, 
+(c) $s$-coordinate: terrain following representation, 
+(d) hybrid $s-z$ coordinate, 
+(e) hybrid $s-z$ coordinate with partial step, and 
+(f) same as (e) but with variable volume associated with the non-linear free surface. 
+Note that the variable volume option (\key{vvl}) can be used with any of the 
+5 coordinates (a) to (e).}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-The choice of a vertical coordinate among all those offered in NEMO, even if it is made through a namelist parameter, must be done once of all at the beginning of an experiment. It is not intended as an option which can be 
-enabled or disabled in the middle of an experiment. Three main choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step bathymetry (\np{ln\_zco}=true), $z$-coordinate with partial step bathymetry (\np{ln\_zps}=true), or generalized, $s$-coordinate (\np{ln\_sco}=true). Hybridation of the three main 
-coordinates are available: hybrid $s-z$ or $s-zps$ coordinate (Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable volume option \key{vvl}), the coordinate follow the time-variation of the free surface so that the transformation is time dependent: $z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step bathymetry or $s$-coordinates (hybride and partial step coordinates not yet implemented in NEMO v2.3). 
-
-Contrary to the horizontal grid, the vertical grid is computed in the code and no provision is made for reading it from a file. The only input file is the bathymetry (in meters). After reading the bathymetry, the algorithm for vertical grid definition differs between the different options:
+The choice of a vertical coordinate, even if it is made through a namelist parameter, 
+must be done once of all at the beginning of an experiment. It is not intended as an 
+option which can be enabled or disabled in the middle of an experiment. Three main 
+choices are offered (Fig.~\ref{Fig_z_zps_s_sps}a to c): $z$-coordinate with full step 
+bathymetry (\np{ln\_zco}=true), $z$-coordinate with partial step bathymetry 
+(\np{ln\_zps}=true), or generalized, $s$-coordinate (\np{ln\_sco}=true). 
+Hybridation of the three main coordinates are available: $s-z$ or $s-zps$ coordinate 
+(Fig.~\ref{Fig_z_zps_s_sps}d and \ref{Fig_z_zps_s_sps}e). When using the variable 
+volume option \key{vvl}) ($i.e.$ non-linear free surface), the coordinate follow the 
+time-variation of the free surface so that the transformation is time dependent: 
+$z(i,j,k,t)$ (Fig.~\ref{Fig_z_zps_s_sps}f). This option can be used with full step 
+bathymetry or $s$-coordinate (hybride and partial step coordinates have not 
+yet been tested in NEMO v2.3). 
+
+Contrary to the horizontal grid, the vertical grid is computed in the code and no 
+provision is made for reading it from a file. The only input file is the bathymetry 
+(in meters)\footnote{N.B. in full step $z$-coordinate, a \textit{bathy\_level} file can 
+replace the \textit{bathy\_meter} file, so that the computation of the number of 
+wet ocean point in each water column is by-passed}. After reading the bathymetry, 
+the algorithm for vertical grid definition differs between the different options:
 \begin{description}
 \item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
 \item[\textit{zps}] set a reference coordinate transformation $z_0 (k)$, and 
-calculate the height at the deepest levels using the bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
-\item[\textit{sco}] Smooth the bathymetry to fullfill the hydrostatic consistency criteria and set the three-dimensional transformation.
-\item[\textit{s-z} and \textit{s-zps}] Smooth the bathymetry to fullfill the hydrostatic consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and possibly introduce masking of extra land points to better fit the original bathymetry file
+calculate the thickness of the deepest level at each $(i,j)$ point using the 
+bathymetry, to obtain the final three-dimensional depth and scale factor arrays.
+\item[\textit{sco}] smooth the bathymetry to fulfil the hydrostatic consistency 
+criteria and set the three-dimensional transformation.
+\item[\textit{s-z} and \textit{s-zps}] smooth the bathymetry to fulfil the hydrostatic 
+consistency criteria and set the three-dimensional transformation $z(i,j,k)$, and 
+possibly introduce masking of extra land points to better fit the original bathymetry file
 \end{description}
-
-Generally, the arrays describing the grid point depths and vertical scale factors are three dimensional arrays $(i,j,k)$. In the special case of $z$-coordinates with full step bottom topography, it is possible to define 
-those arrays as one-dimensional, in order to save memory. This is performed by defining the \key{zco} C-Pre-Processor (CPP) key. To improve the code readability while providing this flexibility, the vertical coordinates and scale factors are defined as functions of $(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be equal to three-dimensional arrays, or a one dimensional array when \key{zco} is defined. These functions are defined in the file 
-\textit{domzgr\_substitute.h90} of the DOM directory. They are used through the code, and replaced by the corresponding arrays at the time of pre-processing (CPP capability).
+%%%
+\gmcomment{   add the description of the smoothing:  envelop topography...}
+%%%
+
+Generally, the arrays describing the grid point depths and vertical scale factors 
+are three dimensional arrays $(i,j,k)$. In the special case of $z$-coordinates with 
+full step bottom topography, it is possible to define those arrays as one-dimensional, 
+in order to save memory. This is performed by defining the \key{zco} 
+C-Pre-Processor (CPP) key. To improve the code readability while providing this 
+flexibility, the vertical coordinate and scale factors are defined as functions of 
+$(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be either 
+three-dimensional arrays, or a one dimensional array when \key{zco} is defined. 
+These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 
+They are used throughout the code, and replaced by the corresponding arrays at 
+the time of pre-processing (CPP capability).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -304,23 +547,35 @@
 namelist variable \np{ntopo}: 
 \begin{description}
-\item[\np{ntopo} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ is given by the coordinate transformation. The domain can either a closed basin or a periodic channel according to the parameter \jp{jperio}. 
-\item[\np{ntopo} = -1] a domain with a bump of topography at the central latitude and 1/3 of the domain width. This is meant for the "EEL-R5" configuration, a periodic or open boundary channel with a seamount. 
-\item[\np{ntopo} = 1] read a bathymetry. The bathymetry file (Netcdf format) provides the ocean depth (positive, in meters) at each grid point of the model grid. The bathymetry is usually built by interpolating a standard bathymetry product (e.g., ETOPO2) onto the horizontal ocean mesh. The bathymetry file defines the coastline: where the bathymetry is zero, no model levels are defined (all levels are masked).
+\item[\np{ntopo} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
+is given by the coordinate transformation. The domain can either be a closed 
+basin or a periodic channel depending on the parameter \jp{jperio}. 
+\item[\np{ntopo} = -1] a domain with a bump of topography one third of the 
+domain width at the central latitude. This is meant for the "EEL-R5" configuration, 
+a periodic or open boundary channel with a seamount. 
+\item[\np{ntopo} = 1] read a bathymetry. The bathymetry file (Netcdf format) 
+provides the ocean depth (positive, in meters) at each grid point of the model grid. 
+The bathymetry is usually built by interpolating a standard bathymetry product 
+($e.g.$ ETOPO2) onto the horizontal ocean mesh. Defining the bathymetry also 
+defines the coastline: where the bathymetry is zero, no model levels are defined 
+(all levels are masked).
 \end{description}
 
-When using the rigid lid approximation (\key{dynspg\_rl} defined) isolated land masses (islands) must be identified by negative integers in the input bathymetry file (see \S\ref{MISC_solisl}).
-
-When the ocean is coupled to an atmospheric model it is better to represent 
+When using the rigid lid approximation (\key{dynspg\_rl} is defined) isolated land 
+masses (islands) must be identified by negative integers in the input bathymetry file 
+(see \S\ref{MISC_solisl}).
+
+When a global ocean is coupled to an atmospheric model it is better to represent 
 all large water bodies (e.g, great lakes, Caspian sea...) even if the model 
-resolution does not allow to represent their communication with the rest of 
-the ocean. This is unnecessary when the ocean is forced by fixed atmospheric 
-conditions. A possibility is offered to the user to set to zero the 
-bathymetry in rectangular regions covering those closed seas (see \S\ref{MISC_closea}), but the code has to be adapted to the user's configuration. 
+resolution does not allow their communication with the rest of the ocean. 
+This is unnecessary when the ocean is forced by fixed atmospheric conditions, 
+so these seas can be removed from the ocean domain. The user has the option 
+to set the bathymetry in closed seas to zero (see \S\ref{MISC_closea}), but the 
+code has to be adapted to the user's configuration. 
 
 % -------------------------------------------------------------------------------------------------------------
 %        z-coordinate  and reference coordinate transformation
 % -------------------------------------------------------------------------------------------------------------
-\subsection	[$z$-coordinate (\np{ln\_zco}=T or \key{zco})]
-			{$z$-coordinate (\np{ln\_zco}=T or \key{zco}) and reference coordinate}
+\subsection[$z$-coordinate (\np{ln\_zco} or \key{zco})]
+	  	  {$z$-coordinate (\np{ln\_zco}=.true. or \key{zco}) and reference coordinate}
 \label{DOM_zco}
 
@@ -328,13 +583,25 @@
 \begin{figure}[!tb] \label{Fig_zgr}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_zgr.pdf}
-\caption{Default vertical mesh for ORCA2-L30. Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from (III.2.1) in z-coordinates.}
+\caption{Default vertical mesh for ORCA2: 30 ocean levels (L30). Vertical level 
+functions for (a) T-point depth and (b) the associated scale factor as computed 
+from \eqref{DOM_zgr_ana} using \eqref{DOM_zgr_coef} in $z$-coordinate.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-The reference coordinate transformation $z_0 (k)$ defines the arrays \textit{gdept0} and \textit{gdepw0} for $T$- and $w$-points, respectively. As indicated on Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw(1)$ being the ocean surface. There are at most \jp{jpk}-1 $T$-points in the ocean, the additional $T$-point at $jk=jpk$ is below the sea floor and is not used. The vertical location of $w$- and $T$-levels is defined from the analytic expression of the depth $z_0 (k)$ whose analytic derivative with respect to $k$ provides the vertical scale factors. The user must provide the analytical expression of both $z_0 $and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} through statement functions, using parameters provided in the \textit{par\_oce.h90} file. 
+The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
+and $gdepw_0$ for $T$- and $w$-points, respectively. As indicated on 
+Fig.\ref{Fig_index_vert} \jp{jpk} is the number of $w$-levels. $gdepw_0(1)$ is the 
+ocean surface. There are at most \jp{jpk}-1 $T$-points inside the ocean, the 
+additional $T$-point at $jk=jpk$ is below the sea floor and is not used. 
+The vertical location of $w$- and $T$-levels is defined from the analytic expression 
+of the depth $z_0(k)$ whose analytical derivative with respect to $k$ provides the 
+vertical scale factors. The user must provide the analytical expression of both 
+$z_0$ and its first derivative with respect to $k$. This is done in routine \mdl{domzgr} 
+through statement functions, using parameters provided in the \textit{par\_oce.h90} file. 
 
 It is possible to define a simple regular vertical grid by giving zero stretching (\pp{ppacr=0}). In that case, the parameters \jp{jpk} (number of $w$-levels) and \pp{pphmax} (total ocean depth in meters) fully define the grid. 
 
-For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface. The following function is proposed as a standard for $z$-coordinates and partial steps: 
+For climate-related studies it is often desirable to concentrate the vertical resolution 
+near the ocean surface. The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): 
 \begin{equation} \label{DOM_zgr_ana}
 \begin{split}
@@ -343,7 +610,12 @@
 \end{split}
 \end{equation}
-where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an expression allows us to define a nearly uniform vertical location of levels at the ocean top and bottom with a smooth hyperbolic tangent transition in between (Fig.~\ref{Fig_zgr}).
-
-The first grid defined for ORCA2 had $10~m$ ($500~m)$ resolution in the surface (bottom) layers and a depth which varies from 0 at the sea surface to a minimum of $-5000~m$. This leads to the following conditions:
+where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels. Such an 
+expression allows us to define a nearly uniform vertical location of levels at the 
+ocean top and bottom with a smooth hyperbolic tangent transition in between 
+(Fig.~\ref{Fig_zgr}).
+
+The most used vertical grid for ORCA2 has $10~m$ ($500~m)$ resolution in the 
+surface (bottom) layers and a depth which varies from 0 at the sea surface to a 
+minimum of $-5000~m$. This leads to the following conditions:
 \begin{equation} \label{DOM_zgr_coef}
 \begin{split}
@@ -355,17 +627,29 @@
 \end{equation}
 
-With the choice of the stretching $h_{cr} =3$ and the number of levels \jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in \eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is satisfied, through an optimisation procedure using a bisection method. For the first standard 
-ORCA2 vertial grid this led to the following values: $h_{sur} =4762.96$, $h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters \pp{ppsur}, \pp{ppa0}, \pp{ppa1}, \pp{ppkth} in the parameter file \mdl{par\_oce}. 
+With the choice of the stretching $h_{cr} =3$ and the number of levels 
+\jp{jpk}=$31$, the four coefficients $h_{sur}$, $h_{0}$, $h_{1}$, and $h_{th}$ in 
+\eqref{DOM_zgr_ana} have been determined such that \eqref{DOM_zgr_coef} is 
+satisfied, through an optimisation procedure using a bisection method. For the first 
+standard ORCA2 vertical grid this led to the following values: $h_{sur} =4762.96$, 
+$h_0 =255.58, h_1 =245.5813$, and $h_{th} =21.43336$. The resulting depths and 
+scale factors as a function of the model levels are shown in Fig.~\ref{Fig_zgr} and 
+given in Table \ref{Tab_orca_zgr}. Those values correspond to the parameters 
+\pp{ppsur}, \pp{ppa0}, \pp{ppa1}, \pp{ppkth} in the parameter file \mdl{par\_oce}. 
 
 Rather than entering parameters $h_{sur}$, $h_{0}$, and $h_{1}$ directly, it is 
 possible to recalculate them. In that case the user sets 
-\pp{ppsur}=\pp{ppa0}=\pp{ppa1}=\pp{pp\_to\_be\_computed}, in \mdl{par\_oce}, and specifies instead the four following parameters:
+\pp{ppsur}=\pp{ppa0}=\pp{ppa1}=\pp{pp\_to\_be\_computed}, in \mdl{par\_oce}, 
+and specifies instead the four following parameters:
 \begin{itemize}
-\item 	\pp{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger \pp{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
-\item 	\pp{ppkth}=$h_{th} $: is approximately the model level at which maximum stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
+\item 	\pp{ppacr}=$h_{cr} $: stretching factor (nondimensional). The larger 
+\pp{ppacr}, the smaller the stretching. Values from $3$ to $10$ are usual.
+\item 	\pp{ppkth}=$h_{th} $: is approximately the model level at which maximum 
+stretching occurs (nondimensional, usually of order 1/2 or 2/3 of \jp{jpk})
 \item 	\pp{ppdzmin}: minimum thickness for the top layer (in meters)
 \item 	\pp{pphmax}: total depth of the ocean (meters).
 \end{itemize}
-As an example, for the $45$ layers used in DRAKKAR configuration those parameters are: \jp{jpk}=46, \pp{ppacr}=9, \pp{ppkth}=23.563, \pp{ppdzmin}=6m, \pp{pphmax}=5750m.
+As an example, for the $45$ layers used in the DRAKKAR configuration those 
+parameters are: \jp{jpk}=46, \pp{ppacr}=9, \pp{ppkth}=23.563, \pp{ppdzmin}=6m, 
+\pp{pphmax}=5750m.
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -396,15 +680,17 @@
 21	& 	\textbf{732.20}	& 	 611.89 & 	\textbf{261.03} & 	220.35 \\ \hline
 22	& 	\textbf{1033.22}&	 872.87 & 	\textbf{339.39} & 	301.42 \\ \hline
-23	& 	\textbf{1405.70}&	1211.59 & 	\textbf{402.26} & 	373.31 \\ \hline
-24	& 	\textbf{1830.89}&	1612.98 & 	\textbf{444.87} & 	426.00 \\ \hline
-25	& 	\textbf{2289.77}&	2057.13 & 	\textbf{470.55} & 	459.47 \\ \hline
-26	& 	\textbf{2768.24}&	2527.22 & 	\textbf{484.95} & 	478.83 \\ \hline
-27	& 	\textbf{3257.48}&	3011.90 & 	\textbf{492.70} & 	489.44 \\ \hline
-28	& 	\textbf{3752.44}&	3504.46 & 	\textbf{496.78} & 	495.07 \\ \hline
-29	& 	\textbf{4250.40}&	4001.16 & 	\textbf{498.90} & 	498.02 \\ \hline
-30	& 	\textbf{4749.91}&	4500.02 & 	\textbf{500.00} &	499.54 \\ \hline
+23	& 	\textbf{1405.70}&	1211.59 & \textbf{402.26} & 	373.31 \\ \hline
+24	& 	\textbf{1830.89}&	1612.98 & \textbf{444.87} & 	426.00 \\ \hline
+25	& 	\textbf{2289.77}&	2057.13 & \textbf{470.55} & 	459.47 \\ \hline
+26	& 	\textbf{2768.24}&	2527.22 & \textbf{484.95} & 	478.83 \\ \hline
+27	& 	\textbf{3257.48}&	3011.90 & \textbf{492.70} & 	489.44 \\ \hline
+28	& 	\textbf{3752.44}&	3504.46 & \textbf{496.78} & 	495.07 \\ \hline
+29	& 	\textbf{4250.40}&	4001.16 & \textbf{498.90} & 	498.02 \\ \hline
+30	& 	\textbf{4749.91}&	4500.02 & \textbf{500.00} &	499.54 \\ \hline
 31	& 	\textbf{5250.23}&	5000.00 &	\textbf{500.56} &	500.33 \\ \hline
 \end{tabular} \end{center} 
-\caption{Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from \eqref{DOM_zgr_ana} using the coefficients given in \eqref{DOM_zgr_coef}}
+\caption{Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration 
+as computed from \eqref{DOM_zgr_ana} using the coefficients given in 
+\eqref{DOM_zgr_coef}}
 \end{table}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -413,13 +699,14 @@
 %        z-coordinate with partial step
 % -------------------------------------------------------------------------------------------------------------
-\subsection{z-coordinate with partial step (\np{ln\_zps}=T)}
+\subsection   [$z$-coordinate with partial step (\np{ln\_zps})]
+			{$z$-coordinate with partial step (\np{ln\_zps}=.true.)}
 \label{DOM_zps}
-%--------------------------------------------namdom-----------------------------------------------------
+%--------------------------------------------namdom-------------------------------------------------------
 \namdisplay{namdom} 
 %--------------------------------------------------------------------------------------------------------------
 
-In that case, the depths of the model levels are still defined by the 
+In $z$-coordinate partial step, the depths of the model levels are defined by the 
 reference analytical function $z_0 (k)$ as described in the previous 
-section, excepted in the bottom layer. The thickness of the bottom layer is 
+section, \emph{except} in the bottom layer. The thickness of the bottom layer is 
 allowed to vary as a function of geographical location $(\lambda,\varphi)$ to allow a 
 better representation of the bathymetry, especially in the case of small 
@@ -431,5 +718,5 @@
 maximum thickness allowed is $2*e_{3t}(jpk-1)$. This has to be kept in mind when 
 specifying the maximum depth \pp{pphmax} in partial steps: for example, with 
-\pp{pphmax}$=5750~m$ for the DRAKKAR 45 layers grid, the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$). Two 
+\pp{pphmax}$=5750~m$ for the DRAKKAR 45 layer grid, the maximum ocean depth allowed is actually $6000~m$ (the default thickness $e_{3t}(jpk-1)$ being $250~m$). Two 
 variables in the namdom namelist are used to define the partial step 
 vertical grid. The mimimum water thickness (in meters) allowed for a cell 
@@ -443,22 +730,23 @@
 %        s-coordinate
 % -------------------------------------------------------------------------------------------------------------
-\subsection{$s$-coordinate (\np{ln\_sco}=T)}
+\subsection   [$s$-coordinate (\np{ln\_sco})]
+			{$s$-coordinate (\np{ln\_sco}=true)}
 \label{DOM_sco}
-%--------------------------------------------nam_zgr_sco---------------------------------------------------
+%------------------------------------------nam_zgr_sco---------------------------------------------------
 \namdisplay{nam_zgr_sco} 
 %--------------------------------------------------------------------------------------------------------------
-In s-coordinate (\key{sco} defined), the depths of the model 
-levels are defined from the product of a depth field and a stretching 
-function and its derivative, respectively:
+In $s$-coordinate (\key{sco} is defined), the depth and thickness of the model 
+levels are defined from the product of a depth field and either a stretching 
+function or its derivative, respectively:
 \begin{equation} \label{DOM_sco_ana}
 \begin{split}
- z(k) 	&= h(i,j) \; z_0(k)	\\
+ z(k) 		&= h(i,j) \; z_0(k)	\\
  e_3(k)	&= h(i,j) \; z_0'(k)
 \end{split}
 \end{equation}
-where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at $T-$point location 
-in the horizontal and $z_0 (k)$ is a function which varies from $0$ at the sea 
+where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $T$-point 
+location in the horizontal and $z_0(k)$ is a function which varies from $0$ at the sea 
 surface to $1$ at the ocean bottom. The depth field $h$ is not necessary the ocean 
-depth as a mixed step-like and bottom following representation of the 
+depth, since a mixed step-like and bottom-following representation of the 
 topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided (\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent sharp bathymetric gradients.
 
@@ -472,12 +760,12 @@
 \end{split}
 \end{equation}
-where $h_c $is the thermocline depth and $\theta $ and $b$ are the surface and 
+where $h_c$ is the thermocline depth and $\theta$ and $b$ are the surface and 
 bottom control parameters such that $0\leqslant \theta \leqslant 20$, and 
-$0\leqslant b\leqslant 1$. Examples of the stretching function applied to a seamount are given in Fig.~\ref{Fig_sco_function}.
+$0\leqslant b\leqslant 1$. $b$ has been designed to allow surface and/or bottom increase of the vertical resolution (Fig.~\ref{Fig_sco_function}).
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!tb] \label{Fig_sco_function}  \begin{center}
 \includegraphics[width=1.0\textwidth]{./Figures/Fig_sco_function.pdf}
-\caption{examples of the stretching function applied to a sea mont: from left to right, surface, surface and bottom, and bottom intensified resolution}
+\caption{Examples of the stretching function applied to a sea mont; from left to right: surface, surface and bottom, and bottom intensified resolutions}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -489,5 +777,6 @@
 \label{DOM_zgr_vvl}
 
-This option is described in the report by Levier \textit{et al.} (2007), available on the NEMO web site. 
+This option is described in the Report by Levier \textit{et al.} (2007), available on 
+the \NEMO web site. 
 
 %gm% key advantage: minimise the diffusion/dispertion associated with advection in response to high frequency surface disturbances
@@ -511,49 +800,75 @@
 
 Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
-routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points that do not 
-communicate with another ocean point at the same level are eliminated.
-
-In case of rigid-lid approximation and islands in the computational domain (\np{ln\_dynspg\_rl}=true and \key{island} defined), the \textit{mbathy} array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 
-following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $T-$points are land points of the $n^{th}$ island ; $mbathy(i,j) =0$, $T-$points are land points of the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $T$- and $w$-points are ocean points, the others land points. This is used to compute the island barotropic stream function used in rigid lid computation (see \S\ref{MISC_solisl}).
+routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points 
+that do not communicate with another ocean point at the same level are eliminated.
+
+In the case of the rigid-lid approximation when islands occur in the computational 
+domain (\np{ln\_dynspg\_rl}=.true. and \key{island} is defined), the \textit{mbathy} 
+array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 
+following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $T$-points are 
+land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $T$-points are land points 
+on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $T$- and $w$-points 
+are ocean points, the others are points below the ocean floor. 
+
+This is used to compute the island barotropic stream function used in the rigid lid 
+computation (see \S\ref{MISC_solisl}).
 
 From the \textit{mbathy} array, the mask fields are defined as follows:
 \begin{align*}
-tmask(i,j,k) &= \begin{cases}   1&   \text{ if $k\leq mbathy(i,j)$  }    \\
-                                              0&   \text{ if $k\leq mbathy(i,j)$  }    \end{cases}     \\
-umask(i,j,k) &= \; tmask(i,j,k) \;.\; tmask(i+1,j,k)	\\
-umask(i,j,k) &= \; tmask(i,j,k) \;.\; tmask(i,j+1,k)	\\
-umask(i,j,k) &= \; tmask(i,j,k) \;.\; tmask(i+1,j,k)	\\
-                  & \quad . \; tmask(i,j,k) \;.\; tmask(i+1,j,k)
+tmask(i,j,k) &= \begin{cases}   \; 1&   \text{ if $k\leq mbathy(i,j)$  }    \\
+                                                \; 0&   \text{ if $k\leq mbathy(i,j)$  }    \end{cases}     \\
+umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)	\\
+umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+                   & \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k)
 \end{align*}
 
-Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with the numerical 
-indexation used (\S~\ref{DOM_Num_Index}). Moreover, the specification of closed lateral 
-boundaries requires that at least the first and last rows and columns of 
-\textit{mbathy} array are set to zero. In the particular case of an east-west cyclic 
-boundary condition, \textit{mbathy} has its last column equal to the second one and its 
-first column equal to the last but one (and so the mask arrays) (see \S~\ref{LBC_jperio}).
-
-\colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}.
+\gmcomment{ STEVEN: are the dots multiplications?}      
+
+Note that \textit{wmask} is not defined as it is exactly equal to \textit{tmask} with 
+the numerical indexing used (\S~\ref{DOM_Num_Index}). Moreover, the 
+specification of closed lateral boundaries requires that at least the first and last 
+rows and columns of the \textit{mbathy} array are set to zero. In the particular 
+case of an east-west cyclical boundary condition, \textit{mbathy} has its last 
+column equal to the second one and its first column equal to the last but one 
+(and so too the mask arrays) (see \S~\ref{LBC_jperio}).
+
+%%%
+\gmcomment{   \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}.  }
+%%%
 
 % ================================================================
-% Time discretisation
+% Time Discretisation
 % ================================================================
 \section{Time Discretisation}
 \label{DOM_nxt}
 
-The time stepping used in OPA is a three level scheme that can be presented as follows:
+The time stepping used in \NEMO is a three level scheme that can be 
+represented as follows:
 \begin{equation} \label{Eq_DOM_nxt}
    x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t,t,t+\Delta t}
 \end{equation} 
-where $x$ stand for $u$, $v$, $T$ or $S$, RHS is the Right-Hand-Side of the corresponding time 
-evolution equation, $\Delta t$ is the time step and the overscripts indicate 
-the time at which a quantity is evaluated. Each term of the RHS is evaluated at 
-specific time step(s) depending on the physics to which it is associated. 
+where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the 
+corresponding time evolution equation; $\Delta t$ 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(s) 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 the implication in term of starting or restarting a model 
+well as the implications in terms of starting or restarting a model 
 simulation. Note that the time stepping is generally performed in a one step 
-operation. With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in time for each term successively.
-
-The three level scheme requires three arrays for the prognostic variables. For each variable $x$ there is $x_b$ (before) and $x_n$ (now). The third array, although referred to as $x_a$ (after) in the code, is usually not the variable $x_a$ at the next time step; rather, it is used to store the time derivative (RHS in \eqref{Eq_DOM_nxt}) prior to time-stepping the equation. Generally, the time stepping is performed once at each time step in \mdl{tranxt} and \mdl{dynnxt} modules, excepted for implicit vertical diffusion or sea surface height when time-splitting options are used.
+operation. With such a complex and nonlinear system of equations it would be dangerous to let a prognostic variable evolve in time for each term separately.
+%%%
+\gmcomment{ STEVEN  suggest separately instead of successively...  wrong?}
+%%%
+
+The three level scheme requires three arrays for each prognostic variables. 
+For each variable $x$ there is $x_b$ (before) and $x_n$ (now). The third array, 
+although referred to as $x_a$ (after) in the code, is usually not the variable at 
+the next time step; but rather it is used to store the time derivative (RHS in 
+\eqref{Eq_DOM_nxt}) prior to time-stepping the equation. Generally, the time 
+stepping is performed once at each time step in \mdl{tranxt} and \mdl{dynnxt} 
+modules, except for implicit vertical diffusion or sea surface height when 
+time-splitting options are used.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -564,7 +879,7 @@
 
 The time stepping used for non-diffusive processes is the well-known 
-leapfrog scheme. It is a time centred scheme, i.e. the RHS are evaluated at 
+leapfrog scheme. It is a time centred scheme, i.e. the RHS is evaluated at 
 time step $t$, the now time step. It is only used for non-diffusive terms, 
-that is momentum and tracer advection, pressure gradient, and coriolis 
+that is momentum and tracer advection, pressure gradient, and Coriolis 
 terms. This scheme is widely used for advective processes in low-viscosity 
 fluids. It is an efficient method that achieves second-order accuracy with 
@@ -578,37 +893,51 @@
 "time-splitting" \citep{Haltiner1980} that develops when the method 
 is used to model non linear fluid dynamics: the even and odd time steps tend 
-to diverge between a physical and a computational mode. Time splitting can 
+to diverge into a physical and a computational mode. Time splitting can 
 be controlled through the use of an Asselin time filter (first designed by 
-\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}) or by 
+\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), or by 
 periodically reinitialising the leapfrog solution through a single 
-integration step with a two-level scheme. In OPA we follow the first 
+integration step with a two-level scheme. In \NEMO we follow the first 
 strategy:
 \begin{equation} \label{Eq_DOM_nxt_asselin}
 x_F^t  = x^t + \gamma \, \left[ x_f^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
 \end{equation} 
-where the subscript $f$ denotes filtered values and $\gamma$ is the asselin coefficient. $\gamma$ is initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp}=0.1.  This default value causes a significant dissipation of high frequency motions. Recommanded values in idealized studies of shallow water turbulence are two order of magnitude lower (\citep{Farge1987}). Both strategies do, nevertheless, degrade the accuracy of the calculation from second to first order. The leapfrog scheme associated to a Robert-Asselin 
+where the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
+coefficient. $\gamma$ is initialized as \np{atfp} (namelist parameter). 
+Its default value is \np{atfp}=0.1.  This default value causes a significant dissipation 
+of high frequency motions. Recommended values in idealized studies of shallow 
+water turbulence are two orders of magnitude smaller (\citep{Farge1987}). 
+Both strategies do, nevertheless, degrade the accuracy of the calculation from 
+second to first order. The leapfrog scheme combined with a Robert-Asselin 
 time filter has been preferred to other time differencing schemes such as 
-predictor corrector or trapezoidal schemes because the user can better 
-control the magnitude and the spatial structure of the time diffusion of the 
-scheme. In association with the centred space discretisation of the 
+predictor corrector or trapezoidal schemes, because the user has an explicit 
+and simple control of the magnitude of the time diffusion of the scheme. 
+In association with the 2nd order centred space discretisation of the 
 advective terms in the momentum and tracer equations, it avoids implicit 
-numerical diffusion in both the time and space discretisation of the 
-advective term: they are both set explicitly by the user through the Robert-Asselin filter parameter and the viscous and diffusive coefficients.
-
+numerical diffusion in both the time and space discretisations of the 
+advective term: they are both set explicitly by the user through the Robert-Asselin 
+filter parameter and the viscous and diffusive coefficients.
+
+\gmcomment{
 %gm - reflexion about leapfrog: ongoing work with Matthieu Leclair
 % to be updated latter with addition of new time stepping strategy
-\amtcomment{
 \colorbox{yellow}{Note}: 
-1- There is no reason why one should apply a same value of $\gamma$ on both momentum and tracer equations. In climate applications, one could found useful to use a lower value on tracer (quantity that one wants to conserve) than on the dynamics. We never explore this possibility.
-The Robert-Asselin time filter slightly departs from a simple second order time diffusive operator computed with a forward time stepping due to the presence of $x_f^{t-\Delta t}$ in the right hand side of  \ref{Eq_DOM_nxt_asselin}. The original willing of Robert1966 and Asselin1972 was to design a time filter that allow much larger parameter than 0.5.   is due to computer saving consideration. In the original asselin filter, $x^{t-\Delta t}$ is used instead:
+The Robert-Asselin time filter slightly departs from a simple second order time 
+diffusive operator computed with a forward time stepping due to the presence of 
+$x_f^{t-\Delta t}$ in the right hand side of  \ref{Eq_DOM_nxt_asselin}. The original 
+willing of Robert1966 and Asselin1972 was to design a time filter that allow much 
+larger parameter than 0.5.   is due to computer saving consideration. In the original 
+asselin filter, $x^{t-\Delta t}$ is used instead:
  \begin{equation} \label{Eq_DOM_nxt_asselin_true}
 x_f^t  = x^t + \gamma \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
 \end{equation} 
-Applying a "true" Asselin time filter is nothing more than adding a harmonic diffusive operator in time. Indeed, equations \ref{Eq_DOM_nxt} and \ref{Eq_DOM_nxt_asselin_true} can be rewritten together as:
+Applying a "true" Asselin time filter is nothing more than adding a harmonic 
+diffusive operator in time. Indeed, equations \ref{Eq_DOM_nxt} and 
+\ref{Eq_DOM_nxt_asselin_true} can be rewritten together as:
 \begin{equation} \label{Eq_DOM_nxt2}
 \begin{split}
   \frac{ x^{t+\Delta t} - x^{t-\Delta t} } { 2 \,\Delta t } 
   &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} + \frac{ x_f^t  - x^t }{2 \,\Delta t} \\
-  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} + \gamma\ \frac{  \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right] }{2 \,\Delta t}  \\
+  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
+    + \gamma\ \frac{  \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right] }{2 \,\Delta t}  \\
   &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
   + 2 \Delta t \ \gamma \ \frac{1}{{2 \Delta t}^2}  
@@ -621,5 +950,12 @@
 \end{equation} 
 
-Equations  \ref{Eq_DOM_nxt2} and \ref{Eq_DOM_nxt3} suggest several remarks. First the Asselin filter is definitively a second order time diffusive operator which is evaluated at centered time step. The magnitude of this diffusion is proportional to the time step (with $\gamma$ usually taken between $10^{-1}$ to $10^{-3}$) . Second, this term has to be taken into account in all budgets of the equations (mass, heat content, salt content, kinetic energy...). Nevertheless,we stress here that it is small and does not create systematic biases. Indeed let us evaluate how it contributes to the time evolution of $x$ between $t_o$ and $t_1$:
+Equations  \ref{Eq_DOM_nxt2} and \ref{Eq_DOM_nxt3} suggest several remarks. 
+First the Asselin filter is definitively a second order time diffusive operator which is 
+evaluated at centered time step. The magnitude of this diffusion is proportional to 
+the time step (with $\gamma$ usually taken between $10^{-1}$ to $10^{-3}$). 
+Second, this term has to be taken into account in all budgets of the equations 
+(mass, heat content, salt content, kinetic energy...). Nevertheless,we stress here 
+that it is small and does not create systematic biases. Indeed let us evaluate how 
+it contributes to the time evolution of $x$ between $t_o$ and $t_1$:
 \begin{equation} \label{Eq_DOM_nxt4}
 \begin{split}
@@ -640,14 +976,15 @@
 \label{DOM_nxt_forward_imp}
 
-The leapfrog differencing is unsuitable for the representation of diffusive 
-and damping processes. For $D$, a horizontal diffusive terms and/or the 
-restoring terms to a tracer climatology (when they are present, see 
-\S~\ref{TRA_dmp}), a forward time differencing scheme is used :
+The leapfrog differencing scheme is unsuitable for the representation of 
+diffusive and damping processes. For a tendancy $D_x$, representing a 
+diffusive term or a restoring term to a tracer climatology 
+(when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme
+ is used :
 \begin{equation} \label{Eq_DOM_nxt_euler}
-   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t}
+   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \ {D_x}^{t-\Delta t}
 \end{equation} 
 
 This is diffusive in time and conditionally stable. For example, the 
-condition of stability for a second and fourth order horizontal diffusions are \citep{Griffies2004}:
+conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies2004}:
 \begin{equation} \label{Eq_DOM_nxt_euler_stability}
 A^h < \left\{
@@ -658,12 +995,12 @@
 \right.
 \end{equation}
-where $e$ is the smallest grid size in the two horizontal direction and $A^h$ the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} is a necessary condition, but not sufficient. If it is not satisfied, even mildly, then the model soon becomes wildly unstable. The instability can be removed by either reducing the time steps or reducing the mixing coefficient.
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} is a necessary condition, but not sufficient. If it is not satisfied, even mildly, then the model soon becomes wildly unstable. The instability can be removed by either reducing the length of the time steps or reducing the mixing coefficient.
 
 For the vertical diffusion terms, a forward time differencing scheme can be 
 used, but usually the numerical stability condition implies a strong 
-constraint on the time step. Two solutions are available in OPA to overcome 
+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}=T) or $(b)$ a backward (or implicit) 
-time differencing scheme by \np{ln\_zdfexp}=F). In $(a)$, the master 
+time splitting technique (\np{ln\_zdfexp}=.true.) or $(b)$ a backward (or implicit) 
+time differencing scheme by \np{ln\_zdfexp}=.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 done as 
@@ -678,5 +1015,7 @@
 \end{split}
 \end{equation}
-with DF a vertical diffusion term. The number of fractional time steps, $N$, is given by setting \np{n\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally stable but diffusive. It can be written as follows:
+with DF a vertical diffusion term. The number of fractional time steps, $N$, is given 
+by setting \np{n\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 
+stable but diffusive. It can be written as follows:
 \begin{equation} \label{Eq_DOM_nxt_imp}
    x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t+\Delta t}
@@ -692,5 +1031,5 @@
 \right]
 \end{equation}
-where RHS is the right hand side of the equation except the vertical diffusion term. We rewrite \eqref{Eq_DOM_nxt_imp} as:
+where RHS is the right hand side of the equation except for the vertical diffusion term. We rewrite \eqref{Eq_DOM_nxt_imp} as:
 \begin{equation} \label{Eq_DOM_nxt_imp_mat}
 -c(k+1)\;u^{t+1}(k+1)+d(k)\;u^{t+1}(k)-\;c(k)\;u^{t+1}(k-1) \equiv b(k)
@@ -703,6 +1042,9 @@
 \end{align*}
 
-\eqref{Eq_DOM_nxt_imp_mat} is a linear system of equations. All the elements of the corresponding matrix vanish except those on the diagonals. Moreover, $c(k)$ 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}).
+\eqref{Eq_DOM_nxt_imp_mat} is a linear system of equations which associated 
+matrix is tridiagonal. Moreover, $c(k)$ 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}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -714,5 +1056,10 @@
 \namdisplay{namrun}         
 %--------------------------------------------------------------------------------------------------------------
-The first time step of this three level scheme when starting from initial conditions is a forward step (Euler time integration): $x^1 = x^0 + \Delta t \ \text{RHS}^0$.
+
+The first time step of this three level scheme when starting from initial conditions 
+is a forward step (Euler time integration): 
+\begin{equation} \label{Eq_DOM_euler}
+	x^1 = x^0 + \Delta t \ \text{RHS}^0
+\end{equation}
 
 It is also possible to restart from a previous computation, by using a 
@@ -720,8 +1067,17 @@
 restartability of the code: the user should obtain the same results to 
 machine precision either by running the model for $2N$ time steps in one go, 
-or by performing two consecutive experiments of $N$ steps with a restart. This 
-requires saving two time levels and many auxiliary data in the restart files 
-in double precision. 
-
+or by performing two consecutive experiments of $N$ steps with a restart. 
+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 \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
+added in the restart file to ensure an exact restartability. This is done only optionally 
+via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of restart file can be obtained when the restartability is not a key issue (operational oceanography or ensemble simulation for seasonal forcast).
+%%%
+\gmcomment{add here how to force the restart to contain only one time step for operational purposes}
+%%%
+
+\gmcomment{       % add a subsection here  
 
 %-------------------------------------------------------------------------------------------------------------
@@ -735,4 +1091,4 @@
  \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}
 
-
-
+}     	%% end add
+
Index: trunk/DOC/BETA/Chapters/Chap_DYN.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_DYN.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_DYN.tex	(revision 817)
@@ -11,5 +11,10 @@
 $\ $\newline      %force an empty line
 
-Using the representation described in Chap.~\ref{DOM}, several semi-discrete space forms of the dynamical equations are available depending on the vertical coordinate used and on the conservative properties of the vorticity term. In all the equations presented here, the masking has been omitted for simplicity. One must be aware that all the quantities are masked fields and that each time a mean or difference operator is used, the resulting field is multiplied by a mask.
+Using the representation described in Chap.\ref{DOM}, several semi-discrete 
+space forms of the dynamical equations are available depending on the vertical 
+coordinate used and on the conservation properties of the vorticity term. In all 
+the equations presented here, the masking has been omitted for simplicity. 
+One must be aware that all the quantities are masked fields and that each time a 
+average or difference operator is used, the resulting field is multiplied by a mask.
 
 The prognostic ocean dynamics equation can be summarized as follows:
@@ -20,11 +25,36 @@
 \end{equation*}
 
-NXT stands for next, referring to the time-stepping. The first group of terms on the rhs of the momentum equations corresponds to the Coriolis and advection terms that are decomposed in a vorticity part (VOR), a kinetic energy part (KEG) and the vertical advection (ZAD) in the vector invariant formulation, and into Coriolis and advection (COR+ADV) in the flux formulation. The following terms are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, and SPG, Surface Pressure Gradient). Contributions from lateral diffusion and vertical diffusion are added to the rhs in the \mdl{dynldf} and \mdl{dynzdf} modules; the latter includes the surface and bottom stresses. The external forcings and parameterisations require complex inputs (surface wind stress calculation using bulk formulae, estimation of mixing coefficients) that are carried out in modules of the SBC, LDF and ZDF categories and described in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. 
-
-In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence and curl of the velocities (\emph{divcur} module) as well as the vertical velocity (\emph{wzvmod} module).
-
-The different options available to the user are managed by namelist variables. For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx}, where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding code can be found in the \textit{dynttt\_xxx} module, in the DYN directory, and it is usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
-
-The user has the option of extracting each tendency term on the rhs of the 3D momentum equation (\key{trddyn} defined) and of the 2D barotropic vorticity balance (\key{trdvor} defined), as described in Chap.~\ref{MISC}.
+NXT stands for next, referring to the time-stepping. The first group of terms on 
+the rhs of the momentum equations corresponds to the Coriolis and advection 
+terms that are decomposed into a vorticity part (VOR), a kinetic energy part (KEG) 
+and, a vertical advection part (ZAD) in the vector invariant formulation or a Coriolis 
+and advection part(COR+ADV) in the flux formulation. The terms following these 
+are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, 
+and SPG, Surface Pressure Gradient); and contributions from lateral diffusion 
+(LDF) and vertical diffusion (ZDF), which are added to the rhs in the \mdl{dynldf} 
+and \mdl{dynzdf} modules. The vertical diffusion term includes the surface and 
+bottom stresses. The external forcings and parameterisations require complex 
+inputs (surface wind stress calculation using bulk formulae, estimation of mixing 
+coefficients) that are carried out in modules SBC, LDF and ZDF and are described 
+in Chapters \ref{SBC}, \ref{LDF} and \ref{ZDF}, respectively. 
+
+In the present chapter we also describe the diagnostic equations used to compute 
+the horizontal divergence and curl of the velocities (\emph{divcur} module) as well 
+as the vertical velocity (\emph{wzvmod} module).
+
+The different options available to the user are managed by namelist variables. 
+For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 
+If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding 
+code can be found in the \textit{dynttt\_xxx} module in the DYN directory, and it is 
+usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
+
+The user has the option of extracting each tendency term of both the rhs of the 
+3D momentum equation (\key{trddyn} defined) for output, as described in 
+Chap.\ref{MISC}.  Furthermore, the tendency terms associated to the 2D 
+barotropic vorticity balance (\key{trdvor} defined) can be derived on-line from the 
+3D terms.
+%%%
+\gmcomment{STEVEN: not quite sure I've got the sense of the last sentence. does MISC correspond to "extracting tendency terms" or "vorticity balance"?}
 
 % ================================================================
@@ -37,11 +67,18 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The vector invariant form of the momentum equations is the most often used in applications of the ocean model. The flux form option (next section) has been introduced recently in version $2$ of NEMO. Coriolis and momentum 
-advection terms are evaluated using a leapfrog scheme, i.e. the velocity appearing in their 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}.
+The vector invariant form of the momentum equations is the one most 
+often used in applications of \NEMO ocean model. The flux form option 
+(see next section) has been recently introduced in version $2$. 
+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}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Vorticity term 
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Vorticity term (\mdl{dynvor})}
+\subsection   [Vorticity term (\textit{dynvor}) ]
+			{Vorticity term (\mdl{dynvor})}
 \label{DYN_vor}
 %------------------------------------------nam_dynvor----------------------------------------------------
@@ -49,14 +86,22 @@
 %-------------------------------------------------------------------------------------------------------------
 
-Different discretisations of the vorticity term (selected by the \textit{ln\_dynvor\_xxx} namelist variable to true) are available, that conserve potential enstrophy of horizontally non-divergent flow, horizontal kinetic energy, or potential enstrophy for the relative vorticity term and horizontal kinetic energy for the planetary vorticity term (see  \colorbox{yellow} { appendix C } ). The vorticity terms are given below for the general case ($s$-coordinate or partial step topography), but note that in full step $z$-coordinate (\key{zco} defined), $e_{3u} =e_{3v} =e_{3f}$ so that the vertical scale factors disappear.
+Different discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=.true.) are 
+available: conserving potential enstrophy of horizontally non-divergent flow; 
+conserving horizontal kinetic energy; or conserving potential enstrophy for the 
+relative vorticity term and horizontal kinetic energy for the planetary vorticity term 
+(see  Appendix~\ref{Apdx_C}). The vorticity terms are given below for the general 
+case, but note that in the full step $z$-coordinate (\key{zco} is defined), 
+$e_{3u} =e_{3v} =e_{3f}$ so that the vertical scale factors disappear.
 
 %-------------------------------------------------------------
 %                 enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{enstrophy conserving scheme (\np{ln\_dynvor\_ens}=T)}
+\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=.true.)}
 \label{DYN_vor_ens}
 
-In this case, the discrete formulation of the vorticity term provides a global conservation of the enstrophy ($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent flow (i.e. $\chi=0$), but does not conserve of the total kinetic energy. It is given by:
-
+In the enstrophy conserving case (ENS scheme), the discrete formulation of the 
+vorticity term provides a global conservation of the enstrophy 
+($ [ (\zeta +f ) / e_{3f} ]^2 $ in $s$-coordinates) for a horizontally non-divergent 
+flow ($i.e.$ $\chi=0$), but does not conserve the total kinetic energy. It is given by:
 \begin{equation} \label{Eq_dynvor_ens}
 \left\{ 
@@ -71,9 +116,9 @@
 %                 energy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{energy conserving scheme (\np{ln\_dynvor\_ene}=T)}
+\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=.true.)}
 \label{DYN_vor_ene}
 
-The kinetic energy conserving scheme conserves the global kinetic energy but 
-not the global enstrophy. It is given by:
+The kinetic energy conserving scheme (ENE scheme) conserves the global 
+kinetic energy but not the global enstrophy. It is given by:
 \begin{equation} \label{Eq_dynvor_ene}
 \left\{ {
@@ -90,8 +135,11 @@
 %                 mix energy/enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=T) }
+\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=.true.) }
 \label{DYN_vor_mix}
 
-In this case, a mixture of the two previous schemes is used. It consists of the enstrophy conserving scheme (\ref{Eq_dynvor_ens}) applied to the relative vorticity term and of the horizontal kinetic energy conserving scheme (\ref{Eq_dynvor_ene}) applied to the planetary vorticity term.
+The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 
+two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) 
+to the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 
+to the planetary vorticity term.
 \begin{equation} \label{Eq_dynvor_mix}
 \left\{ {
@@ -112,16 +160,26 @@
 %                 energy and enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=T) }
+\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=.true.) }
 \label{DYN_vor_een}
 
-In this case, the vorticity term is evaluated using the vorticity advection scheme of \citet{Arakawa1990}. This scheme conserves both total energy and potential enstrophy in the limit of horizontally nondivergent flow (i.e. $\chi$=0), While this scheme is more complicated that the vorticity advection scheme of \citet{Sadourny1975} and does not conserve potential enstrophy and total energy in general flow, as does a scheme from \citet{Arakawa1981}, it tolerates arbitrarily thin layers. This feature is essential for simulating either outcropping isopycnals or large amplitude topography. 
-
-The \citet{Arakawa1990} vorticity advection scheme for a single layer is modified for spherical coordinates as described by \citet{Arakawa1981}. 
-
-The potential vorticity, defined at $f$-point, is: 
+In the energy and enstrophy conserving scheme (EEN scheme), the vorticity term 
+is  evaluated using the vorticity advection scheme of \citet{Arakawa1990}. 
+This scheme conserves both total energy and potential enstrophy in the limit of 
+horizontally nondivergent flow ($i.e. \ \chi=0$). While EEN is more complicated 
+than ENS or ENE and does not conserve potential enstrophy and total energy in 
+general flow, it tolerates arbitrarily thin layers. This feature is essential for 
+$z$-coordinate with partial step. 
+%%%
+\gmcomment{gm :   it actually conserve kinetic energy  !   show that in appendix C }
+%%%
+
+The \citet{Arakawa1990} vorticity advection scheme for a single layer is modified 
+for spherical coordinates as described by \citet{Arakawa1981} to obtain the EEN 
+scheme. The potential vorticity, defined at an $f$-point, is: 
 \begin{equation} \label{Eq_pot_vor}
 q_f  = \frac{\zeta +f} {e_{3f} }
 \end{equation}
-where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $F$-points is: 
+where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter 
+is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 
 \begin{equation} \label{Eq_een_e3f}
 e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
@@ -132,10 +190,18 @@
 \begin{center}
 \includegraphics[width=0.70\textwidth]{./Figures/Fig_DYN_een_triad.pdf}
-\caption{Triads used in the energy and enstrophy conserving scheme (een) for u-component (upper panel) and v-component (lower panel).}
+\caption{Triads used in the energy and enstrophy conserving scheme (een) for 
+$u$-component (upper panel) and $v$-component (lower panel).}
 \end{center}
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and \textbf{j}- directions uses the masked vertical scale factor but is always divided by $4$, not by the sum of the mask at $T$-point. 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 $e_{3f}$ in the land areas. 
+Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and 
+\textbf{j}- directions uses the masked vertical scale factor but is always divided by 
+$4$, not by the sum of the mask at $T$-point. 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 $e_{3f}$ in the land areas. 
+%%%
+\gmcomment{this has to be further investigate in case of several step topography}
+%%%
 
 The vorticity terms are represented as: 
@@ -163,5 +229,6 @@
 } \right.
 \end{equation} 
-where $a$, $b$, $c$ and $d$ are triad combinations of the neighbouring potential vorticities (Fig. \ref{Fig_DYN_een_triad}): 
+where $a$, $b$, $c$ and $d$ are triad combinations of the neighbouring 
+potential vorticities (Fig. \ref{Fig_DYN_een_triad}): 
 \begin{equation} \label{Eq_een_triads}
 \left\{ 
@@ -181,8 +248,11 @@
 %           Kinetic Energy Gradient term
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Kinetic Energy Gradient term (\mdl{dynkeg})}
+\subsection   [Kinetic Energy Gradient term (\textit{dynkeg})]
+			{Kinetic Energy Gradient term (\mdl{dynkeg})}
 \label{DYN_keg}
 
-There is a single discrete formulation of the kinetic energy gradient term, that conserves the total kinetic energy together with the formulation chosen for the vertical advection (see below). 
+As demonstarted in Appendix~\ref{Apdx_C}, there is a single discrete formulation 
+of the kinetic energy gradient term that, together with the formulation chosen for 
+the vertical advection (see below), conserves the total kinetic energy:
 \begin{equation} \label{Eq_dynkeg}
 \left\{ \begin{aligned}
@@ -197,8 +267,12 @@
 %           Vertical advection term
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Vertical advection term (\mdl{dynzad}) }
+\subsection   [Vertical advection term (\textit{dynzad}) ]
+			{Vertical advection term (\mdl{dynzad}) }
 \label{DYN_zad}
 
-The discrete formulation of the vertical advection term conserves the total kinetic energy together with the formulation chosen for the gradient of kinetic energy (KE). Indeed, the change of KE due to the vertical advection is exactly balanced by the change of KE due to the gradient of KE (see \colorbox{yellow}{Annexe C}).
+The discrete formulation of the vertical advection, together with the formulation 
+chosen for the gradient of kinetic energy (KE) term, conserves the total kinetic 
+energy. Indeed, the change of KE due to the vertical advection is exactly 
+balanced by the change of KE due to the gradient of KE (see Appendix~\ref{Apdx_C}).
 \begin{equation} \label{Eq_dynzad}
 \left\{ 		\begin{aligned}
@@ -219,5 +293,9 @@
 %-------------------------------------------------------------------------------------------------------------
 
-In the flux form (as in the vector invariant form), the Coriolis and momentum advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity appearing in their 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}.
+In the flux form (as in the vector invariant form), the Coriolis and momentum 
+advection terms are evaluated using a leapfrog scheme, $i.e.$ the velocity 
+appearing in their 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}.
 
 
@@ -225,8 +303,11 @@
 %           Coriolis plus curvature metric terms
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Coriolis plus curvature metric terms (\mdl{dynvor}) }
+\subsection   [Coriolis plus curvature metric terms (\textit{dynvor}) ]
+			{Coriolis plus curvature metric terms (\mdl{dynvor}) }
 \label{DYN_cor_flux}
 
-In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term. This altered Coriolis parameter is thus discretised at $F$-points. It is given by: 
+In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis 
+parameter has been modified to account for the "metric" term. This altered 
+Coriolis parameter is thus discretised at $f$-points. It is given by: 
 \begin{multline} \label{Eq_dyncor_metric}
 f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\
@@ -236,10 +317,15 @@
 \end{multline} 
 
-Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) schemes can be used to compute the product of the Coriolis parameter and the vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has exclusively been used to date. This term is evaluated using a leapfrog scheme, i.e. the velocity is centred in time (\textit{now} velocity).
+Any of the (\ref{Eq_dynvor_ens}), (\ref{Eq_dynvor_ene}) and (\ref{Eq_dynvor_een}) 
+schemes can be used to compute the product of the Coriolis parameter and the 
+vorticity. However, the energy-conserving scheme  (\ref{Eq_dynvor_een}) has 
+exclusively been used to date. This term is evaluated using a leapfrog scheme, 
+$i.e.$ the velocity is centred in time (\textit{now} velocity).
 
 %--------------------------------------------------------------------------------------------------------------
 %           Flux form Advection term
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Flux form Advection term (\mdl{dynadv}) }
+\subsection   [Flux form Advection term (\textit{dynadv}) ]
+			{Flux form Advection term (\mdl{dynadv}) }
 \label{DYN_adv_flux}
 
@@ -261,10 +347,17 @@
 \end{equation}
 
-Two advection schemes ($2^{nd}$ order centered finite difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS). The latter is described in \citet{Sacha2005}. The schemes are selected using the namelist variable \textit{ln\_dynadv\_xxx} In flux form, the advection schemes differ by the choice of a space and time interpolation to define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, i.e. at the $T$-, $f$-, and $uw$-points and $f$-, $T$- and $vw$-points for $u$ and $v$, respectively. 
+Two advection schemes are available: a $2^{nd}$ order centered finite 
+difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 
+The latter is described in \citet{Sacha2005}. The schemes are selected using 
+the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. In flux 
+form, the schemes differ by the choice of a space and time interpolation to 
+define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, 
+$i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and 
+$vw$-points for $v$. 
 
 %-------------------------------------------------------------
 %                 2nd order centred scheme
 %-------------------------------------------------------------
-\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=T)}
+\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=.true.)}
 \label{DYN_adv_cen2}
 
@@ -274,5 +367,5 @@
 \left\{ 		\begin{aligned}
  u_T^{cen2} &=\overline u^{i }		\quad & 	
- u_F^{cen2} &=\overline u^{j+1/2}		\quad &
+  u_F^{cen2} &=\overline u^{j+1/2}		\quad &
  u_{uw}^{cen2} &=\overline u^{k+1/2}      \\
  v_F^{cen2} &=\overline v ^{i+1/2}		\quad &
@@ -282,10 +375,14 @@
 \end{equation} 
 
-The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive (i.e. it may create false extrema). It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to produce a sensible solution. The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $u$ and $v$ are the \emph{now} velocities.
+The scheme is non diffusive (i.e. conserves the kinetic energy) but dispersive 
+($i.e.$ it may create false extrema). It is therefore notoriously noisy and must 
+be used in conjunction with an explicit diffusion operator to produce a sensible 
+solution. The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $u$ and $v$ are the \emph{now} 
+velocities.
 
 %-------------------------------------------------------------
 %                 UBS scheme
 %-------------------------------------------------------------
-\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=T)}
+\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=.true.)}
 \label{DYN_adv_ubs}
 
@@ -299,19 +396,43 @@
 \end{cases}
 \end{equation}
-where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error \citep{Sacha2005}. The overall performance of the advection scheme is similar to that reported in \citet{Farrow1995}. It is a relatively good compromise between accuracy and smoothness. It is not a \emph{positive} scheme, meaning that false extrema are permitted but their amplitude is significantly reduced over the centred second order method. 
-
-The UBS scheme is not used in all the direction. In the vertical, it has been preferred to keep the centred $2^{nd}$ order evaluation of the advection, i.e. $u_{uw}^{ubs}$ and $u_{vw}^{ubs}$ in (\ref{Eq_dynadv_cen2}) are used. Indeed, UBS is diffusive, it is associated with vertical mixing of momentum. Since vertical mixing of momentum is source term of the TKE equation{\ldots}
-
-For stability reasons, in (\ref{Eq_dynadv_ubs}), the first term which corresponds to a second order centred scheme is evaluated using the \textit{now} velocity (centred in time) while the second term which is the diffusive part of the scheme, is 
-evaluated using the \textit{before} velocity (forward in time). This is discussed by \citet{Webb1998} in the context of the Quick advection scheme.
-
-NB 1: UBS and Quadratic Upstream Interpolation for Convective Kinematics (QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in (\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded. Nevertheless it is quite easy to make the substitution in \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
-
-NB 2 : In the current version of \mdl{dynadv\_ubs}, there is also a possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS. This is an error and should be suppressed soon.
+where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results 
+in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Sacha2005}. 
+The overall performance of the advection scheme is similar to that reported in 
+\citet{Farrow1995}. It is a relatively good compromise between accuracy and 
+smoothness. It is not a \emph{positive} scheme, meaning that false extrema are 
+permitted. But the amplitudes of the false extrema are significantly reduced over 
+those in the centred second order method. 
+
+The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
+order evaluation of the advection is preferred, $i.e.$ $u_{uw}^{ubs}$ and 
+$u_{vw}^{ubs}$ in \eqref{Eq_dynadv_cen2} are used. UBS is diffusive and is 
+associated with vertical mixing of momentum. \gmcomment{ gm  pursue the 
+sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
+
+For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds 
+to a second order centred scheme, is evaluated using the \textit{now} velocity 
+(centred in time), while the second term, which is the diffusive part of the scheme, 
+is evaluated using the \textit{before} velocity (forward in time). This is discussed 
+by \citet{Webb1998} in the context of the Quick advection scheme.
+
+Note that the UBS and Quadratic Upstream Interpolation for Convective Kinematics 
+(QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in 
+(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
+This option is not available through a namelist parameter, since the $1/6$ coefficient 
+is hard coded. Nevertheless it is quite easy to make the substitution in 
+\mdl{dynadv\_ubs} module and obtain a QUICK scheme.
+
+Note also that in the current version of \mdl{dynadv\_ubs}, there is also the 
+possibility of using a $4^{th}$ order evaluation of the advective velocity as in 
+ROMS. This is an error and should be suppressed soon.
+%%%
+\gmcomment{action :  this have to be done}
+%%%
 
 % ================================================================
 %           Hydrostatic pressure gradient term
 % ================================================================
-\section{Hydrostatic pressure gradient (\mdl{dynhpg})}
+\section  [Hydrostatic pressure gradient (\textit{dynhpg})]
+		{Hydrostatic pressure gradient (\mdl{dynhpg})}
 \label{DYN_hpg}
 %------------------------------------------nam_dynhpg---------------------------------------------------
@@ -319,17 +440,31 @@
 \namdisplay{namflg} 
 %-------------------------------------------------------------------------------------------------------------
-\colorbox{red} {Suppress the namflg namelist and incorporate it in the namhpg namelist}
-
-The key distinction between the different algorithms is the vertical coordinate used, since HPG is a horizontal pressure gradient, i.e. computed along geopotential surfaces. As a result, any tilt of the surface of the computational levels will require a specific treatment to compute the hydrostatic pressure gradient.
-
-The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, i.e. the density appearing in its expression is centred in time (\emph{now} velocity), or a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip boundary conditions are applied.
+%%%
+\gmcomment{Suppress the namflg namelist and incorporate it in the namhpg namelist}
+%%%
+
+The key distinction between the different algorithms used for the hydrostatic 
+pressure gradient is the vertical coordinate used, since HPG is a \emph{horizontal} 
+pressure gradient, $i.e.$ computed along geopotential surfaces. As a result, any 
+tilt of the surface of the computational levels will require a specific treatment to 
+compute the hydrostatic pressure gradient.
+
+The hydrostatic pressure gradient term is evaluated either using a leapfrog scheme, 
+$i.e.$ the density appearing in its expression is centred in time (\emph{now} rho), or 
+a semi-implcit scheme. At the lateral boundaries either free slip, no slip or partial slip 
+boundary conditions are applied.
 
 %--------------------------------------------------------------------------------------------------------------
 %           z-coordinate with full step
 %--------------------------------------------------------------------------------------------------------------
-\subsection{\textit{z}-coordinate with full step (\np{ln\_dynhpg\_zco}=T)}
+\subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
+			{$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=.true.)}
 \label{DYN_hpg_zco}
 
-The hydrostatic pressure can be obtained by integrating the hydrostatic equation vertically from the surface. Nevertheless, the pressure is quite large at great depth while its horizontal gradient is several orders of magnitude smaller. This may lead to large truncation error in the pressure gradient terms. Thus, the two horizontal components of the hydrostatic pressure gradient are computed directly as follows:
+The hydrostatic pressure can be obtained by integrating the hydrostatic equation 
+vertically from the surface. However, the pressure is large at great depth while its 
+horizontal gradient is several orders of magnitude smaller. This may lead to large 
+truncation errors in the pressure gradient terms. Thus, the two horizontal components 
+of the hydrostatic pressure gradient are computed directly as follows:
 
 for $k=km$ (surface layer, $jk=1$ in the code)
@@ -355,53 +490,89 @@
 \end{equation} 
 
-Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface level ($z=0)$. 
+Note that the $1/2$ factor in (\ref{Eq_dynhpg_zco_surf}) is adequate because of 
+the definition of $e_{3w}$ as the vertical derivative of the scale factor at the surface 
+level ($z=0)$. 
 
 %--------------------------------------------------------------------------------------------------------------
 %           z-coordinate with partial step
 %--------------------------------------------------------------------------------------------------------------
-\subsection{\textit{z}-coordinate with partial step (\np{ln\_dynhpg\_zps}=T )}
+\subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
+			{$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=.true.)}
 \label{DYN_hpg_zps}
 
-With partial bottom cells, tracers in horizontally adjacent cells generally live at different depths. Before taking horizontal gradients between these tracers, a linear interpolation is used to approximate the deeper tracer as if it actually lived at the depth of the shallower tracer point. 
-
-It is necessary to do so at the last ocean level; otherwise the horizontal hydrostatic pressure gradient evaluated in z-coordinate with partial step is exactly as in pure z-coordinate case. As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure effects in the equation of state is such that it is better to interpolate vertically temperature and salinity before computing the density. Horizontal gradients of and salinity are needed for the TRA modules, which is the reason why the horizontal gradients of density at the deepest model level are computed in module \mdl{zpsdhe} located in the TRA directory and described in \S\ref{TRA_zpshde}.
+With partial bottom cells, tracers in horizontally adjacent cells generally live at 
+different depths. Before taking horizontal gradients between these tracer points, 
+a linear interpolation is used to approximate the deeper tracer as if it actually lived 
+at the depth of the shallower tracer point. 
+
+Apart from this modification, the horizontal hydrostatic pressure gradient evaluated 
+in the $z$-coordinate with partial step is exactly as in the pure $z$-coordinate case. 
+As explained in detail in section \S\ref{TRA_zpshde}, the nonlinearity of pressure 
+effects in the equation of state is such that it is better to interpolate temperature and 
+salinity vertically before computing the density. Horizontal gradients of temperature 
+and salinity are needed for the TRA modules, which is the reason why the horizontal 
+gradients of density at the deepest model level are computed in module \mdl{zpsdhe} 
+located in the TRA directory and described in \S\ref{TRA_zpshde}.
 
 %--------------------------------------------------------------------------------------------------------------
 %           s- and s-z-coordinates
 %--------------------------------------------------------------------------------------------------------------
-\subsection{\textit{s}- and \textit{s-z} coordinates}
+\subsection{$s$- and $z$-$s$-coordinates}
 \label{DYN_hpg_sco}
 
-Pressure gradient formulations in $s$ coordinates have been the subject of a vast literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure gradient options are coded, but they are not yet fully documented nor tested. 
-
-Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}, \np{ln\_dynhpg\_hel})
+Pressure gradient formulations in $s$-coordinate have been the subject of a vast 
+literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure 
+gradient options are coded, but they are not yet fully documented or tested. 
+
+$\bullet$ Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}=.true., 
+\np{ln\_dynhpg\_hel}=.true.)
 \begin{equation} \label{Eq_dynhpg_sco}
 \left\{ \begin{aligned}
- - \frac{1}    						{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  p^h  \right] 
+ - \frac{1}    					{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  p^h  \right] 
 + \frac{g\; \overline {\rho}^{i+1/2}}	{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  z_T  \right]    \\
- - \frac{1}    						{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  p^h  \right]  
+ - \frac{1}    					{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  p^h  \right]  
 + \frac{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  z_T  \right]    \\
 \end{aligned} \right.
 \end{equation} 
 
-Where the first term is the pressure gradient along coordinates, computed as in (II.3.2), and $z_T$ is the depth of $T$-point evaluated from the sum of the vertical scale factor at W-point ($e_{3w})$. The version \np{ln\_dynhpg\_hel} has been added by Aike Beckmann and involves a redefinition of the relative position of $T$ points relative to $w$ points. 
-
-Weighted density Jacobian (wdj) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=T)
-
-Density Jacobian with cubic polynomial scheme (djc) \citep{Sacha2003} (\np{ln\_dynhpg\_djc}=T)
-
-Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=T)
-
-Note that expression (\ref{Eq_dynhpg_sco}) is used when the variable volume formulation is activated (key\_vvl) because in that case, even with a flat bottom, the coordinate surface are not horizontal but follow the free surface \citep{Levier2007}. The other pressure gradient options are not yet available.
+Where the first term is the pressure gradient along coordinates, computed as in 
+\eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of 
+the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 
+($e_{3w}$). The version \np{ln\_dynhpg\_hel}=.true. has been added by Aike 
+Beckmann and involves a redefinition of the relative position of $T$-points relative 
+to $w$-points. 
+
+$\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=.true.)
+
+$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Sacha2003} 
+(\np{ln\_dynhpg\_djc}=.true.)
+
+$\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.)
+
+Note that expression \eqref{Eq_dynhpg_sco} is used when the variable volume 
+formulation is activated (\key{vvl}) because in that case, even with a flat bottom, 
+the coordinate surfaces are not horizontal but follow the free surface 
+\citep{Levier2007}. The other pressure gradient options are not yet available.
 
 %--------------------------------------------------------------------------------------------------------------
 %           Time-scheme
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Time-scheme (\np{ln\_dynhpg\_imp}=T/F)}
+\subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
+			{Time-scheme (\np{ln\_dynhpg\_imp}=.true./.false.)}
 \label{DYN_hpg_imp}
 
-The default time differencing scheme used for the horizontal pressure gradient is a leapfrog scheme and therefore the density used in all discrete expressions given above is the  \textit{now} density, computed from the \textit{now} temperature and salinity. In some specific cases (usually high resolution simulations over a ocean domain including weakly stratified regions) the physical phenomena that control the time-step are internal gravity waves (IGWs). A semi-implicit scheme for doubling the stability limit associated with IGWs can be used \citep{Brown1978, Maltrud1998}. It consists in evaluating the hydrostatic pressure gradient as an average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ (i.e.  \textit{before},  \textit{now} and  \textit{after} time-steps), rather than at central time level $t$ only as in standard leapfrog scheme. 
-
-leapfrog scheme (\np{ln\_dynhpg\_imp}=F):
+The default time differencing scheme used for the horizontal pressure gradient is 
+a leapfrog scheme and therefore the density used in all discrete expressions given 
+above is the  \textit{now} density, computed from the \textit{now} temperature and 
+salinity. In some specific cases (usually high resolution simulations over an ocean 
+domain which includes weakly stratified regions) the physical phenomenum that 
+controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for 
+doubling the stability limit associated with IGWs can be used \citep{Brown1978, 
+Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an 
+average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ ($i.e.$  
+\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at central 
+time level $t$ only, as in the standard leapfrog scheme. 
+
+$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=.true.):
 
 \begin{equation} \label{Eq_dynhpg_lf}
@@ -410,5 +581,5 @@
 \end{equation}
 
-semi-implicit scheme (\np{ln\_dynhpg\_imp}=T):
+$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=.true.):
 \begin{equation} \label{Eq_dynhpg_imp}
 \frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
@@ -417,33 +588,71 @@
 \end{equation}
 
-The semi-implicit time scheme (\ref{Eq_dynhpg_imp}) is made possible without significant additional computation as the density can be updated to time level $t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can be easily shown that the Courant-Friedrichs-Lewy (CFL) limit associated to the hydrostatic pressure gradient double using (\ref{Eq_dynhpg_imp}) compared to the standard leapfrog scheme (\ref{Eq_dynhpg_lf}). Note that (\ref{Eq_dynhpg_imp}) is equivalent to applying a time filter to the pressure gradient to eliminate high frequency IGWs. Obviously, when using (\ref{Eq_dynhpg_imp}), the doubling of the time-step is achievable only if no other factors control the time-step, such as the CFL limits associated with advection or diffusion.
-
-In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=T.
- In this case, we choose to apply the time filter to temperature and salinity used in the equation of state, instead of applying it to the hydrostatic pressure or to the density, so that no additional storage array has to be defined. The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
+The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without 
+significant additional computation since the density can be updated to time level 
+$t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can 
+be easily shown that the stability limit associated with the hydrostatic pressure 
+gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the 
+standard leapfrog scheme \eqref{Eq_dynhpg_lf}. Note that \eqref{Eq_dynhpg_imp} 
+is equivalent to applying a time filter to the pressure gradient to eliminate high 
+frequency IGWs. Obviously, when using \eqref{Eq_dynhpg_imp}, the doubling of 
+the time-step is achievable only if no other factors control the time-step, such as 
+the stability limits associated with advection or diffusion.
+
+In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=.true.. 
+In this case, we choose to apply the time filter to temperature and salinity used in 
+the equation of state, instead of applying it to the hydrostatic pressure or to the 
+density, so that no additional storage array has to be defined. The density used to 
+compute the hydrostatic pressure gradient (whatever the formulation) is evaluated 
+as follows:
 \begin{equation} \label{Eq_rho_flt}
 	\rho^t = \rho( \widetilde{T},\widetilde {S},z_T)
  \quad	  \text{with}	\quad 
-	\widetilde{\bullet} = \frac{  \bullet^{t+\Delta t} +2 \,\bullet^t + \bullet^{t-\Delta t}  } {4}
+	\widetilde{\,\cdot\,} = \frac{  \,\cdot\,^{t+\Delta t} +2 \,\,\cdot\,^t + \,\cdot\,^{t-\Delta t}  } {4}
 \end{equation}
-
-Note that in the semi-implicit case, it is necessary to save one extra three-dimentional field in the restart file to restart the model with exact reproducibility, namely, the filtered density. This option is controlled by the namelist parameter \np{nn\_dynhpg\_rst}.
+\gmcomment{STEVEN: bullets look odd in this, could use X}
+
+Note that in the semi-implicit case, it is necessary to save the filtered density, an 
+extra three-dimensional field, in the restart file to restart the model with exact 
+reproducibility. This option is controlled by the namelist parameter 
+\np{nn\_dynhpg\_rst}=.true..
 
 % ================================================================
 % Surface Pressure Gradient
 % ================================================================
-\section{Surface pressure gradient (\mdl{dynspg})}
+\section  [Surface pressure gradient (\textit{dynspg}) ]
+		{Surface pressure gradient (\mdl{dynspg})}
 \label{DYN_hpg_spg}
 %-----------------------------------------nam_dynspg----------------------------------------------------
 \namdisplay{nam_dynspg} 
 %------------------------------------------------------------------------------------------------------------
-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 or rigid lid) and the variable volume case (nonlinear free surface, \key{vvl} is active). In the linear free surface case (\S\ref{PE_free_surface}) and rigid lid (\S\ref{PE_rigid_lid}), the vertical scale factors $e_{3}$ are fixed in time, while in the nonlinear case (\S\ref{PE_free_surface}) they are time-dependent. 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 form of the surface pressure gradient term is dependent on the representation 
+of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed 
+volume case (linear free surface or rigid lid) and the variable volume case 
+(nonlinear free surface, \key{vvl} is defined). In the linear free surface case 
+(\S\ref{PE_free_surface}) and the rigid lid case (\S\ref{PE_rigid_lid}), the vertical 
+scale factors $e_{3}$ are fixed in time, whilst in the nonlinear case 
+(\S\ref{PE_free_surface}) they are time-dependent. 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 method, which involves a modification of the continuous 
+equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface method 
+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}.
 
 %--------------------------------------------------------------------------------------------------------------
 % Linear free surface formulation
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Linear free surface formulation (\textbf{key\_exp} or \textbf{\_ts} or \textbf{\_flt})}
+\subsection{Linear free surface formulation (\key{exp} or \textbf{\_ts} or \textbf{\_flt})}
 \label{DYN_spg_linear}
 
-In the linear free surface formulation, the sea surface height is assumed to be small compared to the thickness of the ocean levels, so that $(a)$ the time evolution of the sea surface height becomes a linear equation, and $(b)$ the thickness of the ocean levels is assumed to be constant with time. As mentioned in (\S\ref{PE_free_surface}) the linearization affects the conservation of tracers. 
+In the linear free surface formulation, the sea surface height is assumed to be 
+small compared to the thickness of the ocean levels, so that $(a)$ the time 
+evolution of the sea surface height becomes a linear equation, and $(b)$ the 
+thickness of the ocean levels is assumed to be constant with time. 
+As mentioned in (\S\ref{PE_free_surface}) the linearization affects the 
+conservation of tracers. 
 
 %-------------------------------------------------------------
@@ -453,5 +662,7 @@
 \label{DYN_spg_exp}
 
-In the explicit free surface formulation, the model time step is chosen small enough to describe the external gravity waves (typically a few ten seconds). The sea surface height is given by :
+In the explicit free surface formulation, the model time step is chosen to be 
+small enough to describe the external gravity waves (typically a few tens of 
+seconds). The sea surface height is given by :
 \begin{equation} \label{Eq_dynspg_ssh}
 \frac{\partial \eta }{\partial t}\equiv \frac{\text{EMP}}{\rho _w }+\frac{1}{e_{1T} 
@@ -460,7 +671,14 @@
 \end{equation}
 
-where EMP is the surface freshwater budget (evaporation minus precipitation, and minus river runoffs (if the later are introduced as a surface freshwater flux, see \S\ref{SBC}) expressed in $Kg.m^{-2}.s^{-1}$, and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter, i.e. the velocity appearing in (\ref{Eq_dynspg_ssh}) is centred in time (\textit{now} velocity). 
-
-The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by :
+where EMP is the surface freshwater budget, expressed in $Kg.m^{-2}.s^{-1}$, 
+and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water. If river 
+runoff is expressed as a surface freshwater flux, see \S\ref{SBC}, then EMP 
+can be written as the evaporation minus precipitation, minus the river runoff. 
+The sea-surface height is evaluated using a leapfrog scheme in combination 
+with an Asselin time filter, $i.e.$ the velocity appearing in 
+\eqref{Eq_dynspg_ssh} is centred in time (\textit{now} velocity). 
+
+The surface pressure gradient, also evaluated using a leap-frog scheme, is 
+then simply given by :
 \begin{equation} \label{Eq_dynspg_exp}
 \left\{ \begin{aligned}
@@ -471,5 +689,6 @@
 \end{equation} 
 
-Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in (\ref{Eq_dynspg_exp}). 
+Consistent with the linearization, a factor of $\left. \rho \right|_{k=1} / \rho _o$ 
+is omitted in \eqref{Eq_dynspg_exp}. 
 
 %-------------------------------------------------------------
@@ -481,5 +700,10 @@
 \namdisplay{namdom} 
 %--------------------------------------------------------------------------------------------------------------
-The split-explicit free surface formulation used in OPA follows the one proposed by \citet{Griffies2004}. The general idea is to solve the free surface equation with a small time step, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} (Figure III.3). 
+
+The split-explicit free surface formulation used in \NEMO follows the one 
+proposed by \citet{Griffies2004}. The general idea is to solve the free surface 
+equation with a small time step \np{rdtbt}, while the three dimensional 
+prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt} 
+(Fig.\ref {Fig_DYN_dynspg_ts}). 
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -487,10 +711,29 @@
 \begin{center}
 \includegraphics[width=0.90\textwidth]{./Figures/Fig_DYN_dynspg_ts.pdf}
-\caption{Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes, after \citet{Griffies2004}. Time increases to the right. Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. The curved line represents a leap-frog time step, and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. The vertically integrated forcing \textbf{M}(t) computed at baroclinic time step t represents the interaction between the barotropic and baroclinic motions. While keeping the total depth, tracer, and freshwater forcing fields fixed, a leap-frog integration carries the surface height and vertically integrated velocity from t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. Time averaging the barotropic fields over the N+1 time steps (endpoints included) centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$. A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of the time averaged vertically integrated velocity taken from baroclinic time step t. }
+\caption{Schematic of the split-explicit time stepping scheme for the barotropic 
+and baroclinic modes, after \citet{Griffies2004}. Time increases to the right. 
+Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
+The curved line represents a leap-frog time step, and the smaller barotropic time 
+steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line. The vertically 
+integrated forcing \textbf{M}(t) computed at the baroclinic time step $t$ 
+represents the interaction between the barotropic and baroclinic motions. 
+While keeping the total depth, tracer, and freshwater forcing fields fixed, a 
+leap-frog integration carries the surface height and vertically integrated velocity 
+from $t$ to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$. 
+Time averaging the barotropic fields over the N+1 time steps (endpoints 
+included) centers the vertically integrated velocity at the baroclinic timestep 
+$t+\Delta t$. A baroclinic leap-frog time step carries the surface height to 
+$t+\Delta t$ using the convergence of the time averaged vertically integrated 
+velocity taken from baroclinic time step t. }
+%%%
+\gmcomment{STEVEN: what does convergence mean in this context?}
+%%%
 \end{center}
 \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 
-The split-explicit formulation has a damping effect on external gravity waves, which is weaker than the filtered free surface but still significant as shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
+The split-explicit formulation has a damping effect on external gravity waves, 
+which is weaker damping than for the filtered free surface but still significant as 
+shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
 
 %-------------------------------------------------------------
@@ -500,8 +743,11 @@
 \label{DYN_spg_flt}
 
-The filtered formulation follows the \citet{Roullet2000} implementation. The extra term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic solvers available in the code are 
-documented in \S\ref{MISC}. The amplitude of the extra term is given by the namelist variable \np{rnu}. The default value is 1, as recommended by \citet{Roullet2000}
-
-\colorbox{red}{\np{rnu}=1 to be suppressed from namelist !}
+The filtered formulation follows the \citet{Roullet2000} implementation. The extra 
+term introduced in the equations (see {\S}I.2.2) is solved implicitly. The elliptic 
+solvers available in the code are documented in \S\ref{MISC}. The amplitude of 
+the extra term is given by the namelist variable \np{rnu}. The default value is 1, 
+as recommended by \citet{Roullet2000}
+
+\gmcomment{\np{rnu}=1 to be suppressed from namelist !}
 
 %-------------------------------------------------------------
@@ -511,5 +757,11 @@
 \label{DYN_spg_vvl}
 
-In the non-linear free surface formulation, the variations of volume are fully taken into account. This option is presented in a report \citep{Levier2007} available on the NEMO web site. The three time-stepping methods (explicit, split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except that the ocean depth is now time-dependent. In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
+In the non-linear free surface formulation, the variations of volume are fully 
+taken into account. This option is presented in a report \citep{Levier2007} 
+available on the \NEMO web site. The three time-stepping methods (explicit, 
+split-explicit and filtered) are the same as in \S\ref{DYN_spg_linear} except 
+that the ocean depth is now time-dependent. In particular, this means that 
+in the filtered case, the matrix to be inverted has to be recomputed at each 
+time-step.
 
 %--------------------------------------------------------------------------------------------------------------
@@ -531,6 +783,9 @@
 \end{equation}
 
-Here ${\rm {\bf M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, 
-viscous and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn} and the overbar indicates a vertical average over the whole water column (i.e. from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid). The time derivative of $\psi$ is the solution of an elliptic equation:
+Here ${\rm {\bf M}}= \left( M_u,M_v \right)$ represents the collected 
+contributions of nonlinear, viscous and hydrostatic pressure gradient terms in 
+\eqref{Eq_PE_dyn} and the overbar indicates a vertical average over the 
+whole water column (i.e. from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid). 
+The time derivative of $\psi$ is the solution of an elliptic equation:
 \begin{multline} \label{Eq_bsf}
    \delta_{i+1/2} \left[ \frac{e_{2v}}{H_v\;e_{1v}} \delta_{i} \left[  \partial_t \psi \right] \right]
@@ -541,12 +796,12 @@
 \end{multline}
 
-The elliptic solvers available in the code are documented in 
-\S\ref{MISC}). The boundary conditions must be given on all separate 
-landmasses (islands), which is done by integrating the vorticity equation 
-around each island. This requires identifying each island in the bathymetry 
-file, a cumbersome task. This explains why the rigid lid option is not 
-recommended with complex domains such as the global ocean. Parameters jpisl 
-(number of islands) and jpnisl (maximum number of points per island) of the 
-par\_oce.h90 file are related to this option.
+The elliptic solvers available in the code are documented in \S\ref{MISC}). 
+The boundary conditions must be given on all separate landmasses (islands), 
+which is done by integrating the vorticity equation around each island. This 
+requires identifying each island in the bathymetry file, a cumbersome task. 
+This explains why the rigid lid option is not recommended with complex 
+domains such as the global ocean. Parameters jpisl (number of islands) and 
+jpnisl (maximum number of points per island) of the \hf{par\_oce} file are 
+related to this option.
 
 
@@ -554,23 +809,26 @@
 % Lateral diffusion term
 % ================================================================
-\section{Lateral diffusion term (\mdl{dynldf})}
+\section  [Lateral diffusion term (\textit{dynldf})]
+		{Lateral diffusion term (\mdl{dynldf})}
 \label{DYN_ldf}
 %------------------------------------------nam_dynldf----------------------------------------------------
 \namdisplay{nam_dynldf} 
 %-------------------------------------------------------------------------------------------------------------
-The options available for lateral diffusion are laplacian (rotated or not) 
-or biharmonic operators. The coefficients may be constant or spatially 
-variable; the description of the coefficients is found in the chapter on lateral
-physics (Chap.~\ref{LDF}). The lateral diffusion of momentum is evaluated using a 
-forward scheme, i.e. the velocity appearing in its expression is the 
-\textit{before} velocity in time, except for the pure vertical component that appears when 
-a tensor of rotation is used. This latter term is solved implicitly together 
-with the vertical diffusion term (see \S\ref{DOM_nxt}) 
+
+The options available for lateral diffusion are for the choice of  laplacian 
+(rotated or not) or biharmonic operators. The coefficients may be constant 
+or spatially variable; the description of the coefficients is found in the chapter 
+on lateralphysics (Chap.\ref{LDF}). The lateral diffusion of momentum is 
+evaluated using a forward scheme, i.e. the velocity appearing in its expression 
+is the \textit{before} velocity in time, except for the pure vertical component 
+that appears when a tensor of rotation is used. This latter term is solved 
+implicitly together with the vertical diffusion term (see \S\ref{DOM_nxt}) 
 
 At the lateral boundaries either free slip, no slip or partial slip boundary 
-conditions are applied following user's choice (see Chap. \S\ref{LBC}).
-
-% ================================================================
-\subsection{Iso-level laplacian operator (\np{ln\_dynldf\_lap}=T)}
+conditions are applied according to the user's choice (see Chap.\ref{LBC}).
+
+% ================================================================
+\subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
+			{Iso-level laplacian operator (\np{ln\_dynldf\_lap}=.true.)}
 \label{DYN_ldf_lap}
 
@@ -590,22 +848,23 @@
 As explained in \S\ref{PE_ldf}, this formulation (as the gradient of a divergence 
 and curl of the vorticity) preserves symmetry and ensures a complete 
-separation between the vorticity and divergence parts. Note that in pure 
-z-coordinate (\key{zco} defined), $e_{3u} =e_{3v} =e_{3f}$ so that they disappear 
-from the rotational part of \eqref{Eq_dynldf_lap}.
+separation between the vorticity and divergence parts. Note that in the full step 
+$z$-coordinate (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they 
+cancel in the rotational part of \eqref{Eq_dynldf_lap}.
 
 %--------------------------------------------------------------------------------------------------------------
 %           Rotated laplacian operator
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Rotated laplacian operator (\np{ln\_dynldf\_iso}=T)}
+\subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
+			{Rotated laplacian operator (\np{ln\_dynldf\_iso}=.true.)}
 \label{DYN_ldf_iso}
 
-A rotation of lateral momentum diffusive operator is needed for isoneutral 
-diffusion in $z$-coordinates (\np{ln\_dynldf\_iso}=T) and for either 
-isoneutral (\np{ln\_dynldf\_iso}=T) or geopotential 
-(\np{ln\_dynldf\_hor}=T) diffusion in $s$-coordinates. In the partial step 
+A rotation of the lateral momentum diffusive operator is needed in several cases: 
+for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=.true.) and for 
+either iso-neutral (\np{ln\_dynldf\_iso}=.true.) or geopotential 
+(\np{ln\_dynldf\_hor}=.true.) diffusion in $s$-coordinate. In the partial step 
 case, coordinates are horizontal excepted at the deepest level and no 
-rotation is performed when \np{ln\_dynldf\_hor}=T. The diffusive operator 
+rotation is performed when \np{ln\_dynldf\_hor}=.true.. The diffusive operator 
 is defined simply as the divergence of down gradient momentum fluxes on each 
-momentum components. It must be emphasized that this formulation ignores 
+momentum component. It must be emphasized that this formulation ignores 
 constraints on the stress tensor such as symmetry. The resulting discrete 
 representation is:
@@ -657,27 +916,30 @@
  \end{split}
 \end{equation}
-where $r_1$ and $r_2$ are the slopes 
-between the surface along which the diffusive operator acts and the surface 
-of computation ($z$- or $s$-surfaces). The way these slopes are evaluated is given 
-in lateral physics chapter (Chap.~\S\ref{LDF}).
+where $r_1$ and $r_2$ are the slopes between the surface along which the 
+diffusive operator acts and the surface of computation ($z$- or $s$-surfaces). 
+The way these slopes are evaluated is given in the lateral physics chapter 
+(Chap.\ref{LDF}).
 
 %--------------------------------------------------------------------------------------------------------------
 %           Iso-level bilaplacian operator
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Iso-level bilaplacian operator}
+\subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
+			{Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=.true.)}
 \label{DYN_ldf_bilap}
 
 The lateral fourth order operator formulation on momentum is obtained by 
-applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on boundary 
-conditions: the first derivative term normal to the coast depends on the 
-free or no-slip lateral boundary conditions chosen, while the third 
-derivative terms normal to the coast are set to zero (see Chap.~\ref{LBC}).
-
-\colorbox{yellow}{add a remark on the the change in the position of the coefficient}
+applying \eqref{Eq_dynldf_lap} twice. It requires an additional assumption on 
+boundary conditions: the first derivative term normal to the coast depends on 
+the free or no-slip lateral boundary conditions chosen, while the third 
+derivative terms normal to the coast are set to zero (see Chap.\ref{LBC}).
+%%%
+\gmcomment{add a remark on the the change in the position of the coefficient}
+%%%
 
 % ================================================================
 %           Vertical diffusion term
 % ================================================================
-\section{Vertical diffusion term (\mdl{dynzdf})}
+\section  [Vertical diffusion term (\mdl{dynzdf})]
+		{Vertical diffusion term (\mdl{dynzdf})}
 \label{DYN_zdf}
 %----------------------------------------------namzdf------------------------------------------------------
@@ -685,9 +947,15 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The large vertical diffusion coefficient found in the surface mixed layer, together with high vertical resolution, implies a too restrictive constraint on the time step in a pure explicit time stepping 
-case. Two time stepping schemes can be used for the vertical diffusion term : $(a)$ a forward time differencing scheme (\np{ln\_zdfexp}=T) using a time splitting technique (\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme (\np{ln\_zdfexp}=F) (see \S\ref{DOM_nxt}). Note that namelist variables \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 
+The large vertical diffusion coefficient found in the surface mixed layer together 
+with high vertical resolution implies that in the case of explicit time stepping there 
+would be too restrictive a constraint on the time step. Two time stepping schemes 
+can be used for the vertical diffusion term : $(a)$ a forward time differencing 
+scheme (\np{ln\_zdfexp}=.true.) using a time splitting technique 
+(\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 
+(\np{ln\_zdfexp}=.false.) (see \S\ref{DOM_nxt}). Note that namelist variables 
+\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 
 
 The formulation of the vertical subgrid scale physics is the same whatever 
-the vertical coordinate is. The vertical diffusive operators given by 
+the vertical coordinate is. The vertical diffusion operators given by 
 \eqref{Eq_PE_zdf} take the following semi-discrete space form:
 \begin{equation} \label{Eq_dynzdf}
@@ -700,6 +968,6 @@
 \end{aligned}   \right.
 \end{equation} 
-where $A_{uw}^{vm} $ and$A_{vw}^{vm} $ are the vertical eddy viscosity and 
-diffusivity coefficients. The way these coefficients can be evaluated 
+where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and 
+diffusivity coefficients. The way these coefficients are evaluated 
 depends on the vertical physics used (see \S\ref{ZDF}).
 
@@ -711,14 +979,14 @@
 	 = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
 \end{equation}
-where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress vector in the 
-(\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients in the 
-surface mixed layer ensure that the surface wind stress is distributed in 
+where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress 
+vector in the (\textbf{i},\textbf{j}) coordinate system. The high mixing coefficients 
+in the surface mixed layer ensure that the surface wind stress is distributed in 
 the vertical over the mixed layer depth. If the vertical mixing coefficient 
 is small (when no mixed layer scheme is used) the surface stress enters only 
 the top model level, as a body force. The surface wind stress is calculated 
-in the surface module routines (SBC, see Chap.~\S\ref{SBC})
-
-The turbulent flux of momentum at the bottom is specified through a bottom 
-friction parameterization (see \S\ref{ZDF_bfr})
+in the surface module routines (SBC, see Chap.\ref{SBC})
+
+The turbulent flux of momentum at the bottom of the ocean is specified through 
+a bottom friction parameterization (see \S\ref{ZDF_bfr})
 
 % ================================================================
@@ -732,12 +1000,15 @@
 enter the dynamical equations. 
 
-One is the effect of atmospheric pressure on the ocean dynamics (to be introduced later). 
-
-Another forcing term is the tidal potential, which will be introduced in the reference version soon. 
+One is the effect of atmospheric pressure on the ocean dynamics (to be 
+introduced later). 
+
+Another forcing term is the tidal potential, which will be introduced in the 
+reference version soon. 
 
 % ================================================================
 % Time evolution term 
 % ================================================================
-\section{Time evolution term (\mdl{dynnxt})}
+\section  [Time evolution term (\textit{dynnxt})]
+		{Time evolution term (\mdl{dynnxt})}
 \label{DYN_nxt}
 
@@ -746,6 +1017,7 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The general framework of dynamics time stepping is a leap-frog scheme, i.e. 
-a three level centred time scheme associated with a Asselin time filter (cf. \S\ref{DOM_nxt})
+The general framework for dynamics time stepping is a leap-frog scheme, 
+$i.e.$ a three level centred time scheme associated with an Asselin time filter 
+(cf. \S\ref{DOM_nxt})
 \begin{equation} \label{Eq_dynnxt}
 \begin{split}
@@ -755,11 +1027,11 @@
 \end{split}
 \end{equation} 
-where RHS is the right hand side of the momentum equation, the subscript $f$ denotes 
-filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 
-initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp}=0.1.
-
-Note that whith the filtered free surface, the update of the next velocities 
-is done in the \mdl{dynsp\_flt} module, and nly the swap of arrays 
-and Asselin filter is done in \mdl{dynnxt.}
+where RHS is the right hand side of the momentum equation, the subscript $f$ 
+denotes filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is 
+initialized as \np{atfp} (namelist parameter). Its default value is \np{atfp} = 0.1.
+
+Note that whith the filtered free surface, the update of the \textit{next} velocities 
+is done in the \mdl{dynsp\_flt} module, and only the swap of arrays 
+and Asselin filtering is done in \mdl{dynnxt.}
 
 % ================================================================
@@ -770,18 +1042,17 @@
 
 %--------------------------------------------------------------------------------------------------------------
-%           z-coordinate with partial step
-%--------------------------------------------------------------------------------------------------------------
-\subsection{horizontal divergence and relative vorticity (\mdl{divcur})}
+%           Horizontal divergence and relative vorticity
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Horizontal divergence and relative vorticity (\textit{divcur})]
+			{Horizontal divergence and relative vorticity (\mdl{divcur})}
 \label{DYN_divcur}
 
-The vorticity is defined at F-point (i.e. corner point) as follows:
-
+The vorticity is defined at an $f$-point ($i.e.$ corner point) as follows:
 \begin{equation} \label{Eq_divcur_cur}
 \zeta =\frac{1}{e_{1f}\,e_{2f} }\left( {\;\delta _{i+1/2} \left[ {e_{2v}\;v} \right]
-									     -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
-\end{equation} 
-
-The horizontal divergence is defined at T-point. It is given by:
-
+						        -\delta _{j+1/2} \left[ {e_{1u}\;u} \right]\;} \right)
+\end{equation} 
+
+The horizontal divergence is defined at a $T$-point. It is given by:
 \begin{equation} \label{Eq_divcur_div}
 \chi =\frac{1}{e_{1T}\,e_{2T}\,e_{3T} }
@@ -790,15 +1061,24 @@
 \end{equation} 
 
-Note that in z-coordinate with full step (\key{zco} defined), $e_{3u} =e_{3v} =e_{3f}$ so that they disappear from \eqref{Eq_divcur_div}.
-
-Note also that whereas the vorticity have the same discrete expression in $z$- and $s$-coordinate, its physical meaning is not identical. $\zeta$ is a pseudo vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along geopotential surfaces, but are no more necessary defined at the same depth).
-
-The vorticity and divergence at the \textit{before} step are used in the computation of the horizontal diffusion of momentum. Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities, the \textit{before} vorticity and divergence arrays must be included 
-in the restart file to ensure perfect restartability. The vorticity and divergence at the \textit{now} time step are used respectively for the nonlinear advection and the computation of the vertical velocity. 
+Note that in the $z$-coordinate with full step (\key{zco} is defined), $e_{3u} =e_{3v} =e_{3f}$ so that they cancel in \eqref{Eq_divcur_div}.
+
+Note also that whereas the vorticity have the same discrete expression in $z$- 
+and $s$-coordinate, its physical meaning is not identical. $\zeta$ is a pseudo 
+vorticity along $s$-surfaces (only pseudo because $(u,v)$ are still defined along 
+geopotential surfaces, but are no more necessary defined at the same depth).
+
+The vorticity and divergence at the \textit{before} step are used in the computation 
+of the horizontal diffusion of momentum. Note that because they have been 
+calculated prior to the Asselin filtering of the \textit{before} velocities, the 
+\textit{before} vorticity and divergence arrays must be included in the restart file 
+to ensure perfect restartability. The vorticity and divergence at the \textit{now} 
+time step are used for the computation of the nonlinear advection and of the 
+vertical velocity respectively. 
 
 %--------------------------------------------------------------------------------------------------------------
 %           Vertical Velocity
 %--------------------------------------------------------------------------------------------------------------
-\subsection{Vertical velocity (\mdl{wzvmod})}
+\subsection   [Vertical velocity (\textit{wzvmod})]
+			{Vertical velocity (\mdl{wzvmod})}
 \label{DYN_wzv}
 
@@ -814,18 +1094,18 @@
 \end{equation} 
 
-With a free surface, the top vertical velocity is non-zero, due the 
-freshwater forcing and the variations of the free surface elevation. When 
-the free surface is linear or with a rigid lid, the upper boundary condition 
-applies at a fixed level $z=0$. Note that in the rigid-lid case (\key{dynspg\_rl} defined), the surface 
-boundary condition ($\left. w \right|_\text{surface} =0)$ is automatically 
-achieved at least at the computer accuracy, due to the discrete expression 
-of the surface pressure gradient (\colorbox{yellow}{Appendix C}).
+With a free surface, the top vertical velocity is non-zero, due to the 
+freshwater forcing and the variations of the free surface elevation. With a 
+linear free surface or with a rigid lid, the upper boundary condition 
+applies at a fixed level $z=0$. Note that in the rigid-lid case (\key{dynspg\_rl} 
+is defined), the surface boundary condition ($\left. w \right|_\text{surface}=0)$ is 
+automatically achieved at least at computer accuracy, due to the the way the 
+surface pressure gradient is expressed in discrete form (Appendix~\ref{Apdx_C}).
 
 Note also that whereas the vertical velocity has the same discrete 
-expression in $z$- and $s$-coordinate, its physical meaning is not the same: in 
-the second case, $w$ is the velocity normal to the $s$-surfaces. 
+expression in $z$- and $s$-coordinate, its physical meaning is not the same: 
+in the second case, $w$ is the velocity normal to the $s$-surfaces. 
 
 With the variable volume option, the calculation of the vertical velocity is 
-modified (see \citet{Levier2007}, report available on the NEMO web site). 
-
-% ================================================================
+modified (see \citet{Levier2007}, report available on the \NEMO web site). 
+
+% ================================================================
Index: trunk/DOC/BETA/Chapters/Chap_LBC.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_LBC.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_LBC.tex	(revision 817)
@@ -29,16 +29,15 @@
 }{e_{1u} }\delta _{i+1 / 2} \left[ T \right]\;\;mask_u 
 \end{equation}
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t] \label{Fig_LBC_uv}  \begin{center}
-\includegraphics[width=0.90\textwidth]{./Figures/Fig_LBC_uv.pdf}
-\caption {Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
-\end{center}   \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
 where mask$_{u}$ is the mask array at $u$-point, ensures that the heat flux is 
 zero inside land and at the boundaries as mask$_{u}$ is zero at solid 
 boundaries defined at $u$-points in this case (normal velocity $u$ remains zero at 
 the coast) (Fig.~\ref{Fig_LBC_uv}). 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_LBC_uv}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./Figures/Fig_LBC_uv.pdf}
+\caption {Lateral boundary (thick line) at T-level. The velocity normal to the boundary is set to zero.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 On momentum the situation is a bit more complex as two boundary conditions must be provided along the coast (one on the normal velocity and the other on the tangential velocity). The boundary of the ocean in C-grid is defined by the velocity-faces. For example, at a given $T$-level, the lateral boundary (coastline or intersection with the bottom topography) is made of segments 
@@ -138,7 +137,8 @@
 
 % ====================================================================
-% Sub-Domain:Exchange with neighbouring processors 
+% Exchange with neighbouring processors 
 % ====================================================================
-\section{Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
+\section  [Exchange with neighbouring processors (\textit{lbclnk}, \textit{lib\_mpp})]
+		{Exchange with neighbouring processors (\mdl{lbclnk}, \mdl{lib\_mpp})}
 \label{LBC_mpp}
 
@@ -185,5 +185,5 @@
 
 
-	The OPA model computes equation terms with the help of mask arrays (0 onto land points and 1 onto sea points). It is easily readable and very efficient in the context of the vectorial architecture. But in the case of scalar processor, computations over the land regions becomes more expensive in term of CPU time. It is all the more so when we use a complex configuration with a realistic bathymetry like the global ocean where more than 50 \% of points are land points. For this reason, a pre-processing tool allows to search in the mpp domain decomposition strategy if a splitting can be found with a maximum number of only land points processors which could be eliminated: the mpp\_optimiz tools (available from the DRAKKAR web site). This optimisation is made with the knowledge of the specific bathymetry. The user chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with $jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ land processors. When those parameters are specified in module \mdl{par\_oce}, the algorithm in the \rou{inimpp2} routine set each processor name and neighbour parameters (nbound, nono, noea,...) so that the land processors are not taken into account. 
+	The OPA model computes equation terms with the help of mask arrays (0 onto land points and 1 onto sea points). It is easily readable and very efficient in the context of the vectorial architecture. Nevertheless, in the case of scalar processor, computations over the land regions becomes more expensive in term of CPU time. It is all the more so when we use a complex configuration with a realistic bathymetry like the global ocean where more than 50 \% of points are land points. For this reason, a pre-processing tool allows to search in the mpp domain decomposition strategy if a splitting can be found with a maximum number of only land points processors which could be eliminated: the mpp\_optimiz tools (available from the DRAKKAR web site). This optimisation is made with the knowledge of the specific bathymetry. The user chooses optimal parameters \jp{jpni}, \jp{jpnj} and \jp{jpnij} with $jpnij < jpni \times jpnj$, leading to the elimination of $jpni \times jpnj - jpnij$ land processors. When those parameters are specified in module \mdl{par\_oce}, the algorithm in the \rou{inimpp2} routine set each processor name and neighbour parameters (nbound, nono, noea,...) so that the land processors are not taken into account. 
 
 \colorbox{yellow}{Note that the inimpp2 routine is general so that the original inimpp routine should be suppressed from the code.}
Index: trunk/DOC/BETA/Chapters/Chap_LDF.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_LDF.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_LDF.tex	(revision 817)
@@ -22,5 +22,5 @@
 % Lateral Mixing Coefficients
 % ================================================================
-\section{Lateral Mixing Coefficient (key\_ldftra\_c.d and key\_ldfdyn\_c.d)}
+\section{Lateral Mixing Coefficient (\textbf{key\_ldftra\_c.d} - \textbf{key\_ldfdyn\_c.d)} }
 \label{LDF_coef}
 
@@ -61,7 +61,9 @@
 \quad \text{comments}
 \end{equation}
-where $e_{max}$ is the max of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameters. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}. %gm  not only that! stability reasons: with non uniform grid size, it is common to face a blow up of the model due to to large diffusive coefficient compare to the smallest grid size... this is especially true for bilaplacian (to be added in the text!)  
-
-Other formulations can be introduced by the user for a given configuration. For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian viscous operator uses \np{ahm0}~=~$4.10^4 m^2.s^{-1}$ poleward of 20$^{\circ}$ north and south and decreases to \np{aht0}~=~$2.10^3 m^2.s^{-1}$ at the equator \citep{Madec1996, Delecluse2000}. This specification can be found in \rou{ldf\_dyn\_c2d\_orca} routine defined in \mdl{ldfdyn\_c2d}. Similar specific horizontal variation can be found for Antarctic or Arctic sub-domain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
+where $e_{max}$ is the max of $e_1$ and $e_2$ taken over the whole masked ocean domain, and $A_o^l$ is \np{ahm0} (momentum) or \np{aht0} (tracer) namelist parameters. This variation is intended to reflect the lesser need for subgrid scale eddy mixing where the grid size is smaller in the domain. It was introduced in the context of the DYNAMO modelling project \citep{Willebrand2001}. 
+%%%
+\gmcomment { not only that! stability reasons: with non uniform grid size, it is common to face a blow up of the model due to to large diffusive coefficient compare to the smallest grid size... this is especially true for bilaplacian (to be added in the text!)  }
+
+Other formulations can be introduced by the user for a given configuration. For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian viscous operator uses \np{ahm0}~=~$4.10^4 m^2.s^{-1}$ poleward of 20$^{\circ}$ north and south and decreases to \np{aht0}~=~$2.10^3 m^2.s^{-1}$ at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This specification can be found in \rou{ldf\_dyn\_c2d\_orca} routine defined in \mdl{ldfdyn\_c2d}. Similar specific horizontal variation can be found for Antarctic or Arctic sub-domain of ORCA2 and ORCA05 (\key{antarctic} or \key{arctic} defined, see \hf{ldfdyn\_antarctic} and \hf{ldfdyn\_arctic}).
 
 \subsubsection{Space Varying Mixing Coefficients (\key{ldftra\_c3d} and \key{ldfdyn\_c3d})}
@@ -97,8 +99,10 @@
 % Direction of lateral Mixing
 % ================================================================
-\section{Direction of Lateral Mixing (\mdl{ldfslp})}
+\section  [Direction of Lateral Mixing (\textit{ldfslp})]
+		{Direction of Lateral Mixing (\mdl{ldfslp})}
 \label{LDF_slp}
 
-%gm%  we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme.
+%%%
+\gmcomment{  we should emphasize here that the implementation is a rather old one. Better work can be achieved by using \citet{Griffies1998, Griffies2004} iso-neutral scheme. }
 
 
@@ -270,7 +274,7 @@
 % Eddy Induced Mixing
 % ================================================================
-\section{Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
+\section  [Eddy Induced Velocity (\textit{traadv\_eiv}, \textit{ldfeiv})]
+		{Eddy Induced Velocity (\mdl{traadv\_eiv}, \mdl{ldfeiv})}
 \label{LDF_eiv}
-
 
 When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), an eddy induced tracer advection term is added, the formulation of which depends on the slopes of iso-neutral surfaces. Contrary to iso-neutral mixing, the slopes use here are referenced to the geopotential surfaces, i.e. \eqref{Eq_ldfslp_geo} is used in $z$-coordinates, and the sum \eqref{Eq_ldfslp_geo}  + \eqref{Eq_ldfslp_iso} in $s$-coordinates. The eddy induced velocity is given by:
Index: trunk/DOC/BETA/Chapters/Chap_MISC.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_MISC.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_MISC.tex	(revision 817)
@@ -195,5 +195,5 @@
 % Elliptic solvers (SOL)
 % ================================================================
-\section{Elliptic solvers (SOL)}
+\section{Elliptic solvers (SOL directory)}
 \label{MISC_sol}
 %--------------------------------------------namdom-------------------------------------------------------
@@ -452,5 +452,5 @@
 %       Boundary Conditions --- Islands
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Boundary Conditions --- Islands (key{\_}islands defined)}
+\subsection{Boundary Conditions --- Islands (\key{islands})}
 \label{MISC_solisl}
 
Index: trunk/DOC/BETA/Chapters/Chap_Model_Basics.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_Model_Basics.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_Model_Basics.tex	(revision 817)
@@ -22,5 +22,5 @@
 
 
-The ocean is a fluid that can be described to a good approximation by the primitive equations, i.e. the Navier-Stokes equations along with a nonlinear equation of state which couples the two active tracers (temperature and salinity) to the fluid velocity, plus the following additional assumptions made from scale considerations:
+The ocean is a fluid that can be described to a good approximation by the primitive equations, $i.e.$ the Navier-Stokes equations along with a nonlinear equation of state which couples the two active tracers (temperature and salinity) to the fluid velocity, plus the following additional assumptions made from scale considerations:
 
 \textit{(1) spherical earth approximation: }the geopotential surfaces are assumed to be spheres so that gravity (local vertical) is parallel to the earth's radius
@@ -32,10 +32,10 @@
 \textit{(4) Boussinesq hypothesis:} density variations are neglected except in their contribution to the buoyancy force
 
-\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a balance between the vertical pressure gradient and buoyancy force (this removes convective processes from 
-the initial Navier-Stokes equations: they must be parameterized)
+\textit{(5) Hydrostatic hypothesis: }the vertical momentum equation is reduced to a balance between the vertical pressure gradient and the buoyancy force (this removes convective processes from 
+the initial Navier-Stokes equations and so convective processes must be parameterized instead)
 
 \textit{(6) Incompressibility hypothesis: }the three dimensional divergence of the velocity vector is assumed to be zero.
 
-Because the gravitational force is so dominant in the equations of large-scale motions, it is quite useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, i.e. tangent to the geopotential surfaces. Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ (the subscript $h$ denotes the local horizontal vector, i.e. over the (\textbf{i},\textbf{j}) plan), $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system provides the following six equations (namely the momentum balance, the hydrostatic equilibrium, the incompressibility, the heat and salt conservation and an equation of state):
+Because the gravitational force is so dominant in the equations of large-scale motions, it is useful to choose an orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ tangent to the geopotential surfaces. Let us define the following variables: \textbf{U} the vector velocity, $\textbf{U}=\textbf{U}_h + w\, \textbf{k}$ (the subscript $h$ denotes the local horizontal vector, $i.e.$ over the (\textbf{i},\textbf{j}) plane), $T$ the potential temperature, $S$ the salinity, \textit{$\rho $} the \textit{in situ} density. The vector invariant form of the primitive equations in the (\textbf{i},\textbf{j},\textbf{k}) vector system provides the following six equations (namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation, the heat and salt conservation equations and an equation of state):
 \begin{subequations} \label{Eq_PE}
   \begin{equation}     \label{Eq_PE_dyn}
@@ -44,5 +44,5 @@
             +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h
  -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h 
--\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}}
+-\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
   \end{equation}
   \begin{equation}     \label{Eq_PE_hydrostatic}
@@ -53,8 +53,8 @@
   \end{equation}
 \begin{equation} \label{Eq_PE_tra_T}
-\frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T
+\frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T
   \end{equation}
   \begin{equation}     \label{Eq_PE_tra_S}
-\frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S
+\frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S
   \end{equation}
   \begin{equation}     \label{Eq_PE_eos}
@@ -62,5 +62,6 @@
   \end{equation}
 \end{subequations}
-where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ the time, $z$ the vertical coordinate, $\rho $ the \textit{in situ} density given by the equation of state (\ref{Eq_PE_eos}), $\rho_o$ a reference density, $p$ the pressure, $f=2 \bf \Omega \cdot \bf k$ the Coriolis acceleration (where $\bf \Omega$ is the Earth angular velocity vector), and $g$ the gravitational acceleration. ${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterizations of small-scale physics for momentum, temperature and salinity, including surface forcing terms. Their nature and formulation are discussed in \S\ref{PE_zdf_ldf}, page~\pageref{PE_zdf_ldf}.
+where $\nabla$ is the generalised derivative vector operator in $(\bf i,\bf j, \bf k)$ directions, $t$ is the time, $z$ is the vertical coordinate, $\rho $ is the \textit{in situ} density given by the equation of state (\ref{Eq_PE_eos}), $\rho_o$ is a reference density, $p$ the pressure, $f=2 \bf \Omega \cdot \bf k$ is the Coriolis acceleration (where $\bf \Omega$ is the Earth's angular velocity vector), and $g$ is the gravitational acceleration. 
+${\rm {\bf D}}^{\rm {\bf U}}$, $D^T$ and $D^S$ are the parameterizations of small-scale physics for momentum, temperature and salinity, and ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$ surface forcing terms. Their nature and formulation are discussed in \S\ref{PE_zdf_ldf} and page \S\ref{PE_boundary_condition}.
 
 .
@@ -72,5 +73,5 @@
 \label{PE_boundary_condition}
 
-An ocean is bounded by complex coastlines and bottom topography at its base and by an air-sea or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ the height of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, the continental surfaces, the sea ice and the atmosphere. However, some of these fluxes are so weak that even on climatic time scales of thousands of years they can be neglected. In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and the other components of the earth system.
+An ocean is bounded by complex coastlines, bottom topography at its base and an air-sea or ice-sea interface at its top. These boundaries can be defined by two surfaces, $z=-H(i,j)$ and $z=\eta(i,j,k,t)$, where $H$ is the depth of the ocean bottom and $\eta$ is the height of the sea surface. Both $H$ and $\eta$ are usually referenced to a given surface, $z=0$, chosen as a mean sea surface (Fig.~\ref{Fig_ocean_bc}). Through these two boundaries, the ocean can exchange fluxes of heat, fresh water, salt, and momentum with the solid earth, the continental margins, the sea ice and the atmosphere. However, some of these fluxes are so weak that even on climatic time scales of thousands of years they can be neglected. In the following, we briefly review the fluxes exchanged at the interfaces between the ocean and the other components of the earth system.
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -81,11 +82,12 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
+
 \begin{description}
-\item[Land - ocean interface:] the major flux between continental surfaces and the ocean is a mass exchange of fresh water through river runoff. Such an exchange modifies locally the sea surface salinity especially in the vicinity of major river mouths. It can be neglected for short range integrations but has to be taken into account for long term integrations as it influences the characteristics of water masses formed (especially at high latitudes). It is required to close the water cycle of the climatic system. It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths.
-\item[Solid earth - ocean interface:] heat and salt fluxes across the sea floor are negligibly small, except in special areas of little extent. They are always neglected in the model. The boundary condition is thus set to no flux of heat and salt across solid boundaries. For momentum, the situation is different. There is no flow across solid boundaries, i.e. the velocity normal to the ocean bottom and coastlines is zero (in other words, the bottom velocity is parallel to solid boundaries). This kinematic boundary condition can be expressed as:
+\item[Land - ocean interface:] the major flux between continental margins and the ocean is a mass exchange of fresh water through river runoff. Such an exchange modifies the sea surface salinity especially in the vicinity of major river mouths. It can be neglected for short range integrations but has to be taken into account for long term integrations as it influences the characteristics of water masses formed (especially at high latitudes). It is required in order to close the water cycle of the climate system. It is usually specified as a fresh water flux at the air-sea interface in the vicinity of river mouths.
+\item[Solid earth - ocean interface:] heat and salt fluxes through the sea floor are small, except in special areas of little extent. They are usually neglected in the model \footnote{In fact, it has been shown that the heat flux associated with the solid Earth cooling ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean (see \ref{TRA_bbc}).}. The boundary condition is thus set to no flux of heat and salt across solid boundaries. For momentum, the situation is different. There is no flow across solid boundaries, $i.e.$ the velocity normal to the ocean bottom and coastlines is zero (in other words, the bottom velocity is parallel to solid boundaries). This kinematic boundary condition can be expressed as:
 \begin{equation} \label{Eq_PE_w_bbc}
 w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
 \end{equation}
-In addition, the ocean exchanges momentum with the earth through friction processes. Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized in terms of turbulent fluxes through bottom and/or lateral boundary conditions. Its specification depends on the nature of the physical parameterization used for ${\rm {\bf D}}^{\rm {\bf U}}$ in \eqref{Eq_PE_dyn}. They are discussed in  in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.
+In addition, the ocean exchanges momentum with the earth through frictional processes. Such momentum transfer occurs at small scales in a boundary layer. It must be parameterized in terms of turbulent fluxes using bottom and/or lateral boundary conditions. Its specification depends on the nature of the physical parameterization used for ${\rm {\bf D}}^{\rm {\bf U}}$ in \eqref{Eq_PE_dyn}. It is discussed in \S\ref{PE_zdf}, page~\pageref{PE_zdf}.% and Chap. III.6 to 9.
 \item[Atmosphere - ocean interface:] the kinematic surface condition plus the mass flux of fresh water PE  (the precipitation minus evaporation budget) leads to: 
 \begin{equation} \label{Eq_PE_w_sbc}
@@ -95,5 +97,5 @@
 \end{equation}
 The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system) leads to the continuity of pressure across the interface $z=\eta$. The atmosphere and ocean also exchange horizontal momentum (wind stress), and heat.
-\item[Sea ice - ocean interface:] the two media exchange heat, salt, fresh water and momentum. The sea surface temperature is constrained to be at the freezing point at the interface. Sea ice salinity is very low ($\sim5 \,psu$) compared to those of the ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected.
+\item[Sea ice - ocean interface:] the ocean and sea ice exchange heat, salt, fresh water and momentum. The sea surface temperature is constrained to be at the freezing point at the interface. Sea ice salinity is very low ($\sim4-6 \,psu$) compared to those of the ocean ($\sim34 \,psu$). The cycle of freezing/melting is associated with fresh water and salt fluxes that cannot be neglected.
 \end{description}
 
@@ -111,10 +113,10 @@
 \label{PE_p_formulation}
 
-The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a reference geopotential surface ($z=0$) and an hydrostatic pressure $p_h$ such that: $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). The hydrostatic pressure is then given by:
+The total pressure at a given depth $z$ is composed of a surface pressure $p_s$ at a reference geopotential surface ($z=0$) and a hydrostatic pressure $p_h$ such that: $p(i,j,k,t)=p_s(i,j,t)+p_h(i,j,k,t)$. The latter is computed by integrating (\ref{Eq_PE_hydrostatic}), assuming that pressure in decibars can be approximated by depth in meters in (\ref{Eq_PE_eos}). The hydrostatic pressure is then given by:
 \begin{equation} \label{Eq_PE_pressure}
 p_h \left( {i,j,z,t} \right)
  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,z} \right)\;d\varsigma } 
 \end{equation}
-The surface pressure requires a more specific treatment. Two strategies can be considered: $(a)$ the introduction of a new variable $\eta$, the free-surface elevation, for which a prognostic equation can be established and solved; $(b)$ the assumption that the ocean surface is a rigid lid, on which the pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, a solution of the free-surface elevation consists in the excitation of external gravity waves. The flow is barotropic and the surface moves up and down with gravity as the restoring force. The phase speed of such waves is high (some hundreds of metres per second) so that the time step would have to be very short if they were present in the model. The latter strategy filters these waves as the rigid lid approximation implies $\eta=0$, i.e. the sea surface is the surface $z=0$. This well known approximation increases the surface wave speed to infinity and modifies certain other longwave dynamics (e.g. barotropic Rossby or planetary waves). In the present release of OPA, both strategies are still available. They are further described in the next two sub-sections.
+ Two strategies can be considered for the surface pressure term: $(a)$ introduce of a new variable $\eta$, the free-surface elevation, for which a prognostic equation can be established and solved; $(b)$ assume that the ocean surface is a rigid lid, on which the pressure (or its horizontal gradient) can be diagnosed. When the former strategy is used, one solution of the free-surface elevation consists of the excitation of external gravity waves. The flow is barotropic and the surface moves up and down with gravity as the restoring force. The phase speed of such waves is high (some hundreds of metres per second) so that the time step would have to be very short if they were present in the model. The latter strategy filters out these waves since the rigid lid approximation implies $\eta=0$, $i.e.$ the sea surface is the surface $z=0$. This well known approximation increases the surface wave speed to infinity and modifies certain other longwave dynamics ($e.g.$ barotropic Rossby or planetary waves). In the present release of \NEMO, both strategies are still available. They are further described in the next two sub-sections.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -124,5 +126,5 @@
 \label{PE_free_surface}
 
-In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced which describes the shape of the air-sea interface. This variable is solution of a prognostic equation which is established by forming the vertical average of the kinematics surface condition (\ref{Eq_PE_w_bbc}):
+In the free surface formulation, a variable $\eta$, the sea-surface height, is introduced which describes the shape of the air-sea interface. This variable is solution of a prognostic equation which is established by forming the vertical average of the kinematic surface condition (\ref{Eq_PE_w_bbc}):
 \begin{equation} \label{Eq_PE_ssh}
 \frac{\partial \eta }{\partial t}=-D+P-E
@@ -138,5 +140,5 @@
 $\bullet$ If one is interested in EGWs, in particular the tides and their interaction 
 with the baroclinic structure of the ocean (internal waves) possibly in 
-shallow seas, then a non linear free surface is the most adequate: this 
+shallow seas, then a non linear free surface is the most appropriate. This 
 means that no approximation is made in (\ref{Eq_PE_ssh}) and that the variation of the 
 ocean volume is fully taken into account. Note that in order to study the 
@@ -147,15 +149,16 @@
 
 $\bullet$ If one is not interested in EGW but rather sees them as high frequency 
-noise, it is possible to apply a filter to slow down the fastest waves while 
-not altering the slow barotropic Rossby waves. In that case it is also 
-generally sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which allows 
+noise, it is possible to apply an explicit filter to slow down the fastest waves while 
+not altering the slow barotropic Rossby waves. If further, an approximative conservation 
+of heat and salt contents is sufficient for the problem solved, then it is 
+sufficient to solve a linearized version of (\ref{Eq_PE_ssh}), which still allows 
 to take into account freshwater fluxes applied at the ocean surface \citep{Roullet2000}.
 
 $\bullet$ For process studies not involving external waves nor surface freshwater 
 fluxes, it is possible to use the rigid lid approximation see (next 
-section). The ocean surface is considered as a fixed surface, so that all 
+section). The ocean surface is then considered as a fixed surface, so that all 
 external waves are removed from the system. 
 
-The filtering of EGWs in models with a free surface is usually a matter of discretisation of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} or the implicit scheme \citep{Dukowicz1994}. In OPA, we use a slightly different approach developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 
+The filtering of EGWs in models with a free surface is usually a matter of discretisation of the temporal derivatives, using the time splitting method \citep{Killworth1991, Zhang1992} or the implicit scheme \citep{Dukowicz1994}. In \NEMO, we use a slightly different approach developed by \citet{Roullet2000}: the damping of EGWs is ensured by introducing an additional force in the momentum equation. \eqref{Eq_PE_dyn} becomes: 
 \begin{equation} \label{Eq_PE_flt}
 \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
@@ -163,14 +166,14 @@
 - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 
 \end{equation}
-where $T_c$, is a parameter homogeneous to a time which characterizes the force, $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and   represents the collected contributions of the Coriolis, hydrostatic pressure gradient, non-linear and viscous terms in \eqref{Eq_PE_dyn}.
-
-The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagates, $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which has to be computed implicitly. This is not surprising since the use of a large time step has a necessarily numerical cost. Two gains arise in comparison with the previous formulations. Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as soon as $T_c > \Delta t$.
-
-When the variations of free surface elevation are small compared to the thickness of the model layers, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the conservation of salt in the model. With the nonlinear free surface equation, the time evolution of the total salt content is 
+where $T_c$, is a parameter with dimensions of time which characterizes the force, $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density, and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, non-linear and viscous terms in \eqref{Eq_PE_dyn}.
+
+The new force can be interpreted as a diffusion of vertically integrated volume flux divergence. The time evolution of $D$ is thus governed by a balance of two terms, $-g$ \textbf{A} $\eta$ and $g \, T_c \,$ \textbf{A} $D$, associated with a propagative regime and a diffusive regime in the temporal spectrum, respectively. In the diffusive regime, the EGWs no longer propagate, $i.e.$ they are stationary and damped. The diffusion regime applies to the modes shorter than $T_c$. For longer ones, the diffusion term vanishes. Hence, the temporally unresolved EGWs can be damped by choosing $T_c > \Delta t$. \citet{Roullet2000} demonstrate that (\ref{Eq_PE_flt}) can be integrated with a leap frog scheme except the additional term which has to be computed implicitly. This is not surprising since the use of a large time step has a necessarily numerical cost. Two gains arise in comparison with the previous formulations. Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as soon as $T_c > \Delta t$.
+
+When the variations of free surface elevation are small compared to the thickness of the first model layer, the free surface equation (\ref{Eq_PE_ssh}) can be linearized. As emphasized by \citet{Roullet2000} the linearization of (\ref{Eq_PE_ssh}) has consequences on the conservation of salt in the model. With the nonlinear free surface equation, the time evolution of the total salt content is 
 \begin{equation} \label{Eq_PE_salt_content}
 \frac{\partial }{\partial t}\int\limits_{D\eta } {S\;dv} =\int\limits_S 
 {S\;(-\frac{\partial \eta }{\partial t}-D+P-E)\;ds}
 \end{equation}
-where $S$ is the salinity, and the total salt is integrated in the whole ocean volume $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) is satisfied, so that the salt is perfectly conserved. When the free surface equation is linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed volume $D$ (bounded by the surface $z=0$) is no longer conserved:
+where $S$ is the salinity, and the total salt is integrated over the whole ocean volume $D_\eta$ bounded by the time-dependent free surface. The right hand side (which is an integral over the free surface) vanishes when the nonlinear equation (\ref{Eq_PE_ssh}) is satisfied, so that the salt is perfectly conserved. When the free surface equation is linearized, \citet{Roullet2000} show that the total salt content integrated in the fixed volume $D$ (bounded by the surface $z=0$) is no longer conserved:
 \begin{equation} \label{Eq_PE_salt_content_linear}
 \frac{\partial }{\partial t}\int\limits_D {S\;dv} =-\int\limits_S 
@@ -186,5 +189,5 @@
 
 Although the total salt content is not exactly conserved with the linearized free surface, its variations are driven by correlations of the time variation of surface salinity with the sea surface height, which is a negligible term. This situation contrasts with 
-the case of the rigid lid approximation (following section) in which case freshwater forcing is represented by a virtual salt flux, leading to spurious sources or sinks of salt \citep{Roullet2000}.
+the case of the rigid lid approximation (following section) in which case freshwater forcing is represented by a virtual salt flux, leading to a spurious source of salt at the ocean surface \citep{Roullet2000}.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -194,5 +197,5 @@
 \label{PE_rigid_lid}
 
-With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface is equal to zero. From the continuity equation (\ref{Eq_PE_continuity}) and the kinematic condition at the bottom (\ref{Eq_PE_w_bbc}) (no flux across the bottom), it can be shown that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the overbar indicates a vertical average over the whole water column, i.e. from $z=-H$ , the ocean bottom, to $z=0$ , the rigid-lid). Thus,  can be derived from a volume transport streamfunction $\psi$:
+With the rigid lid approximation, we assume that the ocean surface ($z=0$) is a rigid lid on which a pressure $p_s$ is exerted. This implies that the vertical velocity at the surface is equal to zero. From the continuity equation \eqref{Eq_PE_continuity} and the kinematic condition at the bottom \eqref{Eq_PE_w_bbc} (no flux across the bottom), it can be shown that the vertically integrated flow $H{\rm {\bf \bar {U}}}_h$ is non-divergent (where the overbar indicates a vertical average over the whole water column, i.e. from $z=-H$, the ocean bottom, to $z=0$ , the rigid-lid). Thus, $\rm {\bf \bar {U}}_h$ can be derived from a volume transport streamfunction $\psi$:
 \begin{equation} \label{Eq_PE_u_psi}
 \overline{\vect{U}}_h =\frac{1}{H}\left(   \vect{k} \times \nabla \psi   \right)
@@ -215,8 +218,7 @@
 \end{equation}
 
-Using the proper boundary conditions, (\ref{Eq_PE_psi}) can be solved to find $\partial_t \psi$ and thus using (\ref{Eq_PE_u_barotrope}) the horizontal surface pressure gradient. It should be noted that $p_s$ can be computed by taking the divergence of (\ref{Eq_PE_u_barotrope}) and solving the resulting elliptic equation. Thus the surface pressure is a diagnostic quantity that can be recovered for analysis purposes.
-
-A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. The boundary condition on velocity is that there is no flow normal to a solid wall, i.e. the coastlines are streamlines. Therefore (I.2.7 is solved with the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each coastline of the same continent or of the same island. When all the coastlines are connected (there are no islands), the constant value of $\partial_t \psi$ along the coast can be arbitrarily chosen to be zero. When islands are present in the domain, the value of the barotropic streamfunction will generally be different for each island and for the continent, and will vary with respect to time. So the boundary condition is: $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), where $Q$ is the number of islands present in the domain and $\mu_n$ is a time dependent variable. A time evolution equation of the unknown $\mu_n$ can be found by evaluating the circulation of the time derivative of the vertical average (barotropic) velocity field along a closed contour around each island. Since the circulation of a 
-gradient field along a closed contour is zero, from (\ref{Eq_PE_u_barotrope}) we have:
+Using the proper boundary conditions, \eqref{Eq_PE_psi} can be solved to find $\partial_t \psi$ and thus using \eqref{Eq_PE_u_barotrope} the horizontal surface pressure gradient. It should be noted that $p_s$ can be computed by taking the divergence of \eqref{Eq_PE_u_barotrope} and solving the resulting elliptic equation. Thus the surface pressure is a diagnostic quantity that can be recovered for analysis purposes.
+
+A difficulty lies in the determination of the boundary condition on $\partial_t \psi$. The boundary condition on velocity is that there is no flow normal to a solid wall, $i.e.$ the coastlines are streamlines. Therefore \eqref{Eq_PE_psi} is solved with the following Dirichlet boundary condition: $\partial_t \psi$ is constant along each coastline of the same continent or of the same island. When all the coastlines are connected (there are no islands), the constant value of $\partial_t \psi$ along the coast can be arbitrarily chosen to be zero. When islands are present in the domain, the value of the barotropic streamfunction will generally be different for each island and for the continent, and will vary with respect to time. So the boundary condition is: $\psi=0$ along the continent and $\psi=\mu_n$ along island $n$ ($1 \leq n \leq Q$), where $Q$ is the number of islands present in the domain and $\mu_n$ is a time dependent variable. A time evolution equation of the unknown $\mu_n$ can be found by evaluating the circulation of the time derivative of the vertical average (barotropic) velocity field along a closed contour around each island. Since the circulation of a gradient field along a closed contour is zero, from \eqref{Eq_PE_u_barotrope} we have:
 \begin{equation} \label{Eq_PE_isl_circulation}
 \oint_n {\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( 
@@ -226,12 +228,17 @@
 \end{equation}
 
-Since (\ref{Eq_PE_psi}) is linear, its solution \textit{$\psi $} can be decomposed as follows:
+Since (\ref{Eq_PE_psi}) is linear, its solution \textit{$\psi $} can be decomposed 
+as follows:
 \begin{equation} \label{Eq_PE_psi_isl}
 \psi =\psi _o +\sum\limits_{n=1}^{n=Q} {\mu _n \psi _n } 
 \end{equation}
-where $\psi _o$ is the solution of (\ref{Eq_PE_psi}) with $\psi _o=0$ long all the coastlines, and where $\psi _n$ is the solution of (\ref{Eq_PE_psi}) with the right-hand side equal to $0$, and with $\psi _n =1$ long the island $n$, $\psi _n =0$ along the other boundaries. The function $\psi _n$ is thus independent of time. Introducing (\ref{Eq_PE_psi_isl}) into (\ref{Eq_PE_isl_circulation}) yields:
+where $\psi _o$ is the solution of \eqref{Eq_PE_psi} with $\psi _o=0$ long all 
+the coastlines, and where $\psi _n$ is the solution of \eqref{Eq_PE_psi} with 
+the right-hand side equal to $0$, and with $\psi _n =1$ long the island $n$, 
+$\psi _n =0$ along the other boundaries. The function $\psi _n$ is thus 
+independent of time. Introducing \eqref{Eq_PE_psi_isl} into 
+\eqref{Eq_PE_isl_circulation} yields:
 \begin{multline} \label{Eq_PE_psi_isl_circulation}
-\left[ {\oint_n {\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \psi _m } 
-\right]\cdot {\rm {\bf d}}\ell } } \right]_{1\leq m\leqslant Q \atop 1\leq n\leqslant Q  }
+\left[ {\oint_n {\frac{1}{H}  \left[ {{\rm {\bf k}}\times \nabla \psi _m } \right]\cdot {\rm {\bf d}}\ell } } \right]_{1\leq m\leqslant Q \atop 1\leq n\leqslant Q  }
  \left( {\frac{\partial \mu _n }{\partial t}} 
 \right)_{1\leqslant n\leqslant Q}        \\
@@ -246,5 +253,5 @@
 \right)_{1\leqslant n\leqslant Q} ={\rm {\bf B}}
 \end{equation}
-where \textbf{A} is a $Q  \times Q$ matrix and \textbf{B} is a time dependent vector. As \textbf{A} is independent of time, it can be calculated and inverted once. The time derivative of the  streamfunction when islands are present is thus given by:
+where \textbf{A} is a $Q  \times Q$ matrix and \textbf{B} is a time dependent vector. As \textbf{A} is independent of time, it can be calculated and inverted once. The time derivative of the streamfunction when islands are present is thus given by:
 \begin{equation} \label{Eq_PE_psi_isl_dt}
 \frac{\partial \psi }{\partial t}=\frac{\partial \psi _o }{\partial 
@@ -261,5 +268,4 @@
 
 
-
 % -------------------------------------------------------------------------------------------------------------
 % Tensorial Formalism
@@ -268,8 +274,7 @@
 \label{PE_tensorial}
 
-In many ocean circulation problems, the flow field has regions of enhanced dynamics (i.e. surface layers, western boundary currents, equatorial currents, or ocean fronts). The representation of such dynamical processes can be improved by specifically increasing the model resolution in these regions. As well, it may be convenient to use a lateral boundary-following coordinate system to better represent coastal dynamics. Moreover, the common geographical coordinate system has a singular point at the North Pole that cannot be easily treated in a global model without filtering. A solution consists in introducing an appropriate coordinate transformation that shifts the singular point on land \citep{MadecImb1996, Murray1996}. As a conclusion, it is important to solve the primitive equations in various curvilinear coordinate systems. An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. This formalism is suited to any multidimensional curvilinear coordinate system. Ocean modellers mainly use three-dimensional orthogonal grids on the sphere, with conservation of the local vertical. Here we give the simplified equations for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics.
-
-Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, i.e. along geopotential surfaces (\ref{Fig_referential}). Let $(\lambda,\varphi,z)$ be the geographical coordinates system in which a position is defined by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of the 
-earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea level (\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, the three scale factors:
+In many ocean circulation problems, the flow field has regions of enhanced dynamics ($i.e.$ surface layers, western boundary currents, equatorial currents, or ocean fronts). The representation of such dynamical processes can be improved by specifically increasing the model resolution in these regions. As well, it may be convenient to use a lateral boundary-following coordinate system to better represent coastal dynamics. Moreover, the common geographical coordinate system has a singular point at the North Pole that cannot be easily treated in a global model without filtering. A solution consists of introducing an appropriate coordinate transformation that shifts the singular point onto land \citep{MadecImb1996, Murray1996}. As a consequence, it is important to solve the primitive equations in various curvilinear coordinate systems. An efficient way of introducing an appropriate coordinate transform can be found when using a tensorial formalism. This formalism is suited to any multidimensional curvilinear coordinate system. Ocean modellers mainly use three-dimensional orthogonal grids on the sphere (spherical earth approximation), with preservation of the local vertical. Here we give the simplified equations for this particular case. The general case is detailed by \citet{Eiseman1980} in their survey of the conservation laws of fluid dynamics.
+
+Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere associated with the positively oriented orthogonal set of unit vectors (\textbf{i},\textbf{j},\textbf{k}) linked to the earth such that \textbf{k} is the local upward vector and (\textbf{i},\textbf{j}) are two vectors orthogonal to \textbf{k}, $i.e.$ along geopotential surfaces (Fig.\ref{Fig_referential}). Let $(\lambda,\varphi,z)$ be the geographical coordinate system in which a position is defined by the latitude $\varphi(i,j)$, the longitude $\lambda(i,j)$ and the distance from the centre of the earth $a+z(k)$ where $a$ is the earth's radius and $z$ the altitude above a reference sea level (Fig.\ref{Fig_referential}). The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$, the three scale factors:
 \begin{equation} \label{Eq_scale_factors}
 \begin{aligned}
@@ -325,6 +330,5 @@
 \end{equation}
 \end{subequations}
-where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in 
-the $(i,j,k)$ coordinate system.
+where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -334,5 +338,5 @@
 \label{PE_zco_Eq}
 
-In order to express the primitive equations in tensorial formalism, it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
+In order to express the Primitive Equations in tensorial formalism, it is necessary to compute the horizontal component of the non-linear and viscous terms of the equation using \eqref{Eq_PE_grad}) to \eqref{Eq_PE_lap_vector}. Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
 \begin{equation} \label{Eq_PE_curl_Uh}
 \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 
@@ -346,5 +350,5 @@
 \end{equation}
 
-Using the fact that horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that $e_3$  is a function of the single variable $k$, the nonlinear term of \eqref{Eq_PE_dyn} can be transformed as follows:
+Using the fact that the horizontal scale factors $e_1$ and $e_2$ are independent of $k$ and that $e_3$  is a function of the single variable $k$, the nonlinear term of \eqref{Eq_PE_dyn} can be transformed as follows:
 \begin{flalign*}
 &\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
@@ -352,5 +356,5 @@
 \end{flalign*}
 \begin{flalign*}
-&=\left( {{\begin{array}{*{20}c}
+&\qquad=\left( {{\begin{array}{*{20}c}
  {\left[ 	{	 \frac{1}{e_3}	\frac{\partial u	}{\partial k}
 			-\frac{1}{e_1}	\frac{\partial w	}{\partial i} } \right] w - \zeta \; v } 	 	\\
@@ -364,5 +368,5 @@
 \end{flalign*}
 \begin{flalign*}
-&=\left( {{	\begin{array}{*{20}c}
+& \qquad =\left( {{	\begin{array}{*{20}c}
  {-\zeta \; v} \hfill \\
  { \zeta \; u} \hfill \\
@@ -394,15 +398,15 @@
 \end{equation}
 
-This is the so-called \textit{vector invariant form} of the momentum advection. For some purposes, it can be advantageous to write this term in the so-called flux form, i.e. to write it as the divergence of fluxes. For example, the first component of (\ref{Eq_PE_vector_form}) (the $i$-component) is transformed as follows:
+This is the so-called \textit{vector invariant form} of the momentum advection term. For some purposes, it can be advantageous to write this term in the so-called flux form, $i.e.$ to write it as the divergence of fluxes. For example, the first component of \eqref{Eq_PE_vector_form} (the $i$-component) is transformed as follows:
 \begin{flalign*}
 &{	\begin{array}{*{20}l}
 \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
-          +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i    \\
-\\
+          +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\
+%\\
      = - \zeta \;v 
      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
      + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} 			\\
 \\
-=\frac{1}{e_1 \; e_2} \left( 	-v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
+\qquad =\frac{1}{e_1 \; e_2} \left( 	-v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
 							+v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}	 \right)
 +\frac{1}{e_1 e_2 }\left( 	+e_2 \; u\frac{\partial u}{\partial i}
@@ -413,5 +417,5 @@
 \begin{flalign*}
 &{	\begin{array}{*{20}l}
-=\frac{1}{e_1 \; e_2} 	\left\{ 
+\qquad =\frac{1}{e_1 \; e_2} 	\left\{ 
  -\left(   	    v^2 	\frac{\partial e_2                                }{\partial i} 
  		+e_2 \,v		\frac{\partial v                                   }{\partial i}  	\right)
@@ -424,5 +428,5 @@
  						\right\} 
 +\frac{1}{e_3} \left(
- 					\frac{\partial \left( {w\,v} \right)         }{\partial k}
+ 					\frac{\partial \left( {w\,u} \right)         }{\partial k}
 		 -u			\frac{\partial w 						   }{\partial k} 	\right) \\
 \end{array} } 		&
@@ -430,8 +434,8 @@
 \begin{flalign*}
 &{	\begin{array}{*{20}l}
-=\frac{1}{e_1 \; e_2} 	\left( 
+\qquad =\frac{1}{e_1 \; e_2} 	\left( 
 					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
 		+			\frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}	\right)
-+\frac{1}{e_3 }		\frac{\partial \left( {w\,v		 } \right)}{\partial k}
++\frac{1}{e_3 }		\frac{\partial \left( {w\,u		 } \right)}{\partial k}
 \\  \qquad \qquad \quad
 +\frac{1}{e_1 e_2 }		\left( 
@@ -446,6 +450,6 @@
 \begin{flalign*}
 &{	\begin{array}{*{20}l}
-= \nabla \cdot \left( {{\rm {\bf U}}\,u}	\right)
--  \nabla \cdot {\rm {\bf U}} \ u
+\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u}	\right)
+-   \left( \nabla \cdot {\rm {\bf U}} \right) \ u
 +\frac{1}{e_1 e_2 }\left( 
 		-v^2		\frac{\partial e_2 }{\partial i}
@@ -456,5 +460,5 @@
 \begin{flalign*}
 &{	\begin{array}{*{20}l}
-= \nabla \cdot \left( 	{{\rm {\bf U}}\,u}		\right)
+\qquad = \nabla \cdot \left( 	{{\rm {\bf U}}\,u}		\right)
 +	\frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i}
 						       -u \; \frac{\partial e_1}{\partial j} 	\right)  \left( -v \right)	
@@ -462,5 +466,5 @@
 \end{flalign*}
 
-The flux form of the momentum advection is therefore given by:
+The flux form of the momentum advection term is therefore given by:
 \begin{multline} \label{Eq_PE_flux_form}
 		\left[ 
@@ -468,9 +472,9 @@
 +\frac{1}{2}	\nabla \left( 	{{\rm {\bf U}}^2} 	\right)
 		\right]_h 
-=	\nabla \cdot 	\left( {{\begin{array}{*{20}c}	{\rm {\bf U}} \, u 	\hfill \\
+\\
+= \nabla \cdot 	\left( {{\begin{array}{*{20}c}	{\rm {\bf U}} \, u 	\hfill \\
 												{\rm {\bf U}} \, v	\hfill \\
 						\end{array} }} 	
 				\right)
-\\
 +\frac{1}{e_1 e_2 }		\left( 
 		 v\frac{\partial e_2}{\partial i}
@@ -479,5 +483,5 @@
 \end{multline}
 
-The flux form has two terms, the first one is expressed as the divergence of momentum fluxes (so the flux form name given to this formulation) and the second one is due to the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: 
+The flux form has two terms, the first one is expressed as the divergence of momentum fluxes (hence the flux form name given to this formulation) and the second one is due to the curvilinear nature of the coordinate system used. The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: 
 \begin{equation} \label{Eq_PE_cor+metric}
 f \to f + \frac{1}{e_1 \; e_2}	\left( 	 v \frac{\partial e_2}{\partial i}
@@ -485,27 +489,58 @@
 \end{equation}
 
-Note that in the case of geographical coordinate, i.e. when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
-
-The equations solved by the ocean model can be written in the following tensorial formalism:
-
-$\bullet$ vector invariant form of the momentum equations :
+Note that in the case of geographical coordinate, $i.e.$ when $(i,j) \to (\lambda ,\varphi )$ and $(e_1 ,e_2) \to (a \,\cos \varphi ,a)$, we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
+
+To sum up, the equations solved by the ocean model can be written in the following tensorial formalism:
+
+\vspace{+10pt}
+$\bullet$ \textit{momentum equations} :
+
+vector invariant form :
 \begin{subequations} \label{Eq_PE_dyn_vect}
-\begin{equation} \label{Eq_PE_dyn_vect_u}
+\begin{multline} \label{Eq_PE_dyn_vect_u}
 \frac{\partial u}{\partial t}=
-+   \left( {\zeta +f} \right)\,v   -   \frac{1}{e_3 }w\frac{\partial u}{\partial k}
--   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( 
-		{\frac{1}{2}\left( {u^2+v^2} \right)+\frac{p_h+p_s }{\rho _o }}    \right)
-+   D_u^{\vect{U}} 
-\end{equation}
-\begin{equation} \label{Eq_PE_dyn_vect_v}
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} w \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho _o}    \right)    
++   D_u^{\vect{U}}  +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Eq_PE_dyn_vect_v}
 \frac{\partial v}{\partial t}=
--   \left( {\zeta +f} \right)\,u   -   \frac{1}{e_3 }w\frac{\partial v}{\partial k}
--   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( 
-		{\frac{1}{2}\left( {u^2+v^2} \right)+\frac{p_h+p_s }{\rho _o }}    \right)
-+  D_v^{\vect{U}} 
-\end{equation}
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	-   \frac{1}{e_3 }w\frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right)    
++  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+\end{multline}
 \end{subequations}
 
-where $\zeta$ is given by (\ref{Eq_PE_curl_Uh}) and the surface pressure gradient is given by:
+flux form:
+\begin{subequations} \label{Eq_PE_dyn_flux}
+\begin{multline} \label{Eq_PE_dyn_flux_u}
+\frac{\partial u}{\partial t}=
++   \left( { f + \frac{1}{e_1 \; e_2}
+					\left( 	 v \frac{\partial e_2}{\partial i}
+						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
+- \frac{1}{e_1 \; e_2} 	\left( 
+					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+		+			\frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}	\right)
+- \frac{1}{e_3 }		\frac{\partial \left( {w\,u		 } \right)}{\partial k}    \\
+-   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right)
++   D_u^{\vect{U}} +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Eq_PE_dyn_flux_v}
+\frac{\partial v}{\partial t}=
+-   \left( { f + \frac{1}{e_1 \; e_2}
+					\left( 	 v \frac{\partial e_2}{\partial i}
+						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
+ \frac{1}{e_1 \; e_2} 	\left( 
+					\frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
+		+			\frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}	\right)
+- \frac{1}{e_3 }		\frac{\partial \left( {w\,v		 } \right)}{\partial k}    \\
+-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right)
++  D_v^{\vect{U}} +  F_v^{\vect{U}} 
+\end{multline}
+\end{subequations}
+where $\zeta$ is given by \eqref{Eq_PE_curl_Uh} and the surface pressure gradient formulation depends on the one of the free surface:
 
 $*$ free surface formulation
@@ -528,5 +563,5 @@
 \end{equation}
 where ${\vect{M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, 
-viscous and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation:
+viscosity and hydrostatic pressure gradient terms in \eqref{Eq_PE_dyn_vect} and the overbar indicates a vertical average over the whole water column ($i.e.$ from $z=-H$, the ocean bottom, to $z=0$, the rigid-lid), and where the time derivative of $\psi$ is the solution of an elliptic equation:
 \begin{multline} \label{Eq_psi_total}
   \frac{\partial }{\partial i}\left[ {\frac{e_2 }{H\,e_1}\frac{\partial}{\partial i}
@@ -547,18 +582,19 @@
 \end{equation}
 
-where the divergence of the horizontal velocity, $\chi$ is given by (I.3.8).
-
-$\bullet$ tracer equations:
+where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}.
+
+\vspace{+10pt}
+$\bullet$ \textit{tracer equations} :
 \begin{equation} \label{Eq_S}
-\frac{\partial T}{\partial t}=-\frac{1}{e_1 e_2 }\left[ {\frac{\partial 
-\left( {e_2 T\,u} \right)}{\partial i}+\frac{\partial \left( {e_1 T\,v} 
-\right)}{\partial j}} \right]-\frac{1}{e_3 }\frac{\partial \left( {T\,w} 
-\right)}{\partial k}+D^T
+\frac{\partial T}{\partial t} = 
+-\frac{1}{e_1 e_2 }\left[ {	   \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
+						+\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
 \end{equation}
 \begin{equation} \label{Eq_T}
-\frac{\partial S}{\partial t}=-\frac{1}{e_1 e_2 }\left[ {\frac{\partial 
-\left( {e_2 S\,u} \right)}{\partial i}+\frac{\partial \left( {e_1 S\,v} 
-\right)}{\partial j}} \right]-\frac{1}{e_3 }\frac{\partial \left( {S\,w} 
-\right)}{\partial k}+D^S
+\frac{\partial S}{\partial t} = 
+-\frac{1}{e_1 e_2 }\left[ 	  {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
+						+\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
+-\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
 \end{equation}
 \begin{equation} \label{Eq_rho}
@@ -566,6 +602,48 @@
 \end{equation}
 
-The expression of \textbf{D}$^{U}$, $D^{S}$ and$ D^{T}$ depends on the subgrid 
-scale parameterization used. It will be defined in \S\ref{PE_zdf}.
+The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid 
+scale parameterization used. It will be defined in \S\ref{PE_zdf}. The nature and formulation of ${\rm {\bf F}}^{\rm {\bf U}}$, $F^T$ and $F^S$, the surface forcing terms, are discussed in Chapter~\ref{SBC}.
+
+\newpage 
+% ================================================================
+% Curvilinear z*-coordinate System
+% ================================================================
+\section{Curvilinear \textit{z*}-coordinate System}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!b] \label{Fig_z_zstar}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_z_zstar.pdf}
+\caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate \citep{Adcroft_Campin_OM04} ).}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+In that case, the free surface equation is nonlinear, and the variations of volume are fully taken into account. These coordinates systems is presented in a report 
+\citep{Levier2007} available on the \NEMO web site. 
+
+\gmcomment{
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation which allows one to deal with large amplitude free-surface
+variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
+the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
+undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
+but is equally distributed over the full water column. Thus vertical
+levels naturally follow sea-surface variations, with a linear attenuation with
+depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
+fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate
+ \textit{z*}, including the treatment of fresh-water flux at the surface, are
+detailed in Adcroft and Campin (2004). The major points are summarized
+here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
+
+$H +  \textit{z*} = (H + z) / r$ and  $\delta \textit{z*} = \delta z / r$ with $r = \frac{H+\eta} {H}$
+
+Since the vertical displacement of the free surface is incorporated in the vertical
+coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position,  $\textit{z*} = 0$ and  $\textit{z*} = ?H$ respectively. Also the divergence of the flow field is no longer zero as shown by the continuity equation:
+
+$\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
+		\left( r \; w\textit{*} \right) = 0 $
+
+}
+
 
 \newpage 
@@ -581,8 +659,7 @@
 \subsection{Introduction}
 
-Several important aspects of the ocean circulation are influenced by bottom topography. Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and channels that strongly constrain the path of water masses, but more subtle effects exist. For 
-example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. Topographic Rossby waves can be excited and can interact with the mean current. In the $z-$coordinate system presented in the previous section (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom and to large localized depth gradients associated with large localized vertical velocities. The response to such a velocity field often leads to numerical dispersion effects.
-
-A terrain-following coordinate system (hereafter $s-$coordinates) avoids the discretisation error in the depth field since the layers of computation are gradually adjusted with depth to the ocean bottom. Relatively shallow topographic features in the deep ocean, which would be ignored in typical $z-$model applications with the largest grid spacing at greatest depths, can easily be represented (with relatively low vertical resolution) as can gentle, large-scale slopes of the sea floor. A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over a large depth range, which in the framework of the $z-$model would require high vertical resolution over the whole depth range. Moreover, with $s-$coordinates it is possible, at least in principle, to have the bottom and the sea surface as the only boundaries of the domain. Nevertheless, $s-$coordinates also have its drawbacks. Perfectly adapted to a homogeneous ocean, it has strong limitations as soon as stratification is introduced. The main two problems come from the truncation error in the horizontal pressure gradient and a possibly increased diapycnal diffusion. The horizontal pressure force in $s-$coordinates consists of two terms (see \colorbox{yellow}{Appendix A}),
+Several important aspects of the ocean circulation are influenced by bottom topography. Of course, the most important is that bottom topography determines deep ocean sub-basins, barriers, sills and channels that strongly constrain the path of water masses, but more subtle effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary one along continental slopes. Topographic Rossby waves can be excited and can interact with the mean current. In the $z-$coordinate system presented in the previous section (\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom and to large localized depth gradients associated with large localized vertical velocities. The response to such a velocity field often leads to numerical dispersion effects. One solution to strongly reduce this error is to use a partial step representation of bottom topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 
+
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of computation are gradually adjusted with depth to the ocean bottom. Relatively small topographic features as well as  gentle, large-scale slopes of the sea floor in the deep ocean, which would be ignored in typical $z$-model applications with the largest grid spacing at greatest depths, can easily be represented (with relatively low vertical resolution). A terrain-following model (hereafter $s-$model) also facilitates the modelling of the boundary layer flows over a large depth range, which in the framework of the $z$-model would require high vertical resolution over the whole depth range. Moreover, with a $s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface as the only boundaries of the domain (nomore lateral boundary condition to specify). Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a homogeneous ocean, it has strong limitations as soon as stratification is introduced. The main two problems come from the truncation error in the horizontal pressure gradient and a possibly increased diapycnal diffusion. The horizontal pressure force in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
 
 \begin{equation} \label{Eq_PE_p_sco}
@@ -591,25 +668,11 @@
 \end{equation}
 
-The second term in (\ref{Eq_PE_p_sco}) depends on the tilt of the coordinate surface and 
-introduces a truncation error that is not present in a $z-$model. In the special case of $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), \cite{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. It depends on topographic slope, stratification, horizontal and vertical resolution, and the finite difference scheme. This error limits the possible topographic slopes that a model can handle at a given horizontal and vertical resolution. This is a severe restriction for large-scale applications using realistic bottom topography. The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. This problem can be, at least partially, overcome by mixing $s-$coordinates and step-like representation of bottom topography \citep{Madec1996}. However, another problem is then raised in the definition of the model domain. 
-
-\colorbox{yellow}{Aike Beckmann's solution}
-
-A minimum of diffusion along the coordinate surfaces of any finite difference model is always required for numerical reasons. It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. This is the case for a $z-$model as well as for a $s-$model. 
-However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal diffusion in a $s-$model than in a $z-$model. Whereas such a diapycnal diffusion in a $z-$model tends to weaken horizontal density (pressure) gradients and thus the horizontal circulation, it usually reinforces these gradients in a $s-$model, creating spurious circulation. For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. Spurious diffusion along $s-$surfaces will induce a bump of isoneutral surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, the ocean will stay at rest in a $z-$model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the 
-water column (i.e. the main thermocline) \citep{Madec1996}. An alternate solution consists in rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_zdf} and \colorbox{yellow}{Appendix B}.
-
-%AMT
-\amtcomment{ 
-The $s-$coordinates introduced here \citep{Lott1990,Madec1996} 
-\colorbox{yellow}{ to be update $==>$}
-differ mainly in two aspects from 
-similar models. It combines the properties which make OPA suitable for 
-climate applications with a good representation of bottom topography 
-allowing mixed step-like/terrain following topography. It also offers a 
-completely general transformation, $s=s(i,j,z)$, for the vertical coordinate which goes beyond those of previous hybrid models except the GFDL version developed by \citep{Gerdes1993a,Gerdes1993b} 
-which has similar properties as the OPA release presented here.
-\colorbox{yellow}{ <== end update}
-}
+The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface and introduces a truncation error that is not present in a $z$-model. In the special case of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), \citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude of this truncation error. It depends on topographic slope, stratification, horizontal and vertical resolution, the equation of state, and the finite difference scheme. This error limits the possible topographic slopes that a model can handle at a given horizontal and vertical resolution. This is a severe restriction for large-scale applications using realistic bottom topography. The large-scale slopes require high horizontal resolution, and the computational cost becomes prohibitive. This problem can be at least partially overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model domain vertical coordinate becomes then a non-trivial thing for a realistic bottom topography: a envelope topography is defined in $s$-coordinate on which a full or partial step bottom topography is then applied in order to adjust the model depth to the observed one (see \S\ref{DOM_zgr}.
+
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces of any finite difference model. It causes spurious diapycnal mixing when coordinate surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as well as for a $s$-model. 
+However, density varies more strongly on $s-$surfaces than on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a $z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. For example, imagine an isolated bump of topography in an ocean at rest with a horizontally uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column (i.e. the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). 
+
+The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two aspects from similar models:  it allows  a representation of bottom topography with mixed full or partial step-like/terrain following topography ; It also offers a completely general transformation, $s=s(i,j,z)$ for the vertical coordinate.
+
 % -------------------------------------------------------------------------------------------------------------
 % The s-coordinate Formulation
@@ -617,5 +680,5 @@
 \subsection{The \textit{s-}coordinate Formulation}
 
-Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k)$, which includes $z-$ and $\sigma-$coordinates as special cases ($s=z$ and $s=\sigma=z/H$, resp.). A formal derivation of the transformed equations is given in \colorbox{yellow}{Appendix A}. Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k)$ ), and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
+Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ and thus $e_3=1$, we introduce an arbitrary vertical coordinate $s=s(i,j,k,t)$, which includes $z$-, \textit{z*}- and $\sigma-$coordinates as special cases ($s=z$, $s=\textit{z*}$, and $s=\sigma=z/H$ or $=z/\left(H+\eta \right)$, resp.). A formal derivation of the transformed equations is given in Appendix~\ref{Apdx_A}. Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ), and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by :
 \begin{equation} \label{Eq_PE_sco_slope}
 \sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
@@ -623,29 +686,36 @@
 \sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
 \end{equation}
-
-We also introduce a "vertical" velocity $\omega$ defined as the velocity normal to $s-$surfaces:
+We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity relative to the moving $s$-surfaces and normal to them:
 \begin{equation} \label{Eq_PE_sco_w}
-\omega = w - \sigma_1 \; u - \sigma_2 \; v
-\end{equation}
-The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinates can be written as follows:
-
+\omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v    \\
+\end{equation}
+
+The equations solved by the ocean model \eqref{Eq_PE} in $s-$coordinate can be written as follows:
+
+ \vspace{0.5cm}
 * momentum equation:
 \begin{multline} \label{Eq_PE_sco_u}
-\frac{\partial u}{\partial t}=+\left( {\zeta +f} \right)\,v-\frac{1}{e_3 
-}\omega \frac{\partial u}{\partial k}-\frac{1}{e_1 }\frac{\partial 
-}{\partial i}\left( {\frac{1}{2}\left( {u^2+v^2} \right)+\frac{p_h }{\rho _o}} \right)   \\
-+g\frac{\rho }{\rho _o }\sigma _1 -\frac{1}{\rho _o e_1 
-}\frac{\partial p_s }{\partial i}+D_u^{\rm {\bf U}} 
+\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
+	+   \left( {\zeta +f} \right)\,v                                    
+	-   \frac{1}{2\,e_1} \frac{\partial}{\partial i} \left(  u^2+v^2   \right) 
+	-   \frac{1}{e_3} \omega \frac{\partial u}{\partial k}       \\
+	-   \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
+	+  g\frac{\rho }{\rho _o}\sigma _1 
+	+   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
 \end{multline}
 \begin{multline} \label{Eq_PE_sco_v}
-\frac{\partial v}{\partial t}=-\left( {\zeta +f} \right)\,u-\frac{1}{e_3 
-}\omega \frac{\partial v}{\partial k}-\frac{1}{e_2 }\frac{\partial 
-}{\partial j}\left( {\frac{1}{2}\left( {u^2+v^2} \right)+\frac{p_h }{\rho _o}} \right)   \\
-+g\frac{\rho }{\rho _o }\sigma _2 -\frac{1}{\rho _o e_2 
-}\frac{\partial p_s }{\partial j}+D_v^{\rm {\bf U}} 
-\end{multline}
-where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic pressure have the same expressions as in $z-$coordinates although they do not represent exactly the same quantities. $\omega $ is provided by the same equation as w, i.e. (\ref{Eq_w_diag}), with $\chi$, the divergence of the horizontal velocity field given by:
-
-\begin{equation} \label{Eq_PE_sco_div}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+	-   \left( {\zeta +f} \right)\,u   
+	-   \frac{1}{2\,e_2 }\frac{\partial }{\partial j}\left(  u^2+v^2  \right)     	
+	-   \frac{1}{e_3 } \omega \frac{\partial v}{\partial k}         \\
+	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
+	 +  g\frac{\rho }{\rho _o }\sigma _2   
+	+  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
+\end{multline}
+where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient, and the hydrostatic pressure have the same expressions as in $z$-coordinates although they do not represent exactly the same quantities. $\omega$ is provided by the continuity equation (see Appendix~\ref{Apdx_A}):
+
+\begin{equation} \label{Eq_PE_sco_continuity}
+\frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0   
+\qquad \text{with }\;\;  
 \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} 
 \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial 
@@ -653,45 +723,26 @@
 \end{equation}
 
+ \vspace{0.5cm}
 * tracer equations:
-
-\begin{equation} \label{Eq_PE_sco_t}
-\frac{\partial T}{\partial t}=-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial 
-\left( {e_2 e_3 T\,u} \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 
-T\,v} \right)}{\partial j}} \right]-\frac{1}{e_3 }\frac{\partial \left( 
-{T\,\omega } \right)}{\partial k}+D^T
-\end{equation}
-
-\begin{equation} \label{Eq_PE_sco_s}
-\frac{\partial S}{\partial t}=-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial 
-\left( {e_2 e_3 S\,u} \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 
-S\,v} \right)}{\partial j}} \right]-\frac{1}{e_3 }\frac{\partial \left( 
-{S\,\omega } \right)}{\partial k}+D^S
-\end{equation}
-
-The equation of state has the same expression as in $z-$coordinates. The expression of \textbf{D}$^{U}$, $D^{S}$ and$ D^{T}$ depends on the subgrid scale parameterization used. It will be defined in \S\ref{PE_zdf_ldf}. 
-
+\begin{multline} \label{Eq_PE_sco_t}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
+                                           +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\
+-\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad
+\end{multline}
+
+\begin{multline} \label{Eq_PE_sco_s}
+\frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}=
+-\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
+                                           +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\
+-\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad
+\end{multline}
+
+The equation of state has the same expression as in $z$-coordinate, and similar expressions are used for mixing and forcing terms.
+
+\gmcomment{
 \colorbox{yellow}{ to be updated $= = >$}
-The whole set of the continuous equations solved by the model in the $s-$coordinate system is summarised in Table~I.2. 
-
 Add a few works on z and zps and s and underlies the differences between all of them
-\colorbox{yellow}{ $< = =$ end update}
-
-\newpage 
-% ================================================================
-% Curvilinear z*- s*-coordinate System
-% ================================================================
-\section{Curvilinear \textit{z*}- or \textit{s*} coordinate System}
-
-% -------------------------------------------------------------------------------------------------------------
-% ????
-% -------------------------------------------------------------------------------------------------------------
-
-\colorbox{yellow}{ to be updated $= = >$}
-
-In that case, the free surface equation is nonlinear, and the variations of 
-volume are fully taken into account. These coordinates systems is presented in a report 
-\citep{Levier2007} available on the \NEMO web site. 
-
-\colorbox{yellow}{ $< = =$ end update}
+\colorbox{yellow}{ $< = =$ end update}  }
 
 \newpage 
@@ -705,9 +756,9 @@
 space and time scales larger than a few kilometres in the horizontal, a few 
 meters in the vertical and a few minutes. They are usually solved at larger 
-scales, the specified grid spacing and time step of the numerical model. The 
+scales: the specified grid spacing and time step of the numerical model. The 
 effects of smaller scale motions (coming from the advective terms in the 
 Navier-Stokes equations) must be represented entirely in terms of 
 large-scale patterns to close the equations. These effects appear in the 
-equations as the divergence of turbulent fluxes (i.e. fluxes associated with 
+equations as the divergence of turbulent fluxes ($i.e.$ fluxes associated with 
 the mean correlation of small scale perturbations). Assuming a turbulent 
 closure hypothesis is equivalent to choose a formulation for these fluxes. 
@@ -718,5 +769,5 @@
 
 The control exerted by gravity on the flow induces a strong anisotropy 
-between the lateral and vertical motions. Therefore subgrid-scale physics  \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in \eqref{Eq_PE_dyn}, \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part  \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part  \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$ . The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
+between the lateral and vertical motions. Therefore subgrid-scale physics  \textbf{D}$^{\vect{U}}$, $D^{S}$ and $D^{T}$  in \eqref{Eq_PE_dyn}, \eqref{Eq_PE_tra_T} and \eqref{Eq_PE_tra_S} are divided into a lateral part  \textbf{D}$^{l \vect{U}}$, $D^{lS}$ and $D^{lT}$ and a vertical part  \textbf{D}$^{vU}$, $D^{vS}$ and $D^{vT}$. The formulation of these terms and their underlying physics are briefly discussed in the next two subsections.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -727,5 +778,5 @@
 
 The model resolution is always larger than the scale at which the major 
-sources of vertical turbulence occurs (shear instability, internal wave 
+sources of vertical turbulence occur (shear instability, internal wave 
 breaking...). Turbulent motions are thus never explicitly solved, even 
 partially, but always parameterized. The vertical turbulent fluxes are 
@@ -746,6 +797,5 @@
    \end{split}
 \end{equation}
-where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must 
-be specified (see Chap.~\ref{SBC}). All the vertical physics is embedded in the specification of the eddy coefficients. They can be assumed to be either constant, or function of the local fluid properties (as Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a turbulent closure model. The choices available in OPA are discussed in \S\ref{ZDF}).
+where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively. At the sea surface and at the bottom, turbulent fluxes of momentum, heat and salt must be specified (see Chap.~\ref{SBC} and \ref{ZDF} and \S\ref{TRA_bbl}). All the vertical physics is embedded in the specification of the eddy coefficients. They can be assumed to be either constant, or function of the local fluid properties ($e.g.$ Richardson number, Brunt-Vais\"{a}l\"{a} frequency...), or computed from a turbulent closure model. The choices available in \NEMO are discussed in \S\ref{ZDF}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -756,28 +806,27 @@
 
 Lateral turbulence can be roughly divided into a mesoscale turbulence 
-associated to eddies which can be solved explicitly if the resolution is 
-sufficient as their underlying physics are included in the primitive 
-equations, and a sub mesoscale turbulence which is never explicitly solved 
+associated with eddies (which can be solved explicitly if the resolution is 
+sufficient since their underlying physics are included in the primitive 
+equations), and a sub mesoscale turbulence which is never explicitly solved 
 even partially, but always parameterized. The formulation of lateral eddy 
 fluxes depends on whether the mesoscale is below or above the grid-spacing 
-(i.e. the model is eddy-resolving or not).
-
-In non-eddy- resolving configurations, the closure is similar to that used 
+($i.e.$ the model is eddy-resolving or not).
+
+In non-eddy-resolving configurations, the closure is similar to that used 
 for the vertical physics. The lateral turbulent fluxes are assumed to depend 
 linearly on the lateral gradients of large-scale quantities. The resulting 
 lateral diffusive and dissipative operators are of second order. 
 Observations show that lateral mixing induced by mesoscale turbulence tends 
-to be along isoneutral surfaces (or more precisely neutral surfaces, i.e. 
-isoneutral surfaces referenced at the local depth) rather than across them. 
-As the slope of isoneutral surfaces is small in the ocean, a common 
+to be along isopycnal surfaces (or more precisely neutral surfaces \cite{McDougall1987}) rather than across them. 
+As the slope of neutral surfaces is small in the ocean, a common 
 approximation is to assume that the `lateral' direction is the horizontal, 
-i.e. the lateral mixing is performed along geopotential surfaces. This leads 
+$i.e.$ the lateral mixing is performed along geopotential surfaces. This leads 
 to a geopotential second order operator for lateral subgrid scale physics. 
 This assumption can be relaxed: the eddy-induced turbulent fluxes can be 
 better approached by assuming that they depend linearly on the gradients of 
-large-scale quantities computed along isoneutral surfaces. In such a case, 
+large-scale quantities computed along neutral surfaces. In such a case, 
 the diffusive operator is an isoneutral second order operator and it has 
 components in the three space directions. However, both horizontal and 
-isoneutral operators have no effect on mean (i.e. large scale) potential 
+isoneutral operators have no effect on mean ($i.e.$ large scale) potential 
 energy whereas potential energy is a main source of turbulence (through 
 baroclinic instabilities). \citet{Gent1990} have proposed a 
@@ -788,5 +837,5 @@
 operator and an eddy induced advective part. In all these lateral diffusive 
 formulations, the specification of the lateral eddy coefficients remains the 
-problematic point as there is no satisfactory formulation of these 
+problematic point as there is no really satisfactory formulation of these 
 coefficients as a function of large-scale features.
 
@@ -796,14 +845,18 @@
 eddies. The role devoted to the subgrid-scale physics is to dissipate the 
 energy that cascades toward the grid scale and thus ensures the stability of 
-the model while not interfering with the solved mesoscale activity.
+the model while not interfering with the solved mesoscale activity. Another approach 
+is becoming more and more popular: instead of specifying explicitly a sub-grid scale 
+term in the momentum and tracer time evolution equations, one uses a advective scheme which is diffusive enough to maintain the model stability. It must be emphasised
+that then, all the sub-grid scale physics is in this case include in the formulation of the
+advection scheme. 
 
 All these parameterizations of subgrid scale physics present advantages and 
-disadvantages. There are not all available in OPA. In the $z-$coordinate 
-formulation, four options are offered for active tracers (temperature and 
+drawbacks. There are not all available in \NEMO. In the $z$-coordinate 
+formulation, five options are offered for active tracers (temperature and 
 salinity): second order geopotential operator, second order isoneutral 
-operator, \citet{Gent1990} parameterization and fourth order 
-geopotential operator. The same options are available for momentum, except 
-\citet{Gent1990} parameterization which only involves tracers. In 
-$s-$coordinate formulation, an additional option is offered for tracers: second 
+operator, \citet{Gent1990} parameterization, fourth order 
+geopotential operator, and various slightly diffusive advection schemes. The same options are available for momentum, except 
+\citet{Gent1990} parameterization which only involves tracers. In the
+$s$-coordinate formulation, additional options are offered for tracers: second 
 order operator acting along $s-$surfaces, and for momentum: fourth order 
 operator acting along $s-$surfaces (see \S\ref{LDF}).
@@ -811,5 +864,5 @@
 \subsubsection{lateral second order tracer diffusive operator}
 
-The lateral second order tracer diffusive operator is defined by (see \colorbox{yellow}{Appendix B}):
+The lateral second order tracer diffusive operator is defined by (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_iso_tensor}
 D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
@@ -821,19 +874,14 @@
 \end{equation}
 where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along 
-which the diffusive operator acts and the surface of computation ($z-$ or 
-$s-$surfaces), and $r_1$ and $r_2$ is the differential operator defined 
-in \S\ref{PE_zco} or \S\ref{PE_sco} depending on the vertical coordinate used. Note that the formulation of $r_1$ and $r_2 $ is exact for the 
-slopes between geopotential and $s-$surfaces, while it is only an approximation 
-for the slopes between isoneutral and $z$ or $s-$surfaces. Indeed, in the latter 
-case, two assumptions are made to simplify $r_1$ and $r_2$ \citep{Cox1987}: the ratio between lateral and vertical diffusive coefficients is 
-known to be several orders of magnitude smaller than unity, and the slopes 
-are, generally less than $10^{2}$ in the ocean (see \colorbox{yellow}{Appendix B}). This leads to 
-the linear tensor (\ref{Eq_PE_iso_tensor}) where the two isoneutral directions of diffusion 
-are independent and where the diapycnal diffusivity contribution is solely along the vertical.
+which the diffusive operator acts and the model level ($e. g.$ $z$- or 
+$s$-surfaces). Note that the formulation \eqref{Eq_PE_iso_tensor} is exact for the 
+rotation between geopotential and $s$-surfaces, while it is only an approximation 
+for the rotation between isoneutral and $z$- or $s$-surfaces. Indeed, in the latter 
+case, two assumptions are made to simplify  \eqref{Eq_PE_iso_tensor} \citep{Cox1987}. First, the horizontal contribution of the dianeutral mixing is neglected since the ratio between iso and dia-neutral diffusive coefficients is known to be several orders of magnitude smaller than unity. Second, the two isoneutral directions of diffusion are assumed to be independent since the slopes are generally less than $10^{-2}$ in the ocean (see Appendix~\ref{Apdx_B}).
 
 For \textit{geopotential} diffusion, $r_1$ and $r_2 $ are the slopes between the 
-geopotential and computational surfaces: in $z$-coordinates they are zero ($r_1$ and $r_2)$ while in $s-$coordinate they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ).
-
-For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. Therefore, they have a same expression in $z-$and $s-$coor-dinates:
+geopotential and computational surfaces: in $z$-coordinates they are zero ($r_1 = r_2 = 0$) while in $s$-coordinate (including $\textit{z*}$ case) they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \eqref{Eq_PE_sco_slope} ).
+
+For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces. Therefore, they have a same expression in $z$- and $s$-coordinates:
 \begin{equation} \label{Eq_PE_iso_slopes}
 r_1 =\frac{e_3 }{e_1 }	\left( {\frac{\partial \rho }{\partial i}} \right)
@@ -861,11 +909,11 @@
 \begin{align} \label{Eq_PE_slopes_eiv}
 \tilde{r}_n = \begin{cases}
-   r_n                  &      \text{in $z-$coordinate}    \\
-   r_n + \sigma_n &      \text{in $s-$coordinate}  
+   r_n                  &      \text{in $z$-coordinate}    \\
+   r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}  
                    \end{cases}
 \quad \text{where } n=1,2
 \end{align}
 
-The normal component of the eddy induced velocity is zero at all the boundaries. this can be achieved by tapering either the eddy coefficient or the slopes to zero in the vicinity of the boundaries. 
+The normal component of the eddy induced velocity is zero at all the boundaries. this can be achieved in a model by tapering either the eddy coefficient or the slopes to zero in the vicinity of the boundaries. The latter strategy is used in \NEMO (cf. Chap.~\ref{LDF}).
 
 \subsubsection{lateral fourth order tracer diffusive operator}
@@ -877,11 +925,11 @@
  \end{equation}
 
-It is the second order operator given by (\ref{Eq_PE_iso_tensor}) applied twice with the eddy 
-diffusion coefficient correctly placed. 
+It is the second order operator given by \eqref{Eq_PE_iso_tensor} applied twice with the eddy diffusion coefficient correctly placed. 
+
 
 \subsubsection{lateral second order momentum diffusive operator}
 
-The second order momentum diffusive operator along $z-$ or $s-$surfaces is found by 
-applying \eqref{Eq_PE_curl} to the horizontal velocity vector (see Appendix B):
+The second order momentum diffusive operator along $z$- or $s$-surfaces is found by 
+applying \eqref{Eq_PE_lap_vector} to the horizontal velocity vector (see Appendix~\ref{Apdx_B}):
 \begin{equation} \label{Eq_PE_lapU}
 \begin{split}
@@ -899,8 +947,7 @@
 
 Such a formulation ensures a complete separation between the vorticity and 
-horizontal divergence fields ({\S}~II.4c). Unfortunately, it is not 
+horizontal divergence fields (see Appendix~\ref{Apdx_C}). Unfortunately, it is not 
 available for geopotential diffusion in $s-$coordinates and for isoneutral 
-diffusion. In these two cases, the $u$ and $v-$fields are considered as independent 
-scalar fields, so that the diffusive operator is given by:
+diffusion in both $z$- and $s$-coordinates ($i.e.$ when a rotation is required). In these two cases, the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by:
 \begin{equation} \label{Eq_PE_lapU_iso}
 \begin{split}
@@ -909,12 +956,11 @@
  \end{split}
  \end{equation}
-where $\Re$ is given by (I.5.2). It is the same expression as those used 
-for diffusive operator on tracers.
+where $\Re$ is given by  \eqref{Eq_PE_iso_tensor}. It is the same expression as those used for diffusive operator on tracers. It must be emphasised that such a formulation is only exact in a Cartesian coordinate system, $i.e.$ on a $f-$ or $\beta-$plane, not on the sphere. It is also a very good approximation in vicinity of the Equator in a geographical coordinate system \citep{Lengaigne_al_JGR03}.
 
 \subsubsection{lateral fourth order momentum diffusive operator}
 
-As for tracers, the fourth order momentum diffusive operator along $z$ or $s-$surfaces is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} with the eddy viscosity coefficient correctly placed:
-
-geopotential diffusion in $z-$coordinates:
+As for tracers, the fourth order momentum diffusive operator along $z$ or $s$-surfaces is a re-entering second order operator \eqref{Eq_PE_lapU} or \eqref{Eq_PE_lapU} with the eddy viscosity coefficient correctly placed:
+
+geopotential diffusion in $z$-coordinate:
 \begin{equation} \label{Eq_PE_bilapU}
 \begin{split}
@@ -928,5 +974,7 @@
 \end{equation}
 
-geopotential diffusion in $s-$coordinates:
+\gmcomment{  change the position of the coefficient, both here and in the code}
+
+geopotential diffusion in $s$-coordinate:
 \begin{equation} \label{Eq_bilapU_iso}
    \left\{   \begin{aligned}
Index: trunk/DOC/BETA/Chapters/Chap_SBC.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_SBC.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_SBC.tex	(revision 817)
@@ -30,5 +30,5 @@
 (CORE or CLIO bulk formulae) and a coupled formulation (exchanges with a 
 atmospheric model via OASIS coupler). In addition, the resulting fields can 
-be further modified on used demand via several namelist option. These option 
+be further modified on used demand via several namelist options. These options 
 control the addition of a surface restoring term to observed SST and/or SSS, 
 the modification of fluxes below ice-covered area (using observed ice-cover 
@@ -37,5 +37,5 @@
 mean sea-level drift.
 
-In this chapter we first discuss where the surface boundary condition 
+In this chapter, we first discuss where the surface boundary condition 
 appears in the model equations. Then we present the four ways of providing 
 the surface boundary condition. Finally, the different options that modify 
@@ -110,5 +110,5 @@
 
 In the real ocean, EMP=EMP$_S$ and the ocean salt content is conserved, 
-but it exist several numerical reason why this equality should be broken. 
+but it exist several numerical reasons why this equality should be broken. 
 For example:
 
@@ -117,5 +117,5 @@
 
 When a sea-ice model is considered, the water exchanged between ice and 
-ocean is not fresh as mean ice salinity is $\sim $\textit{4 psu}. In this case, 
+ocean is very lightly salty (mean sea-ice salinity is $\sim $\textit{4 psu}). In this case, 
 EMP$_{S}$ take into account both concentration/dilution effect associated with 
 freezing/melting together with salt flux between ice and ocean, while EMP is 
@@ -199,5 +199,6 @@
 % Analytical formulation (sbcana module) 
 % ================================================================
-\section{Analytical formulation (\textit{sbcana} module) }
+\section  [Analytical formulation (\textit{sbcana}) ]
+		{Analytical formulation (\mdl{sbcana} module) }
 \label{SBC_ana}
 
@@ -224,5 +225,6 @@
 % Flux formulation 
 % ================================================================
-\section{Flux formulation (\mdl{sbcflx} module, \key{sbcflx}) }
+\section  [Flux formulation (\textit{sbcflx}) ]
+		{Flux formulation (\mdl{sbcflx} module) }
 \label{SBC_flx}
 
@@ -262,5 +264,6 @@
 % Bulk formulation
 % ================================================================
-\section{Bulk formulation (\mdl{sbcblk\_core} or\mdl{sbcblk\_clio} module) }
+\section  [Bulk formulation (\textit{sbcblk\_core} or \textit{sbcblk\_clio}) ]
+		{Bulk formulation \small{(\mdl{sbcblk\_core} or \mdl{sbcblk\_clio} module)} }
 \label{SBC_blk}
 
@@ -352,5 +355,6 @@
 % Coupled formulation
 % ================================================================
-\section{Coupled formulation (\mdl{sbccpl} module)}
+\section  [Coupled formulation (\textit{sbccpl}) ]
+		{Coupled formulation (\mdl{sbccpl} module)}
 \label{SBC_cpl}
 
@@ -370,5 +374,6 @@
 %        Surface restoring to observed SST and/or SSS
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Surface restoring to observed SST and/or SSS (\mdl{sbcssr})}
+\subsection    [Surface restoring to observed SST and/or SSS (\textit{sbcssr})]
+			{Surface restoring to observed SST and/or SSS (\mdl{sbcssr})}
 \label{SBC_ssr}
 
@@ -420,5 +425,6 @@
 %        Addition of river runoffs
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Addition of river runoffs (\mdl{sbcrnf})}
+\subsection   [Addition of river runoffs (\textit{sbcrnf})]
+			{Addition of river runoffs (\mdl{sbcrnf})}
 \label{SBC_rnf}
 
@@ -433,7 +439,8 @@
 
 % -------------------------------------------------------------------------------------------------------------
-%        Addition of river runoffs
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Freshwater budget control (\mdl{sbcfwb})}
+%        Freshwater budget control 
+% -------------------------------------------------------------------------------------------------------------
+\subsection   [Freshwater budget control (\textit{sbcfwb})]
+			{Freshwater budget control (\mdl{sbcfwb})}
 \label{SBC_fwb}
 %--------------------------------------------namfwb--------------------------------------------------------
@@ -441,7 +448,9 @@
 %--------------------------------------------------------------------------------------------------------------
 
-\colorbox{yellow}{freshwater budget correction{\ldots}}
-
-
-
-
+To be written latter...
+
+\gmcomment{The descrition of the technique used to control the freshwater budget has to be added here}
+
+
+
+
Index: trunk/DOC/BETA/Chapters/Chap_TRA.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_TRA.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_TRA.tex	(revision 817)
@@ -12,9 +12,11 @@
 %			==> addfigure on bbl
 
-
+%STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below
+
+\newpage
 $\ $\newline    % force a new ligne
 
-Using the representation described in Chap.~\ref{DOM}, several semi-discrete space 
-forms of the tracer equations are available depending on the vertical 
+Using the representation described in Chap.~\ref{DOM}, several semi-discrete 
+space forms of the tracer equations are available depending on the vertical 
 coordinate used and on the physics used. In all the equations presented 
 here, the masking has been omitted for simplicity. One must be aware that 
@@ -22,5 +24,6 @@
 operator is used, the resulting field is multiplied by a mask.
 
-The two active tracers are potential temperature and salinity. Their prognostic equation can be summarized as follows:
+The two active tracers are potential temperature and salinity. Their prognostic 
+equations can be summarized as follows:
 \begin{equation*}
 \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
@@ -28,16 +31,40 @@
 \end{equation*}
 
-NXT stands for next, referring to the time-stepping. From left to right, the terms on the rhs of the tracer equations are the advection (ADV), the lateral diffusion (LDF), the vertical diffusion (ZDF), the contributions from the external forcings (SBC: Surface Boundary Condition, QSR: Solar Radiation penetration, and BBC: Bottom Boundary Condition), the contribution from the bottom boundary Layer (BBL) parametrisation, and an internal damping (DMP) term. The last four have been put inside brackets as they are optional. The external forcings and parameterizations require complex inputs and calculations (bulk formulae, estimation of mixing coefficients) that are carried out in modules of the SBC, LDF and ZDF categories and described in chapters \S\ref{SBC}, \S\ref{LDF} and  \S\ref{ZDF}, respectively. Note that \mdl{tranpc}, the non-penetrative convection module,  although (temporarily) located in the NEMO/OPA//TRA directory, is described with the model vertical physics (ZDF).
-
-In the present chapter we also describe the diagnostic equations used to compute the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory.
-
-The different options available to the user are managed by namelist logical or CPP keys. For each equation term ttt, the namelist logicals are \textit{ln\_trattt\_xxx}, where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.
-
-The user has the option of extracting each tendency term on the rhs of the tracer equation (\key {trdtra} defined), as described in Chap.~\ref{MISC}.
+NXT stands for next, referring to the time-stepping. From left to right, the terms 
+on the rhs of the tracer equations are the advection (ADV), the lateral diffusion 
+(LDF), the vertical diffusion (ZDF), the contributions from the external forcings 
+(SBC: Surface Boundary Condition, QSR: penetrative Solar Radiation, and BBC: 
+Bottom Boundary Condition), the contribution from the bottom boundary Layer 
+(BBL) parametrisation, and an internal damping (DMP) term. The terms QSR, 
+BBC, BBL and DMP are optional. The external forcings and parameterizations 
+require complex inputs and complex calculations (e.g. bulk formulae, estimation 
+of mixing coefficients) that are carried out in the SBC, LDF and ZDF modules and 
+described in chapters \S\ref{SBC}, \S\ref{LDF} and  \S\ref{ZDF}, respectively. 
+Note that \mdl{tranpc}, the non-penetrative convection module,  although 
+(temporarily) located in the NEMO/OPA/TRA directory, is described with the 
+model vertical physics (ZDF).
+%%%
+\gmcomment{change the position of eosbn2 in the reference code}
+%%%
+
+In the present chapter we also describe the diagnostic equations used to compute 
+the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and 
+freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} 
+and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory.
+
+The different options available to the user are managed by namelist logical or 
+CPP keys. For each equation term ttt, the namelist logicals are \textit{ln\_trattt\_xxx}, 
+where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. 
+The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be 
+found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.
+
+The user has the option of extracting each tendency term on the rhs of the tracer 
+equation for output (\key{trdtra} is defined), as described in Chap.~\ref{MISC}.
 
 % ================================================================
 % Tracer Advection
 % ================================================================
-\section{Tracer Advection (\mdl{traadv})}
+\section  [Tracer Advection (\textit{traadv})]
+		{Tracer Advection (\mdl{traadv})}
 \label{TRA_adv}
 %------------------------------------------nam_traadv-----------------------------------------------------
@@ -45,5 +72,5 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The advection tendency in flux form is the divergence of the advective 
+The advection tendency of a tracer in flux form is the divergence of the advective 
 fluxes. Its discrete expression is given by :
 \begin{equation} \label{Eq_tra_adv}
@@ -54,5 +81,6 @@
 {w\;\tau _w } \right]
 \end{equation}
-which, in pure z-coordinate (\key{zco} defined), reduces to :
+where $\tau$ is either T or S. In pure $z$-coordinate (\key{zco} is defined), 
+it reduces to :
 \begin{equation} \label{Eq_tra_adv_zco}
 ADV_\tau =-\frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}}\left( {\;\delta _i 
@@ -61,89 +89,123 @@
 e\nolimits_{3T} }\delta _k \left[ {w\;\tau _w } \right]
 \end{equation}
-as the vertical scale factors are function of $k$ only, and thus $e_{3u} 
+since the vertical scale factors are functions of $k$ only, and thus $e_{3u} 
 =e_{3v} =e_{3T} $.
 
-The flux form requires implicitly the use of the continuity equation: 
+The flux form in \eqref{Eq_tra_adv} requires implicitly the use of the continuity equation: 
 $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
-using $\nabla \cdot \vect{U}=0)$ or $\partial _t e_3 +\nabla \cdot \vect{U}=0$ in variable volume case ($i.e.$ \key{vvl} defined). Therefore it is of 
+(using $\nabla \cdot \vect{U}=0)$ or $\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$
+ in variable volume case ($i.e.$ \key{vvl} defined). Therefore it is of 
 paramount importance to design the discrete analogue of the advection 
 tendency so that it is consistent with the continuity equation in order to 
-enforce conservation properties of the continuous equations. In other words, 
-by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover discrete form of the 
-continuity equation which is used to calculate the vertical velocity.
+enforce the conservation properties of the continuous equations. In other words, 
+by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover the discrete form of 
+the continuity equation which is used to calculate the vertical velocity.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \label{Fig_adv_scheme}  \begin{center}
 \includegraphics[width=0.9\textwidth]{./Figures/Fig_adv_scheme.pdf}
-\caption{Schematic representation of some ways used to evaluate the tracer value at $u$-point and the amount of tracer exchanged between two neighbouring grid points. Upsteam biased scheme (ups): the upstream value is used and the black area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation is used and black + dark grey areas is exchanged. Monotonic upstream scheme for conservative laws (muscl):  a parabolic interpolation is used and black + dark grey + grey areas are exchanged. Second order scheme (cen2): the mean value is used and black + dark grey + grey + light grey areas are exchanged. Note that this illustration does not include the flux limiter used in ppm and muscl schemes.}
+\caption{Schematic representation of some ways used to evaluate the tracer value 
+at $u$-point and the amount of tracer exchanged between two neighbouring grid 
+points. Upsteam biased scheme (ups): the upstream value is used and the black 
+area is exchanged. Piecewise parabolic method (ppm): a parabolic interpolation 
+is used and the black and dark grey areas are exchanged. Monotonic upstream 
+scheme for conservative laws (muscl):  a parabolic interpolation is used and black, 
+dark grey and grey areas are exchanged. Second order scheme (cen2): the mean 
+value is used and black, dark grey, grey and light grey areas are exchanged. Note 
+that this illustration does not include the flux limiter used in ppm and muscl schemes.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-The advection schemes used in OPA differ by the choice made in space and 
-time interpolation to define the value of the tracer at the velocity points (\ref{Fig_adv_scheme}). 
+
+The key difference between the advection schemes used in \NEMO is the choice 
+made in space and time interpolation to define the value of the tracer at the 
+velocity points (Fig.~\ref{Fig_adv_scheme}). 
+
 Along solid lateral and bottom boundaries a zero tracer flux is naturally 
 specified, since the normal velocity is zero there. At the sea surface the 
-boundary condition depends on the type of sea surface chosen: (1) in 
-rigid-lid formulation, $w=0$ at the surface, so the advective fluxes through the 
-surface is zero ; (2) in non-linear free surface (variable volume case, 
-\key{vvl} defined), convergence/divergence in the first ocean level moves 
-up/down the free surface: there is no tracer advection through it so that 
-the advective fluxes through the surface is also zero ; (3) in the linear 
-free surface, the first level thickness is constant in time. The vertical 
-boundary condition is applied at the fixed surface $z=0$ rather than on the 
-moving surface $z=\eta$. There is a non-zero advective flux which is set for all 
-advection schemes as the product of surface velocity (at $z=0$) by the first level 
-tracer value: $\left. {\tau _w } \right|_{k=1 \mathord{\left/ {\vphantom {1 
-2}} \right. \kern-\nulldelimiterspace} 2} =T_{k=1} $. This boundary 
-condition retains local conservation of tracer. Strict global conservation 
-is not possible in linear free surface but is achieved to a good 
-approximation since the non-conservative term is the product of the time 
-derivative of the tracer and the free surface height, two quantities that are not correlated (see \S\ref{PE_free_surface}, and also \citet{Roullet2000,Griffies2001,Campin2004}).
-
-The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) is the centred (\textit{now}) \textit{eulerian} ocean velocity (see \S\ref{DYN}). Nevertheless, when advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity (\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} velocity (i.e. the sum of the eulerian, the bbl and/or the eiv velocities) which is used.
+boundary condition depends on the type of sea surface chosen: 
+\begin{description}
+\item  [rigid-lid formulation:] $w=0$ at the surface, so the advective 
+fluxes through the surface are zero.
+\item [linear free surface:] the first level thickness is constant in time: 
+the vertical boundary condition is applied at the fixed surface $z=0$ 
+rather than on the moving surface $z=\eta$. There is a non-zero advective 
+flux which is set for all advection schemes as the product of surface 
+velocity (at $z=0$) by the first level tracer value: 
+$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $. 
+\item [non-linear free surface:] (\key{vvl} is defined) 
+convergence/divergence in the first ocean level moves the free surface 
+up/down. There is no tracer advection through it so that the advective 
+fluxes through the surface are also zero 
+\end{description}
+In all cases, this boundary condition retains local conservation of tracer. 
+Global conservation is obtained in both rigid-lid and non-linear free surface 
+cases, but not in the linear free surface case. Nevertheless, in the latter
+case, it is achieved to a good approximation since the non-conservative 
+term is the product of the time derivative of the tracer and the free surface 
+height, two quantities that are not correlated (see \S\ref{PE_free_surface}, 
+and also \citet{Roullet2000,Griffies2001,Campin2004}).
+
+The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) 
+is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). 
+When advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity 
+(\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} 
+velocity ($i.e.$ the sum of the eulerian, the bbl and/or the eiv velocities) which is used.
 
 The choice of an advection scheme is made in the \np{nam\_traadv} namelist, by 
 setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The 
-corresponding code can be found in \textit{traadv\_xxx.F90} module, where 
-\textit{xxx} is a 3 or 4 letter acronym accounting for each scheme. Details 
+corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 
+\textit{xxx} is a 3 or 4 letter acronym corresponding to each scheme. Details 
 of the advection schemes are given below. The choice of an advection scheme 
 is a complex matter which depends on the model physics, model resolution, 
 type of tracer, as well as the issue of numerical cost. 
 
-Note that (1) cen2, cen4 and TVD schemes require an explicit diffusion 
+Note that 
+(1) cen2, cen4 and TVD schemes require an explicit diffusion 
 operator while the other schemes are diffusive enough so that they do not 
-require additional diffusion ; (2) cen2, cen4, MUSCL2, and UBS are not 
-\textit{positive} schemes, meaning false extrema are permitted. It is not recommended to use 
-them on passive tracers ; (3) It is highly recommended to use the same 
-advection-diffusion scheme on both active and passive tracers. In 
-particular, if a source or sink of a passive tracer depends on a active one, 
-the difference of treatment of active and passive tracers can create very 
-nice-looking frontal structures that are pure numerical artefacts. 
-
+require additional diffusion ; 
+(2) cen2, cen4, MUSCL2, and UBS are not \textit{positive} schemes
+\footnote{negative values can appear in an initially strictly positive tracer field 
+which is advected}
+, implying that false extrema are permitted. Their use is not recommended on passive tracers ; 
+(3) It is highly recommended that the same advection-diffusion scheme is 
+used on both active and passive tracers. Indeed, if a source or sink of a 
+passive tracer depends on an active one, the difference of treatment of 
+active and passive tracers can create very nice-looking frontal structures 
+that are pure numerical artefacts. 
 
 % -------------------------------------------------------------------------------------------------------------
 %        2nd order centred scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=T)}
+\subsection   [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})]
+			{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=.true.)}
 \label{TRA_adv_cen2}
 
 In the centred second order formulation, the tracer at velocity points is 
-evaluated as the mean of the two neighbouring $T$-points. For example, in the 
-$i$-direction :
+evaluated as the mean of the two neighbouring $T$-point values. 
+For example, in the $i$-direction :
 \begin{equation} \label{Eq_tra_adv_cen2}
 \tau _u^{cen2} =\overline T ^{i+1/2}
 \end{equation}
 
-The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to produce a sensible solution. The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.
-
-Note that using cen2 scheme, the overall tracer advection is of second order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) have this order of accuracy.
+The scheme is non diffusive ($i.e.$ it conserves the tracer variance, $\tau^2)$ 
+but dispersive ($i.e.$ it may create false extrema). It is therefore notoriously 
+noisy and must be used in conjunction with an explicit diffusion operator to 
+produce a sensible solution. The associated time-stepping is performed using 
+a leapfrog scheme in conjunction with an Asselin time-filter, so $T$ in 
+(\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value.
+
+Note that using the cen2 scheme, the overall tracer advection is of second 
+order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 
+have this order of accuracy.
 
 % -------------------------------------------------------------------------------------------------------------
 %        4nd order centred scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=T)}
+\subsection   [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]
+			{$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=.true.)}
 \label{TRA_adv_cen4}
 
-In the $4^{th}$ order formulation (to be implemented), tracer is evaluated 
-at velocity points as the $4^{th}$ order interpolation of $T$, and thus use the 
-four neighbouring $T$-points. For example, in the $i$-direction:
+In the $4^{th}$ order formulation (to be implemented), tracer values are 
+evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses 
+the four neighbouring $T$-points. For example, in the $i$-direction:
 \begin{equation} \label{Eq_tra_adv_cen4}
 \tau _u^{cen4} 
@@ -152,34 +214,35 @@
 
 Strictly speaking, the cen4 scheme is not a $4^{th}$ order advection scheme 
-but a $4^{th}$ order evaluation of advective fluxes since the divergence of 
-advective fluxes, (\ref{Eq_tra_adv}), is kept at $2^{nd}$ order. The ``$4^{th}$ order 
-scheme'' denomination used in oceanographic literature is usually associated 
+but a $4^{th}$ order evaluation of advective fluxes, since the divergence of 
+advective fluxes \eqref{Eq_tra_adv} is kept at $2^{nd}$ order. The phrase ``$4^{th}$ 
+order scheme'' used in oceanographic literature is usually associated 
 with the scheme presented here. Introducing a \textit{true} $4^{th}$ order advection 
 scheme is feasible but, for consistency reasons, it requires changes in the 
 discretisation of the tracer advection together with changes in both the 
-continuity equation and the momentum advection. 
+continuity equation and the momentum advection terms.  
 
 A direct consequence of the pseudo-fourth order nature of the scheme is that 
 it is not non-diffusive, i.e. the global variance of a tracer is not 
-preserved through \textit{cen4}. Furthermore, it must be used in conjunction with an 
+preserved using \textit{cen4}. Furthermore, it must be used in conjunction with an 
 explicit diffusion operator to produce a sensible solution. The 
 time-stepping is also performed using a leapfrog scheme in conjunction with 
 an Asselin time-filter, so $T$ in (\ref{Eq_tra_adv_cen4}) is the \textit{now} tracer.
 
-At $T$-grid cell abutted to a boundary (coastline, bottom and surface), an 
+At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface), an 
 additional hypothesis must be made to evaluate $\tau _u^{cen4}$. This 
 hypothesis usually reduces the order of the scheme. Here we choose to set 
 the gradient of $T$ across the boundary to zero. Alternative conditions can be 
-specified such as the reduction to a second order scheme for near boundary 
-grid point.
+specified, such as a reduction to a second order scheme for these near boundary 
+grid points.
 
 % -------------------------------------------------------------------------------------------------------------
 %        TVD scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=T)}
+\subsection   [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]
+			{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=.true.)}
 \label{TRA_adv_tvd}
 
 In the Total Variance Dissipation (TVD) formulation, the tracer at velocity 
-points is evaluated as a combination of upstream and centred scheme. For 
+points is evaluated using a combination of an upstream and a centred scheme. For 
 example, in the $i$-direction :
 \begin{equation} \label{Eq_tra_adv_tvd}
@@ -194,27 +257,29 @@
 \end{equation}
 where $c_u$ is a flux limiter function taking values between 0 and 1. There 
-exists many ways to define $c_u$., each correcponding to a different total 
-variance decreasing scheme. The one chosen in OPA is described in \citet{Zalesak1979}. $c_u$ only departs from $1$ when the advective term produces a local 
-extremum in the tracer field. The resulting scheme is quite expensive but 
-\emph{positive}. It can be used on both active and passive tracers. This scheme is tested 
-and compared with MUSCL and the MPDATA scheme in \citet{Levy2001}; note that 
-in this paper it is referred to as "FCT" (Flux corrected transport) 
-rather than TVD.
-
-For stability reasons in (\ref{Eq_tra_adv_tvd}) $\tau _u^{cen2}$ is evaluated using the 
-\textit{now} velocity (leap-frog environment: centred in time) while $\tau _u^{ups}$ is 
-evaluated using the \textit{before} velocity (diffusive part: forward in time). 
+exist many ways to define $c_u$, each correcponding to a different total 
+variance decreasing scheme. The one chosen in \NEMO is described in \citet{Zalesak1979}. $c_u$ only departs from $1$ when the advective term 
+produces a local extremum in the tracer field. The resulting scheme is quite 
+expensive but \emph{positive}. It can be used on both active and passive tracers. 
+This scheme is tested and compared with MUSCL and the MPDATA scheme in 
+\citet{Levy2001}; note that in this paper it is referred to as "FCT" (Flux corrected 
+transport) rather than TVD.
+
+For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd}) 
+$\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$ 
+is evaluated using the \textit{before} tracer. In other words, the advective part of 
+the scheme is time stepped with a leap-frog scheme while a forward scheme is 
+used for the diffusive part. 
 
 % -------------------------------------------------------------------------------------------------------------
 %        MUSCL scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl}=T)]
+\subsection[MUSCL scheme  (\np{ln\_traadv\_muscl})]
 	{Monotone Upstream Scheme for Conservative Laws (MUSCL) (\np{ln\_traadv\_muscl}=T)}
 \label{TRA_adv_muscl}
 
 The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 
-implemented by \citet{Levy2001}. In its formulation, the tracer at 
-velocity points is evaluated assuming a linear tracer variation between two 
-$T$-points (Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :
+implemented by \citet{Levy2001}. In its formulation, the tracer at velocity points 
+is evaluated assuming a linear tracer variation between two $T$-points 
+(Fig.\ref{Fig_adv_scheme}). For example, in the $i$-direction :
 \begin{equation} \label{Eq_tra_adv_muscl}
    \tau _u^{mus} = \left\{      \begin{aligned}
@@ -225,25 +290,26 @@
    \end{aligned}    \right.
 \end{equation}
-where $\widetilde{\partial _i \tau}$ is the slope of the tracer 
-on which a limitation is imposed to ensure the \textit{positive} character of the scheme.
-
-The time stepping is performed using a forward scheme, that is the \textit{before} tracer 
-field is used to evaluate $\tau _u^{mus}$.
-
-For an ocean grid point abutted to land and where the ocean velocity is 
-toward land, two choices are available: use of an upstream flux 
-(\np{ln\_traadv\_muscl}=T) or use of second order flux 
-(\np{ln\_traadv\_muscl2}=T). Note that the latter choice does not insure the 
-\textit{positive} character of the scheme. Only the former can be used on both active and 
-passive tracers.
+where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation 
+is imposed to ensure the \textit{positive} character of the scheme.
+
+The time stepping is performed using a forward scheme, that is the \textit{before} 
+tracer field is used to evaluate $\tau _u^{mus}$.
+
+For an ocean grid point adjacent to land and where the ocean velocity is 
+directed toward land, two choices are available: an upstream flux 
+(\np{ln\_traadv\_muscl}=.true.) or a second order flux 
+(\np{ln\_traadv\_muscl2}=.true.). Note that the latter choice does not ensure 
+the \textit{positive} character of the scheme. Only the former can be used 
+on both active and passive tracers.
 
 % -------------------------------------------------------------------------------------------------------------
 %        UBS scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Upstream Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=T)}
+\subsection   [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})]
+			{Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=.true.)}
 \label{TRA_adv_ubs}
 
-The UBS advection scheme is an upstream biased third order scheme based on 
-an upstream-biased parabolic interpolation. It is also known as Cell 
+The UBS advection scheme is an upstream-biased third order scheme based on 
+an upstream-biased parabolic interpolation. It is also known as the Cell 
 Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective 
 Kinematics). For example, in the $i$-direction :
@@ -261,40 +327,50 @@
 advection scheme is similar to that reported in \cite{Farrow1995}. 
 It is a relatively good compromise between accuracy and smoothness. It is 
-not a \emph{positive} scheme meaning false extrema are permitted but the 
+not a \emph{positive} scheme, meaning that false extrema are permitted, but the 
 amplitude of such are significantly reduced over the centred second order 
-method. Nevertheless it is not recommended to apply it to a passive tracer 
-that requires positivity. 
+method. Nevertheless it is not recommended that it should be applied to a passive 
+tracer that requires positivity. 
 
 The intrinsic diffusion of UBS makes its use risky in the vertical direction 
 where the control of artificial diapycnal fluxes is of paramount importance. 
-It has therefore been preferred to evaluate the vertical flux using the TVD 
-scheme when \np{ln\_traadv\_ubs}=T.
-
-For stability reasons, in \eqref{Eq_tra_adv_ubs}, the first term which corresponds to a 
-second order centred scheme is evaluated using the \textit{now} velocity (centred in 
-time) while the second term which is the diffusive part of the scheme, is 
-evaluated using the \textit{before} velocity (forward in time. This is discussed by \citet{Webb1998} in the context of the Quick advection scheme. UBS and QUICK 
-schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 
-(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. This 
-option is not available through a namelist parameter, since the 1/6 
-coefficient is hard coded. Nevertheless it is quite easy to make the 
-substitution in \mdl{traadv\_ubs} module and obtain a QUICK scheme
-
-NB 1: When a high vertical resolution $O(1m)$ is used, the model stability can 
+Therefore the vertical flux is evaluated using the TVD 
+scheme when \np{ln\_traadv\_ubs}=.true..
+
+For stability reasons  (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs}, 
+the first term (which corresponds to a second order centred scheme) 
+is evaluated using the \textit{now} tracer (centred in time) while the 
+second term (which is the diffusive part of the scheme), is 
+evaluated using the \textit{before} tracer (forward in time). 
+This is discussed by \citet{Webb1998} in the context of the Quick 
+advection scheme. UBS and QUICK 
+schemes only differ by one coefficient. Replacing 1/6 with 1/8 in 
+\eqref{Eq_tra_adv_ubs} leads to the QUICK advection scheme 
+\citep{Webb1998}. This option is not available through a namelist 
+parameter, since the 1/6 coefficient is hard coded. Nevertheless 
+it is quite easy to make the substitution in the \mdl{traadv\_ubs} module 
+and obtain a QUICK scheme.
+
+Note that :
+
+(1): When a high vertical resolution $O(1m)$ is used, the model stability can 
 be controlled by vertical advection (not vertical diffusion which is usually 
 solved using an implicit scheme). Computer time can be saved by using a 
-time-splitting technique on vertical advection. This possibility have been 
-implemented and validated in ORCA05-L301. It is not currently offered in the 
+time-splitting technique on vertical advection. This case has been 
+implemented and validated in ORCA05 with 301 levels. It is not available in the 
 current reference version. 
 
-NB 2 : In a forthcoming release four options will be proposed for the 
-vertical component used in the UBS scheme. $\tau _w^{ubs}$ will be 
-evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} a TVD 
-scheme, or  \textit{(c)} an interpolation based on conservative parabolic splines 
-following \citet{Sacha2005} implementation of UBS in ROMS, 
-or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an 
+(2) : In a forthcoming release four options will be available for the vertical 
+component used in the UBS scheme. $\tau _w^{ubs}$ will be evaluated 
+using either \textit{(a)} a centred $2^{nd}$ order scheme , or  \textit{(b)} 
+a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
+parabolic splines following the \citet{Sacha2005} implementation of UBS 
+in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case has dispersion properties 
+similar to an eighth-order accurate conventional scheme.
+
+following \citet{Sacha2005} implementation of UBS in ROMS, or  \textit{(d)} 
+an UBS. The $3^{rd}$ case has dispersion properties similar to an 
 eight-order accurate conventional scheme.
 
-NB 3 : It is straight forward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
+(3) : It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
 \begin{equation} \label{Eq_tra_adv_ubs2}
 \tau _u^{ubs} = \left\{	 \begin{aligned}
@@ -309,37 +385,55 @@
 - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
 \end{equation}
-\eqref{Eq_tra_adv_ubs2} has several advantages. First it clearly evidence that the UBS scheme is based on the fourth order scheme to which is added an upstream biased diffusive term. Second, this emphasises that the $4^{th}$ order part have to be evaluated at \emph{now} time step, not only the $2^{th}$ order part as stated above using \eqref{Eq_tra_adv_ubs} and also as it is coded in NEMO v2.3. Third, the diffusive term is in fact a biharmonic operator with a eddy coefficient with is simply proportional to the velocity: $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that the current version of NEMO uses \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_tra_adv_ubs2}. 
-
+\eqref{Eq_tra_adv_ubs2} has several advantages. Firstly, it clearly reveals 
+that the UBS scheme is based on the fourth order scheme to which an 
+upstream-biased diffusion term is added. Secondly, this emphasises that the 
+$4^{th}$ order part (as well as the $2^{nd}$ order part as stated above) has 
+to be evaluated at the \emph{now} time step using \eqref{Eq_tra_adv_ubs}. 
+Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 
+coefficient which is simply proportional to the velocity:
+ $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses 
+ \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_tra_adv_ubs2}. This should be 
+ changed in forthcoming release.
+ %%%
+ \gmcomment{the change in UBS scheme has to be done}
+ %%%
 
 % -------------------------------------------------------------------------------------------------------------
 %        QCK scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=T)}
+\subsection   [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})]
+			{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=.true.)}
 \label{TRA_adv_qck}
 
 The Quadratic Upstream Interpolation for Convective Kinematics with 
-Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} is 
-the third order Godunov scheme. It is associated with ULTIMATE QUICKEST 
+Estimated Streaming Terms (QUICKEST) scheme proposed by \citet{Leonard1979} 
+is the third order Godunov scheme. It is associated with the ULTIMATE QUICKEST 
 limiter \citep{Leonard1991}. It has been implemented in NEMO by G. Reffray 
-(MERCATOR-ocean).
-
-The resulting scheme is quite expensive but \emph{positive}. It can be used on both active 
-and passive tracers.
+(MERCATOR-ocean). 
+The resulting scheme is quite expensive but \emph{positive}. It can be used on both active and passive tracers. Nevertheless, the intrinsic diffusion of QCK makes its use 
+risky in the vertical direction where the control of artificial diapycnal fluxes is of 
+paramount importance. Therefore the vertical flux is evaluated using the CEN2 
+scheme. This no more ensure the positivity of the scheme. The use of TVD in the 
+vertical direction as for the UBS case should be implemented to maintain the property.
+
 
 % -------------------------------------------------------------------------------------------------------------
 %        PPM scheme  
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=T)}
+\subsection   [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]
+			{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=.true.)}
 \label{TRA_adv_ppm}
 
 The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 
-is based on a quadradic piecewise rebuilding. As QCK scheme, it is associated with 
-ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the current reference 
-version.
+is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated 
+with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 
+in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 
+version 2.3.
 
 % ================================================================
 % Tracer Lateral Diffusion
 % ================================================================
-\section{Tracer Lateral Diffusion (\mdl{traldf})}
+\section  [Tracer Lateral Diffusion (\textit{traldf})]
+		{Tracer Lateral Diffusion (\mdl{traldf})}
 \label{TRA_ldf}
 %-----------------------------------------nam_traldf------------------------------------------------------
@@ -347,22 +441,24 @@
 %-------------------------------------------------------------------------------------------------------------
  
-The options available for lateral diffusion are laplacian (rotated or not) 
-or biharmonic operators, that latter being more scale-selective (more 
+The options available for lateral diffusion are a laplacian (rotated or not) 
+or a biharmonic operator, the latter being more scale-selective (more 
 diffusive at small scales). The specification of eddy diffusivity 
-coefficients (either constant, variable in space and time) as well as the 
-computation of the slope along which the operators act are performed in 
+coefficients (either constant or variable in space and time) as well as the 
+computation of the slope along which the operators act, are performed in the 
 \mdl{ldftra} and \mdl{ldfslp} modules, respectively. This is described in Chap.~\ref{LDF}. The lateral diffusion of tracers is evaluated using a forward scheme, 
-i.e. the tracers appearing in its expression are the \textit{before} tracers in time, 
-except for the pure vertical component that appears when a tensor of 
-rotation is used. This latter term is solved implicitly together with the 
-vertical diffusion term (see \S\ref{DOM_nxt}) 
+$i.e.$ the tracers appearing in its expression are the \textit{before} tracers in time, 
+except for the pure vertical component that appears when a rotation tensor 
+is used. This latter term is solved implicitly together with the 
+vertical diffusion term (see \S\ref{DOM_nxt}).
 
 % -------------------------------------------------------------------------------------------------------------
 %        Iso-level laplacian operator
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Iso-level laplacian operator (\mdl{traldf\_lap}, \np{ln\_traldf\_lap}) }
+\subsection   [Iso-level laplacian operator (\textit{traldf\_lap} - \np{ln\_traldf\_lap})]
+			{Iso-level laplacian operator (\mdl{traldf\_lap} - \np{ln\_traldf\_lap}=.true.) }
 \label{TRA_ldf_lap}
 
-A laplacian diffusive operator (i.e. a harmonic operator) acting along the model surfaces is given by: 
+A laplacian diffusion operator (i.e. a harmonic operator) acting along the model 
+surfaces is given by: 
 \begin{equation} \label{Eq_tra_ldf_lap}
 \begin{split}
@@ -377,19 +473,31 @@
 \end{equation}
 
-This lateral operator is a \emph{horizontal} one (i.e. acting along geopotential surfaces) in 
-$z$-coordinate with or without partial step, but it is simply an iso-level operator in $s$-coordinate. 
-It is thus used when, in addition to \np{ln\_traldf\_lap}=T, we have \np{ln\_traldf\_level}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=F. In both cases, it significantly contributes to diapycnal mixing. It is therefore not recommended to use it.
-
-\textit{Notes} : In pure z-coordinate (\key{zco} defined), $e_{3u}=e_{3v}=e_{3T}$, so that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}). 
-
-\textit{Notes} : In partial step $z$-coordinate (\np{ln\_zps}=T), tracers in horizontally adjacent cells are located at different depths in vicinity of the bottom. In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level require a specific treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
+This lateral operator is a \emph{horizontal} one ($i.e.$ acting along 
+geopotential surfaces) in the $z$-coordinate with or without partial step, 
+but is simply an iso-level operator in the $s$-coordinate. 
+It is thus used when, in addition to \np{ln\_traldf\_lap}=.true., we have 
+\np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and 
+\np{ln\_zco}=.false.. In both cases, it significantly contributes to 
+diapycnal mixing. It is therefore not recommended.
+
+Note that 
+(1) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}=e_{3v}=e_{3T}$, so 
+that the vertical scale factors disappear from (\ref{Eq_tra_ldf_lap}). 
+(2) In partial step $z$-coordinate (\np{ln\_zps}=.true.), tracers in horizontally 
+adjacent cells are located at different depths in the vicinity of the bottom. 
+In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level 
+require a specific treatment. They are calculated in the \mdl{zpshde} module, 
+described in \S\ref{TRA_zpshde}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Rotated laplacian operator
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Rotated laplacian operator (\mdl{traldf\_iso}, \np{ln\_traldf\_lap})}
+\subsection   [Rotated laplacian operator (\textit{traldf\_iso} - \np{ln\_traldf\_lap})]
+			{Rotated laplacian operator (\mdl{traldf\_iso} - \np{ln\_traldf\_lap}=.true.)}
 \label{TRA_ldf_iso}
 
-The general form of the second order lateral tracer subgrid scale physics (\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and $s$-coordinates:
+The general form of the second order lateral tracer subgrid scale physics 
+(\ref{Eq_PE_zdf}) takes the following semi-discrete space form in $z$- and 
+$s$-coordinates:
 
 \begin{equation} \label{Eq_tra_ldf_iso}
@@ -422,27 +530,53 @@
  \end{split}
  \end{equation}
-where $r_1$ and $r_2$ are the slopes between the surface of computation ($z$- or $s$-surfaces) and the surface along which the diffusive operator acts ($i.e.$ horizontal or iso-neutral surfaces). 
- It is thus used when, in addition to \np{ln\_traldf\_lap}=T, we have \np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. The way these slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom and lateral boundaries, the turbulent fluxes of heat and salt are set to zero using the mask technique (see \S\ref{LBC_coast}). 
-
-The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical derivatives. For numerical stability, the vertical second derivative must be solved using the same implicit time scheme as those used in the vertical physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term is not computed in \mdl{traldf} module, but in \mdl{trazdf} module where, if iso-neutral mixing is used, the vertical mixing coefficient is simply increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. 
-
-This formulation conserves the tracer but does not ensure the decrease of the tracer variance. Nevertheless the treatment performed on the slopes (see \S\ref{LDF}) allows to run safely without any additional background horizontal diffusion \citep{Guily2001}. An alternate scheme \citep{Griffies1998} which preserves both tracer and its variance is currently been tested in \NEMO.
-
-Note that in partial step $z$-coordinate (\np{ln\_zps}=T), the horizontal derivatives in \eqref{Eq_tra_ldf_iso} at the bottom level require a specific treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
+where $r_1$ and $r_2$ are the slopes between the surface of computation 
+($z$- or $s$-surfaces) and the surface along which the diffusion operator 
+acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when, 
+in addition to \np{ln\_traldf\_lap}=.true., we have \np{ln\_traldf\_iso}=.true., 
+or both \np{ln\_traldf\_hor}=.true. and \np{ln\_zco}=.true.. The way these 
+slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom 
+and lateral boundaries, the turbulent fluxes of heat and salt are set to zero 
+using the mask technique (see \S\ref{LBC_coast}). 
+
+The operator in \eqref{Eq_tra_ldf_iso} involves both lateral and vertical 
+derivatives. For numerical stability, the vertical second derivative must 
+be solved using the same implicit time scheme as that used in the vertical 
+physics (see \S\ref{TRA_zdf}). For computer efficiency reasons, this term 
+is not computed in the \mdl{traldf} module, but in the \mdl{trazdf} module 
+where, if iso-neutral mixing is used, the vertical mixing coefficient is simply 
+increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$. 
+
+This formulation conserves the tracer but does not ensure the decrease 
+of the tracer variance. Nevertheless the treatment performed on the slopes 
+(see \S\ref{LDF}) allows the model to run safely without any additional 
+background horizontal diffusion \citep{Guily2001}. An alternative scheme 
+\citep{Griffies1998} which preserves both tracer and its variance is currently 
+been tested in \NEMO. 
+
+Note that in the partial step $z$-coordinate (\np{ln\_zps}=.true.), the horizontal 
+derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific 
+treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Iso-level bilaplacian operator
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Iso-level bilaplacian operator (\mdl{traldf\_bilap}, \np{ln\_traldf\_bilap})}
+\subsection   [Iso-level bilaplacian operator (\textit{traldf\_bilap} - \np{ln\_traldf\_bilap})]
+			{Iso-level bilaplacian operator (\mdl{traldf\_bilap} - \np{ln\_traldf\_bilap}=.true.)}
 \label{TRA_ldf_bilap}
 
-The lateral fourth order operator formulation on tracers is obtained by applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption on boundary conditions: first and third derivative terms normal to the coast are set to zero.
-
-It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have \np{ln\_traldf\_level}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=F. In both cases, it can contributes to diapycnal mixing even if it should be less than in the laplacian case. It is therefore not recommended to use it.
-
-\textit{Notes:} In the code, the bilaplacian routine does not call twice the laplacian 
-routine but is rather a specific routine. This is due to the fact that we 
+The lateral fourth order bilaplacian operator on tracers is obtained by 
+applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption 
+on boundary conditions: the first and third derivative terms normal to the 
+coast are set to zero.
+
+It is used when, in addition to \np{ln\_traldf\_bilap}=.true., we have 
+\np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and 
+\np{ln\_zco}=.false.. In both cases, it can contribute diapycnal mixing, 
+although less than in the laplacian case. It is therefore not recommended.
+
+Note that in the code, the bilaplacian routine does not call the laplacian 
+routine twice but is rather a separate routine. This is due to the fact that we 
 introduce the eddy diffusivity coefficient, A, in the operator as: $\nabla 
-\cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$ and instead of 
+\cdot \nabla \left( {A\nabla \cdot \nabla T} \right)$, instead of 
 $-\nabla \cdot a\nabla \left( {\nabla \cdot a\nabla T} \right)$ where 
 $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 
@@ -453,16 +587,26 @@
 %        Rotated bilaplacian operator
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Rotated bilaplacian operator (\mdl{traldf\_bilapg}, \np{ln\_traldf\_bilap})}
+\subsection   [Rotated bilaplacian operator (\textit{traldf\_bilapg} - \np{ln\_traldf\_bilap})]
+			{Rotated bilaplacian operator (\mdl{traldf\_bilapg} - \np{ln\_traldf\_bilap}=.true.)}
 \label{TRA_ldf_bilapg}
 
-The lateral fourth order operator formulation on tracers is obtained by applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption on boundary conditions: first and third derivative terms normal to the coast, the bottom and the surface are set to zero.
-
- It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have \np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. Nevertheless, this rotated bilaplacian operator has never been seriously tested. No warranties that it is neither free of bugs or correctly formulated. 
-Moreover, the stability range of such an operator will be probably quite narrow, requiring a significantly smaller time-step than the one used on unrotated operator.
+The lateral fourth order operator formulation on tracers is obtained by 
+applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption 
+on boundary conditions: first and third derivative terms normal to the 
+coast, the bottom and the surface are set to zero.
+
+It is used when, in addition to \np{ln\_traldf\_bilap}=T, we have 
+\np{ln\_traldf\_iso}=T, or both \np{ln\_traldf\_hor}=T and \np{ln\_zco}=T. 
+Nevertheless, this rotated bilaplacian operator has never been seriously 
+tested. No warranties that it is neither free of bugs or correctly formulated. 
+Moreover, the stability range of such an operator will be probably quite 
+narrow, requiring a significantly smaller time-step than the one used on 
+unrotated operator.
 
 % ================================================================
 % Tracer Vertical Diffusion
 % ================================================================
-\section{Tracer Vertical Diffusion (\mdl{trazdf})}
+\section  [Tracer Vertical Diffusion (\textit{trazdf})]
+		{Tracer Vertical Diffusion (\mdl{trazdf})}
 \label{TRA_zdf}
 %--------------------------------------------namzdf---------------------------------------------------------
@@ -470,5 +614,9 @@
 %--------------------------------------------------------------------------------------------------------------
 
-The formulation of the vertical subgrid scale tracer physics is the same for all the vertical coordinates, based on a laplacian operator. The vertical diffusive operator given by (\ref{Eq_PE_zdf}) takes the following semi-discrete space form:
+The formulation of the vertical subgrid scale tracer physics is the same 
+for all the vertical coordinates, and is based on a laplacian operator. 
+The vertical diffusion operator given by (\ref{Eq_PE_zdf}) takes the 
+following semi-discrete space form:
+(\ref{Eq_PE_zdf}) takes the following semi-discrete space form:
 \begin{equation} \label{Eq_tra_zdf}
 \begin{split}
@@ -482,13 +630,28 @@
 
 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity 
-coefficients on Temperature and Salinity, respectively. Generally, $A_w^{vT}=A_w^{vS}$ ecept when double diffusion mixing is parameterised (\key{zdfddm} defined). The way these coefficients can be evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when iso-neutral mixing is used, the both mixing coefficient are increased by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. 
-
-At the surface and bottom boundaries, the turbulent fluxes of momentum, heat and salt must be specified. At the surface they are prescribed from the surface forcing (see \S\ref{TRA_sbc}), while at the bottom they are set to zero for heat and salt, unless a geothermal flux forcing is prescribed as a bottom boundary condition (\S\ref{TRA_bbc}). 
+coefficients on Temperature and Salinity, respectively. Generally, 
+$A_w^{vT}=A_w^{vS}$ except when double diffusive mixing is 
+parameterised (\key{zdfddm} is defined). The way these coefficients 
+are evaluated is given in \S\ref{ZDF} (ZDF). Furthermore, when 
+iso-neutral mixing is used, both mixing coefficients are increased 
+by $\frac{e_{1w}\,e_{2w} }{e_{3w} }\ \left( {r_{1w} ^2+r_{2w} ^2} \right)$ 
+to account for the vertical second derivative of \eqref{Eq_tra_ldf_iso}. 
+
+At the surface and bottom boundaries, the turbulent fluxes of 
+momentum, heat and salt must be specified. At the surface they 
+are prescribed from the surface forcing (see \S\ref{TRA_sbc}), 
+whilst at the bottom they are set to zero for heat and salt unless 
+a geothermal flux forcing is prescribed as a bottom boundary 
+condition (\S\ref{TRA_bbc}). 
 
 The large eddy coefficient found in the mixed layer together with high 
-vertical resolution implies a too restrictive constraint on the time step in 
-explicit time stepping case (\np{ln\_zdfexp}=True). Therefore, the default implicit time stepping is generally preferred for the vertical diffusion as it overcomes the stability 
-constraint. A forward time differencing scheme (\np{ln\_zdfexp}=T) using a time splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. Namelist variables 
-\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 
+vertical resolution implies that in the case of explicit time stepping 
+(\np{ln\_zdfexp}=.true.) there would be too restrictive a constraint on 
+the time step. Therefore, the default implicit time stepping is preferred 
+for the vertical diffusion since it overcomes the stability constraint. 
+A forward time differencing scheme (\np{ln\_zdfexp}=.true.) using a time 
+splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. 
+Namelist variables \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both 
+tracers and dynamics. 
 
 % ================================================================
@@ -501,5 +664,6 @@
 %        surface boundary condition
 % -------------------------------------------------------------------------------------------------------------
-\subsection{surface boundary condition (\mdl{trasbc})}
+\subsection   [Surface boundary condition (\textit{trasbc})]
+			{Surface boundary condition (\mdl{trasbc})}
 \label{TRA_sbc}
 
@@ -510,29 +674,33 @@
 equivalent; the forcing terms in trasbc are the surface fluxes divided by 
 the thickness of the top model layer. Following \citet{Roullet2000} the 
-forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}^C $, the 
+forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}$, the 
 flux of tracer crossing the sea surface and not linked with the water 
-exchange d at the surface with the atmosphere, and $F_{wf}^C$ the forcing 
-on the concentration associated with the water flux. The latter forcing has 
-also two components: a direct effect of change in concentration associated 
-with the tracer carried by the water flux, and an indirect concentration/dilution effect :
+exchange with the atmosphere, $F_{wf}^d$, and $F_{wf}^i$ the forcing 
+on the tracer concentration associated with this water exchange. 
+The latter forcing has two components: a direct effect of change 
+in concentration associated with the tracer carried by the water flux, 
+and an indirect concentration/dilution effect :
 \begin{equation*}
 \begin{split}
  F^C &= F_{ext} + F_{wf}^d                                          +F_{wf}^i  	\\
-\\ 
         &= F_{ext} - \left( c_E \, E - c_p \,P - c_R \,R \right) +c\left( E-P-R \right)
 \end{split}
 \end{equation*} 
 
-Two cases must be distinguished, the nonlinear free surface case (\key{vvl} defined) and the linear free surface case. The first case is simpler, because the indirect concentration/dilution effect is naturally taken into 
-account by letting the vertical scale factors vary in time. The salinity of 
-water exchanged at the surface is assumed to be zero, so there is no salt 
-flux at the free surface, excepted in the presence of sea ice. The heat flux 
-at the free surface is the sum of $F_{ext}$, the direct heating/cooling 
-(by the total non-penetrative heat flux) and $F_{wf}^e$ the heat carried by 
-the water exchanged through the surface (evaporation, precipitation, 
-runoff). The temperature of precipitations is not well known. In the model 
-we assume that this water has the same temperature as the sea surface 
-temperature, The resulting forcing terms for temperature T and salinity S 
-are: 
+\gmcomment{add here a description of the variable names used in the above equation}
+
+Two cases must be distinguished, the nonlinear free surface case 
+(\key{vvl} is defined) and the linear free surface case. The first case 
+is simpler, because the indirect concentration/dilution effect is naturally 
+taken into account by letting the vertical scale factors vary in time. 
+The salinity of water exchanged at the surface is assumed to be zero, 
+so there is no salt flux at the free surface, except in the presence of sea ice. 
+The heat flux at the free surface is the sum of $F_{ext}$, the direct 
+heating/cooling (by the total non-penetrative heat flux) and $F_{wf}^e$ 
+the heat carried by the water exchanged through the surface (evaporation, 
+precipitation, runoff). The temperature of precipitation is not well known. 
+In the model we assume that this water has the same temperature as 
+the sea surface temperature. The resulting forcing terms for temperature 
+T and salinity S are: 
 \begin{equation} \label{Eq_tra_forcing}
 \begin{aligned}
@@ -544,9 +712,20 @@
 \end{equation} 
 
-where EMP is the freshwater budget (evaporation minus precipitation minus river runoff) which forces the ocean volume, $Q_{ns}$ is the non-penetrative part of the net surface heat flux (difference between the total surface heat flux and the fraction of the short wave flux that penetrates in the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ is  the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface salinity (\textit{SSS}). The total salt content is conserved in this formulation (excepted for the effect of the Asselin filter).
-%AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, 
-%AMT  has this been corrected in the code? 
-
-In the second case (linear free surface), the vertical scale factors are fixed in time so that the concentration/dilution effect must be added in trasbc. Because of the hypothesis made for the temperature of precipitation and runoffs, for temperature $F_{wf}^e +F_{wf}^i =0$. The resulting forcing term for temperature is: 
+where EMP is the freshwater budget (evaporation minus precipitation 
+minus river runoff) which forces the ocean volume, $Q_{ns}$ is the 
+non-penetrative part of the net surface heat flux (difference between 
+the total surface heat flux and the fraction of the short wave flux that 
+penetrates into the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ 
+is  the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface 
+salinity (\textit{SSS}). The total salt content is conserved in this formulation 
+(except for the effect of the Asselin filter).
+
+%AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, has this been corrected in the code? 
+
+In the second case (linear free surface), the vertical scale factors are 
+fixed in time so that the concentration/dilution effect must be added in 
+the \mdl{trasbc} module. Because of the hypothesis made for the 
+temperature of precipitation and runoffs, $F_{wf}^e +F_{wf}^i =0$ 
+for temperature. The resulting forcing term for temperature is: 
 
 \begin{equation} \label{Eq_tra_forcing_q}
@@ -554,32 +733,42 @@
 \end{equation} 
 
-The salinity forcing is still given by \eqref{Eq_tra_forcing} but the definition of EMP$_S$ is 
-different: it is the total surface freshwater budget (evaporation minus 
-precipitation minus river runoff plus the rate of change of the sea ice 
-thickness). The total salt content is not exactly conserved (\citet{Roullet2000}, see also 
-\S\ref{PE_free_surface}).
+The salinity forcing is still given by \eqref{Eq_tra_forcing} but the 
+definition of EMP$_S$ is different: it is the total surface freshwater 
+budget (evaporation minus precipitation minus river runoff plus 
+the rate of change of the sea ice thickness). The total salt content 
+is not exactly conserved (\citet{Roullet2000}. 
+See also \S\ref{PE_free_surface}).
 
 In the case of the rigid lid approximation, the surface salinity forcing $F^s$ 
-is also expressed by \eqref{Eq_tra_forcing} but now the global integral of the product 
-EMP*S is not compensated by the advection of fluid through the top level: in 
-the rigid lid case (contrary to the linear free surface), because \textit{w(k=1) = 0}. As a 
-result, even if the budget of \textit{EMP} is zero in average over the whole ocean 
-domain, the associated salt flux is not, as sea-surface salinity and \textit{EMP} are 
-intrinsically correlated (high \textit{SSS} are found where evaporation is strong while 
-low \textit{SSS} is usually associated with high precipitation or river runoff input).
-
-The $Q_{ns} $ and \textit{EMP} fields are defined and updated in \mdl{sbcmod} module (see 
-\S\ref{SBC}).
+is also expressed by \eqref{Eq_tra_forcing}, but now the global integral of 
+the product of EMP and S, is not compensated by the advection of fluid 
+through the top level: this is because in the rigid lid case \textit{w(k=1) = 0} 
+(in contrast to the linear free surface case). As a result, even if the budget 
+of \textit{EMP} is zero on average over the whole ocean domain, the 
+associated salt flux is not, since sea-surface salinity and \textit{EMP} are 
+intrinsically correlated (high \textit{SSS} are found where evaporation is 
+strong whilst low \textit{SSS} is usually associated with high precipitation 
+or river runoff).
+
+The $Q_{ns} $ and \textit{EMP} fields are defined and updated in the 
+\mdl{sbcmod} module (see \S\ref{SBC}).
 
 % -------------------------------------------------------------------------------------------------------------
 %        Solar Radiation Penetration 
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Solar Radiation Penetration (\mdl{traqsr})}
+\subsection   [Solar Radiation Penetration (\textit{traqsr})]
+			{Solar Radiation Penetration (\mdl{traqsr})}
 \label{TRA_qsr}
-%--------------------------------------------namqsr---------------------------------------------------------
+%--------------------------------------------namqsr--------------------------------------------------------
 \namdisplay{namqsr}
 %--------------------------------------------------------------------------------------------------------------
 
-When the penetrative solar radiation option is used (\np{ln\_flxqsr}=T, the solar radiation penetrates the top few meters of the ocean, otherwise all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=F). A term is thus added to the time evolution equation of temperature \eqref{Eq_PE_tra_T} while the surface boundary condition is modified to take into account only the non-penetrative part of the surface heat flux:
+When the penetrative solar radiation option is used (\np{ln\_flxqsr}=.true.), 
+the solar radiation penetrates the top few meters of the ocean, otherwise 
+all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=.false.). 
+Thus, in the former case a term is added to the time evolution equation of 
+temperature \eqref{Eq_PE_tra_T} whilst the surface boundary condition is 
+modified to take into account only the non-penetrative part of the surface 
+heat flux:
 \begin{equation} \label{Eq_PE_qsr}
 \begin{split}
@@ -594,23 +783,50 @@
 \end{equation}
 
-A formulation including extinction coefficients is assumed for the downward irradiance $I$ 
-\citep{Paulson1977}:
+A formulation involving two extinction coefficients is assumed for the 
+downward irradiance $I$ \citep{Paulson1977}:
 \begin{equation} \label{Eq_traqsr_iradiance}
 I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right]
 \end{equation}
 where $Q_{sr}$ is the penetrative part of the surface heat flux, 
-$\xi_1$ and $\xi_2$ are two extinction length scales and $R$ determines the relative 
-contribution of the two terms. The default values used correspond to a Type 
-I water in Jerlov's [1968] classification: $\xi_1 = 0.35m$, $\xi_2 = 0.23m$ and 
-$R = 0.58$ ((corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, respectively). $I$ is masked (no flux through the ocean bottom), so all the solar radiation that reaches the last ocean level is absorbed in that level. The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation is added to the temperature trend and the surface heat flux modified in routine \mdl{traqsr}. Note that in $z$-coordinates, the depth of $T-$levels depends on the single variable $k$. A one dimensional array of the coefficients $gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then be computed once and saved in central memory. Moreover \textit{nksr}, the level at which $gdrs$ becomes negligible (less than the computer precision) is computed once and the trend associated with the penetration of the solar radiation is only added until that level. At last, note that when the ocean is shallow (< 200~m), the part of the solar radiation can reach the ocean floor. In this case, we have chosen that all the radiation is absorbed at the last ocean level ($i.e.$ $I_w$ is masked). 
-
-When coupling with a biology model (PISCES or LOBSTER), it is possible to calculate the light attenuation using information from the biology model. At the time of this writing, reading the light attenuation from a file is not implemented yet in the reference version.
-
-\colorbox{yellow}{case 4 bands and bio-coupling to add !!!}
+$\xi_1$ and $\xi_2$ are two extinction length scales and $R$ 
+determines the relative contribution of the two terms. 
+The default values used correspond to a Type I water in Jerlov's [1968] 
+%
+\gmcomment : Jerlov reference to be added
+%
+classification: $\xi_1 = 0.35m$, $\xi_2 = 0.23m$ and $R = 0.58$ 
+(corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, 
+respectively). $I$ is masked (no flux through the ocean bottom), 
+so all the solar radiation that reaches the last ocean level is absorbed 
+in that level. The trend in \eqref{Eq_tra_qsr} associated with the 
+penetration of the solar radiation is added to the temperature trend, 
+and the surface heat flux is modified in routine \mdl{traqsr}. 
+Note that in the $z$-coordinate, the depth of $T-$levels depends 
+on the single variable $k$. A one dimensional array of the coefficients 
+$gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then 
+be computed once and saved in memory. Moreover \textit{nksr}, 
+the level at which $gdrs$ becomes negligible (less than the 
+computer precision) is computed once, and the trend associated 
+with the penetration of the solar radiation is only added until that level. 
+Finally, note that when the ocean is shallow (< 200~m), part of the 
+solar radiation can reach the ocean floor. In this case, we have 
+chosen that all remaining radiation is absorbed in the last ocean 
+level ($i.e.$ $I_w$ is masked). 
+
+When coupling with a biological model (for example PISCES or LOBSTER), 
+it is possible to calculate the light attenuation using information from 
+the biology model. Without biological model, it is still possible to introduce 
+a horizontal variation of the light attenuation by using the observed ocean 
+surface color. At the time of writing, the latter has not been implemented
+ in the reference version.
+%
+\gmcomment{  {yellow}{case 4 bands and bio-coupling to add !!!}  }
+%
 
 % -------------------------------------------------------------------------------------------------------------
 %        Bottom Boundary Condition
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Bottom Boundary Condition (\mdl{trabbc} + \key{bbc})}
+\subsection   [Bottom Boundary Condition (\textit{trabbc} - \key{bbc})]
+			{Bottom Boundary Condition (\mdl{trabbc} - \key{bbc})}
 \label{TRA_bbc}
 %--------------------------------------------nambbc--------------------------------------------------------
@@ -620,31 +836,33 @@
 \begin{figure}[!t] \label{Fig_geothermal}  \begin{center}
 \includegraphics[width=1.0\textwidth]{./Figures/Fig_TRA_geoth.pdf}
-\caption{Geothermal Heat flux (in $mW.m^{-2}$) as inferred from the age of the sea floor and the formulae of \citet{Stein1992}.}
+\caption{Geothermal Heat flux (in $mW.m^{-2}$) as inferred from the age 
+of the sea floor and the formulae of \citet{Stein1992}.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-Usually it is considered that there is no exchange of heat nor salt through 
-the ocean bottom, i.e. a no flux boundary condition is applied on active 
-tracers at the bottom. This is the default option in NEMO, and it is 
-implemented using the masking technique. Nevertheless, there exists a 
-non-zero heat flux across the seafloor that is associated with the solid 
-earth cooling. This flux is weak compared with surface fluxes --- a mean 
-global value of $\sim0.1\;W/m^2$ \citep{Stein1992} --- but it is 
-systematically positive and it acts only on the densest water masses. Taking 
-this flux into account in a global ocean model increases by a few Sverdrups 
+Usually it is assumed that there is no exchange of heat or salt through 
+the ocean bottom, $i.e.$ a no flux boundary condition is applied on active 
+tracers at the bottom. This is the default option in \NEMO, and it is 
+implemented using the masking technique. Hoever, there is a 
+non-zero heat flux across the seafloor that is associated with solid 
+earth cooling. This flux is weak compared to surface fluxes (a mean 
+global value of $\sim0.1\;W/m^2$ \citep{Stein1992}), but it is 
+systematically positive and acts on the densest water masses. Taking 
+this flux into account in a global ocean model increases
 the deepest overturning cell (i.e. the one associated with the Antarctic 
-Bottom Water). 
-
-The presence or not of a geothermal heating is controlled by the namelist 
-parameter  \np{ngeo\_flux}. Set to 1, a constant geothermal heatingis 
-introducted which value is given by the \np{ngeo\_flux\_const}, also a 
-namelist parameter. Set to 2, a spatially varying geothermal heat flux is 
-introducted which is provided in the geothermal\_heating.nc NetCDF 
-file (Fig.\ref{Fig_geothermal}).
+Bottom Water) by a few Sverdrups. 
+
+The presence or not of geothermal heating is controlled by the namelist 
+parameter  \np{ngeo\_flux}. If this parameter is set to 1, a constant 
+geothermal heating is introduced whose value is given by the 
+\np{ngeo\_flux\_const}, which is also a namelist parameter. If it is set to 2, 
+a spatially varying geothermal heat flux is introduced which is provided 
+in the geothermal\_heating.nc NetCDF file (Fig.\ref{Fig_geothermal}).
 
 % ================================================================
 % Bottom Boundary Layer
 % ================================================================
-\section{Bottom Boundary Layer (\mdl{trabbl} + \key{bbl\_diff} or \key{bbl\_adv})}
+\section  [Bottom Boundary Layer (\textit{trabbl}, \textit{trabbl\_adv} )]
+		{Bottom Boundary Layer (\mdl{trabbl}, \mdl{trabbl\_adv})}
 \label{TRA_bbl}
 %--------------------------------------------nambbl---------------------------------------------------------
@@ -652,62 +870,100 @@
 %--------------------------------------------------------------------------------------------------------------
 
-In z-coordinate configuration, the bottom topography is represented as a 
+In a $z$-coordinate configuration, the bottom topography is represented by a 
 series of discrete steps. This is not adequate to represent gravity driven 
 downslope flows. Such flows arise downstream of sills such as the Strait of 
 Gibraltar, Bab El Mandeb, or Denmark Strait, where dense water formed in 
 marginal seas flows into a basin filled with less dense water. The amount of 
-entrainment that occurs in those gravity plumes is critical to determine the 
-density and volume flux of the densest waters of the ocean, such as the 
-Antarctic Bottom water, or the North Atlantic Deep Water. $z$-coordinate 
-models tend to overestimate the entrainment because the gravity flow is 
+entrainment that occurs in these gravity plumes is critical in determining the 
+density and volume flux of the densest waters of the ocean, such as 
+Antarctic Bottom Water, or North Atlantic Deep Water. $z$-coordinate 
+models tend to overestimate the entrainment, because the gravity flow is 
 mixed down vertically by convection as it goes ``downstairs'' following the 
 step topography, sometimes over a thickness much larger than the thickness 
-of the observed gravity plume. A similar problem occurs in $s$-coordinate when 
+of the observed gravity plume. A similar problem occurs in the $s$-coordinate when 
 the thickness of the bottom level varies in large proportions downstream of 
-a sill \citep{Willebrand2001}, and the thickness of the plume is not 
-resolved. 
-
-The idea of the bottom boundary layer parameterization first introduced by 
+a sill \citep{Willebrand2001}, and the thickness of the plume is not resolved. 
+
+The idea of the bottom boundary layer (BBL) parameterization first introduced by 
 \citet{BeckDos1998} is to allow a direct communication between 
-two adjacent bottom cells at varying level, whenever the densest water is 
-located above the less dense water. The communication can be by diffusive 
-fluxes (diffusive BBL), advective fluxes (advective BBL) or both. Only 
-tracers are modified, not the velocities. Implementing a BBL 
-parameterization for momentum is a more complex problem because of the 
-pressure gradient errors. 
+two adjacent bottom cells at different levels, whenever the densest water is 
+located above the less dense water. The communication can be by a diffusive 
+(diffusive BBL), advective fluxes (advective BBL), or both. In the current 
+implementation of the BBL, only the tracers are modified, not the velocities. 
+Furthermore, it only connects ocean bottom cells, and therefore does not include 
+the improvment proposed by \citet{Campin_Goosse_Tel99}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Diffusive BBL
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Diffusive Bottom Boundary layer (\mdl{trabbl})}
+\subsection{Diffusive Bottom Boundary layer (\key{bbl\_diff})}
 \label{TRA_bbl_diff}
 
-The lateral diffusivity $A_l^\sigma $ in the BBL can be prescribed with a 
-spatial dependence, e.g., in the conditional form
-\begin{equation} \label{Eq_tra_bbl}
+When applying sigma-diffusion (\key{trabbl} is defined), the diffusive flux between 
+two adjacent cells living at the ocean bottom is given by 
+\begin{equation} \label{Eq_tra_bbl_diff}
+{\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
+\end{equation} 
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
+and  $A_l^\sigma $ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
+the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
+\begin{equation} \label{Eq_tra_bbl_coef}
 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
- \mbox{large}\quad if\;\nabla \rho \cdot \nabla H<0 \\ 
+ A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ 
  \\
  0\quad \quad \;\,\mbox{otherwise} \\ 
  \end{array}} \right.
 \end{equation} 
-
-The large value of the coefficient when the diffusive BBL is active is given 
-by the namelist parameter \np{atrbbl.}
+where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 
+parameter \np{atrbbl}. $A_{bbl}$ is usually set to a value much larger 
+than the one used on lateral mixing in open ocean. 
+Note that in practice, \eqref{Eq_tra_bbl_coef} constraint is applied 
+separately in the two horizontal directions, and the density gradient in 
+\eqref{Eq_tra_bbl_coef} is evaluated at $\overline{H}^i$ ($\overline{H}^j$) 
+using the along bottom mean temperature and salinity. 
 
 % -------------------------------------------------------------------------------------------------------------
 %        Advective BBL
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Advective Bottom Boundary Layer (\mdl{trabb\_adv})}
+\subsection   {Advective Bottom Boundary Layer (\key{bbl\_adv})}
 \label{TRA_bbl_adv}
 
-Implemented in NEMO v2. 
-
-\colorbox{yellow} {Documentation to be added here }
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_bbl}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./Figures/Fig_BBL_adv.pdf}
+\caption{Advective Bottom Boundary Layer.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%%%gmcomment   :  this section has to be really written
+
+The advective BBL is in fact not only an advective one but include a diffusive 
+component as we chose an upstream scheme to perform the advection within 
+the BBL. The associated diffusion only act in the stream direction and is 
+proportional to the velocity.
+
+When applying sigma-advection (\key{trabbl\_adv} defined), the advective 
+flux between two adjacent cells living at the ocean bottom is given by 
+\begin{equation} \label{Eq_tra_bbl_fadv}
+{\rm {\bf F}}_\sigma={\rm {\bf U}}_h^\sigma \; \overline{T}^\sigma
+\end{equation} 
+with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
+and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
+the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
+\begin{equation} \label{Eq_tra_bbl_Aadv}
+A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
+ A_{bbl} \quad \quad \mbox{if}     \quad    \nabla_\sigma \rho \cdot \nabla H<0 
+              \quad \quad \mbox{and} \quad         {\rm {\bf U}}_h  \cdot \nabla H<0 \\ 
+ \\
+ 0\quad \quad \;\,\mbox{otherwise} \\ 
+ \end{array}} \right.
+\end{equation} 
 
 % ================================================================
 % Tracer damping
 % ================================================================
-\section{Tracer damping (\mdl{tradmp})}
+\section  [Tracer damping (\textit{tradmp})]
+		{Tracer damping (\mdl{tradmp})}
 \label{TRA_dmp}
 %--------------------------------------------namdmp-----------------------------------------------------
@@ -715,6 +971,6 @@
 %--------------------------------------------------------------------------------------------------------------
 
-In some applications it can be useful to add a Newtonian damping term in the 
-temperature and salinity equations:
+In some applications it can be useful to add a Newtonian damping term 
+into the temperature and salinity equations:
 \begin{equation} \label{Eq_tra_dmp}
 \begin{split}
@@ -724,37 +980,55 @@
  \end{split}
  \end{equation} 
-where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields (usually a climatology). The restoring term is added when \key{tradmp} is defined. It also requires that both \key{temdta} and \key{saldta} are defined ($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient $S_o$ is a three-dimensional array initialized by the user in 
-\rou{dtacof} routine also located in  \mdl{tradmp}. 
-
-The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} the specification of 
-the boundary conditions along artificial walls of a limited domain basin and 
-\textit{(b)} the computation of the velocity field associated with a given $T$-$S$ field 
-(for example to build the initial state of a prognostic simulation, or to 
-use the resulting velocity field for a passive tracer study). The first case 
-applies to regional models that have artificial walls instead of open 
-boundaries. In the vicinity of these walls, $S_o$ takes large values 
-(equivalent to a few day time scale) whereas it is zero in the interior of 
-the model domain. The second case corresponds to the use of the robust 
-diagnostic method \citep{Sarmiento1982}. It allows to find the velocity 
-field consistent with the model dynamics while having a $T$-$S$ field close to a 
-given climatology field ($T_o -S_o$). The time scale associated with 
-$S_o$ is generally not a constant but spatially varying in order to respect 
-some considerations. For example, it is usually set to zero in the mixed 
-layer (defined either on a density or $S_o$ criterion) \citep{Madec1996} and in the equatorial region \citep{Reverdin1991, Fujio1991, MartiTh1992} as those two regions have a small time scale of adjustment, 
-while smaller $S_o$ are used in the deep ocean where the typical time scale 
-is long \citep{Sarmiento1982}. In addition it is reduced (and even 
-zero) along the western boundary to allow the model to reconstruct its own 
-western boundary structure in equilibrium with its physics. The choice of a 
-Newtonian damping acting in the mixed layer or not is controlled by \np{nmldmp} 
-(\textbf{namelist} \np{nmldmp}parameter).
-
-The robust diagnostic method is very efficient to prevent the temperature drift in intermediate waters but it produces artificial sources of heat and salt within the ocean. It has also undesirable effects on the ocean convection. It tends to prevent deep convection and subsequent deep-water formation by stabilising too much the water columns.
-
-An example of computation of $S_o$ for robust diagnostic experiments with the ORCA2 model is provided in the \mdl{tradmp} module (subroutines \rou{dtacof} and \rou{cofdis} which compute coefficient and the distance to the bathymetry, respectively). Those routines are provided as examples and can be customised by the user. 
+where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ 
+are given temperature and salinity fields (usually a climatology). 
+The restoring term is added when \key{tradmp} is defined. 
+It also requires that both \key{temdta} and \key{saldta} are defined 
+($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient 
+$S_o$ is a three-dimensional array initialized by the user in routine 
+\rou{dtacof} also located in module \mdl{tradmp}. 
+
+The two main cases in which \eqref{Eq_tra_dmp} is used are \textit{(a)} 
+the specification of the boundary conditions along artificial walls of a 
+limited domain basin and \textit{(b)} the computation of the velocity 
+field associated with a given $T$-$S$ field (for example to build the 
+initial state of a prognostic simulation, or to use the resulting velocity 
+field for a passive tracer study). The first case applies to regional 
+models that have artificial walls instead of open boundaries. 
+In the vicinity of these walls, $S_o$ takes large values (equivalent to 
+a time scale of a few days) whereas it is zero in the interior of the 
+model domain. The second case corresponds to the use of the robust 
+diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity 
+field consistent with the model dynamics whilst having a $T$-$S$ field 
+close to a given climatological field ($T_o -S_o$). The time scale 
+associated with $S_o$ is generally not a constant but spatially varying 
+in order to respect other properties. For example, it is usually set to zero 
+in the mixed layer (defined either on a density or $S_o$ criterion) 
+\citep{Madec1996} and in the equatorial region 
+\citep{Reverdin1991, Fujio1991, MartiTh1992} since these two regions 
+have a short time scale of adjustment; while smaller $S_o$ are used 
+in the deep ocean where the typical time scale is long \citep{Sarmiento1982}. 
+In addition the time scale is reduced (even to zero) along the western 
+boundary to allow the model to reconstruct its own western boundary 
+structure in equilibrium with its physics. The choice of a 
+Newtonian damping acting in the mixed layer or not is controlled by
+namelist parameter \np{nmldmp}.
+
+The robust diagnostic method is very efficient in preventing temperature 
+drift in intermediate waters but it produces artificial sources of heat and salt 
+within the ocean. It also has undesirable effects on the ocean convection. 
+It tends to prevent deep convection and subsequent deep-water formation, 
+by stabilising the water column too much.
+
+An example of the computation of $S_o$ for robust diagnostic experiments 
+with the ORCA2 model is provided in the \mdl{tradmp} module 
+(subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient 
+and the distance to the bathymetry, respectively). These routines are 
+provided as examples and can be customised by the user. 
 
 % ================================================================
 % Tracer time evolution
 % ================================================================
-\section{Tracer time evolution (\mdl{tranxt})}
+\section  [Tracer time evolution (\textit{tranxt})]
+		{Tracer time evolution (\mdl{tranxt})}
 \label{TRA_nxt}
 %--------------------------------------------namdom-----------------------------------------------------
@@ -762,5 +1036,7 @@
 %--------------------------------------------------------------------------------------------------------------
 
-The general framework of dynamics time stepping is a leap-frog scheme, $i.e.$ a three level centred time scheme associated with a Asselin time filter (cf. \S\ref{DOM_nxt}):
+The general framework for tracer time stepping is a leap-frog scheme, 
+$i.e.$ a three level centred time scheme associated with a Asselin time 
+filter (cf. \S\ref{DOM_nxt}):
 \begin{equation} \label{Eq_tra_nxt}
 \begin{split}
@@ -771,17 +1047,21 @@
 \end{equation} 
 
-where $\text{RHS}_T$ is the right hand side of the temperature equation, the subscript $f$ denotes 
-filtered values and $\gamma$ is the Asselin coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). Its default value is \np{atfp=0.1}. 
-
-When the vertical mixing is solved implicitly, the update of the next tracer 
-fields is done in module \mdl{trazdf}. In that case only the swap of arrays 
-and the Asselin filtering is done in \mdl{tranxt} module.
-
-In order to prepare the computation of the next time step, a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. 
+where $\text{RHS}_T$ is the right hand side of the temperature equation, 
+the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
+coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). 
+Its default value is \np{atfp=0.1}. 
+
+When the vertical mixing is solved implicitly, the update of the \textit{next} tracer 
+fields is done in module \mdl{trazdf}. In this case only the swapping of arrays 
+and the Asselin filtering is done in the \mdl{tranxt} module.
+
+In order to prepare for the computation of the \textit{next} time step, 
+a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. 
 
 % ================================================================
 % Equation of State (eosbn2) 
 % ================================================================
-\section{Equation of State (\mdl{eosbn2}) }
+\section  [Equation of State (\textit{eosbn2}) ]
+		{Equation of State (\mdl{eosbn2}) }
 \label{TRA_eosbn2}
 %--------------------------------------------nameos-----------------------------------------------------
@@ -795,10 +1075,39 @@
 \label{TRA_eos}
 
-It is necessary to know the equation of state for the ocean very accurately to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), particularly in the deep ocean. The ocean density is a non linear empirical function of \textit{in situ }temperature, salinity and pressure. The reference is the equation of state defined by the Joint Panel on Oceanographic Tables 
-and Standards \citep{UNESCO1983}. It was the standard equation of state used in early releases of OPA. Even though this computation is fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total CPU time) as it requires the prior computation of the \textit{in situ} temperature from the model \textit{potential} temperature using the \citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order Runge-Kutta integration scheme. Since OPA6, we have chosen the \citet{JackMcD1995} equation of state for seawater. It allows the computation of the \textit{in situ} ocean density directly as a function of \textit{potential} temperature relative to the sea surface (an OPA variable), the practical salinity (another OPA variable) and the pressure (assuming no pressure variation along geopotential surfaces, i.e. the pressure in decibars is 
-approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have the same expression except that the values of the various coefficients have been adjusted by \citet{JackMcD1995} in order to use directly the \textit{potential} temperature instead of the \textit{in situ} one. This reduces the CPU time of the in situ density computation to about $3${\%} of the total CPU time, while maintaining a quite accurate equation of state.
-
-In the computer code, a \textit{true} density, $d$, is computed, i.e. the ratio of seawater volumic mass over $\rho_o$, a reference volumic mass (\textit{rau0} defined in \mdl{phycst}, usually $rau0= 1,020~Kg/m^3$). The default option (\np{neos}=0) is the \citet{JackMcD1995} equation of state. It is highly recommended to use it. Nevertheless, for process studies, it is often convenient to use a linear approximation of the density$^{\ast}$\footnote{$^{\ast }$ With the linear equation of state there is no longer 
-a distinction between \textit{in situ} and \textit{potential} density. Cabling and thermobaric effects are also removed.}. Two linear formulations are available: a function of $T$ only (\np{neos}=1) and a function of both $T$ and $S$ (\np{neos}=2):
+It is necessary to know the equation of state for the ocean very accurately 
+to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), 
+particularly in the deep ocean. The ocean density is a non linear empirical 
+function of \textit{in situ }temperature, salinity and pressure. The reference 
+equation of state is that defined by the Joint Panel on Oceanographic Tables 
+and Standards \citep{UNESCO1983}. It was the standard equation of state 
+used in early releases of OPA. However, even though this computation is 
+fully vectorised, it is quite time consuming ($15$ to $20${\%} of the total 
+CPU time) since it requires the prior computation of the \textit{in situ} 
+temperature from the model \textit{potential} temperature using the 
+\citep{Bryden1973} polynomial for adiabatic lapse rate and a $4^th$ order 
+Runge-Kutta integration scheme. Since OPA6, we have used the 
+\citet{JackMcD1995} equation of state for seawater instead. It allows the 
+computation of the \textit{in situ} ocean density directly as a function of 
+\textit{potential} temperature relative to the surface (an \NEMO variable), 
+the practical salinity (another \NEMO variable) and the pressure (assuming no 
+pressure variation along geopotential surfaces, i.e. the pressure in decibars is 
+approximated by the depth in meters). Both the \citet{UNESCO1983} and \citet{JackMcD1995} equations of state have exactly the same except that 
+the values of the various coefficients have been adjusted by \citet{JackMcD1995} 
+in order to directly use the \textit{potential} temperature instead of the 
+\textit{in situ} one. This reduces the CPU time of the in situ density computation 
+to about $3${\%} of the total CPU time, while maintaining a quite accurate 
+equation of state.
+
+In the computer code, a \textit{true} density $d$ is computed, $i.e.$ the ratio 
+of seawater volumic mass to $\rho_o$, a reference volumic mass (\textit{rau0} 
+defined in \mdl{phycst}, usually $rau0= 1,020~Kg/m^3$). The default option 
+(namelist prameter \np{neos}=0) is the \citet{JackMcD1995} equation of state. 
+Its use is highly recommended. However, for process studies, it is often 
+convenient to use a linear approximation of the density$^{\ast}$
+\footnote{$^{\ast }$ With the linear equation of state there is no longer 
+a distinction between \textit{in situ} and \textit{potential} density. Cabling 
+and thermobaric effects are also removed.}. 
+Two linear formulations are available: a function of $T$ only (\np{neos}=1) 
+and a function of both $T$ and $S$ (\np{neos}=2):
 \begin{equation} \label{Eq_tra_eos_linear}
 \begin{aligned}
@@ -807,6 +1116,10 @@
 \end{aligned}
 \end{equation} 
-where $\alpha $ and $\beta $ are the thermal and haline expansion coefficients, and $\rho_o$, the reference volumic mass, $rau0$. $\alpha $ and $\beta$ can be modified through \np{ralpha} and \np{rbeta} namelist parameters). Note that when $d$ is a function of $T$ only (\np{neos}=1), the salinity is a passive tracer and can be used as such.
-
+where $\alpha$ and $\beta$ are the thermal and haline expansion 
+coefficients, and $\rho_o$, the reference volumic mass, $rau0$. 
+($\alpha$ and $\beta$ can be modified through the \np{ralpha} and 
+\np{rbeta} namelist parameters). Note that when $d$ is a function 
+of $T$ only (\np{neos}=1), the salinity is a passive tracer and can be 
+used as such.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -816,36 +1129,63 @@
 \label{TRA_bn2}
 
-An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a} frequency) is of paramount importance as it is used in several ocean parameterisations (namely TKE, KPP, Richardson number dependent vertical diffusion, enhanced vertical diffusion, non-penetrative convection, iso-neutral diffusion). In particular, one must be aware that $N^2$ has to be computed with an \textit{in situ} reference. The expression of $N^2$ depends on the type of equation of state used (\np{neos} namelist parameter).
+An accurate computation of the ocean stability (i.e. of $N$, the brunt-Vais\"{a}l\"{a}
+ frequency) is of paramount importance as it is used in several ocean 
+ parameterisations (namely TKE, KPP, Richardson number dependent 
+ vertical diffusion, enhanced vertical diffusion, non-penetrative convection, 
+ iso-neutral diffusion). In particular, one must be aware that $N^2$ has to 
+ be computed with an \textit{in situ} reference. The expression for $N^2$ 
+ depends on the type of equation of state used (\np{neos} namelist parameter).
 
 For \np{neos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 
-polynomial expression is used with the pressure in decibar approximated by 
-the depth in meters: 
-\begin{multline} \label{Eq_tra_bn2}
-N^2 =	\frac{g}{e_{3w}} \; \beta ( \overline{T}^{\,k+1/2},\widetilde{S},z_w )   \\
-		\left\{  \alpha / \beta ( \overline{T}^{\,k+1/2},\widetilde{S},z_w )
-			\ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]
-		 \right\}
-\end{multline} 
-where $T$ is the \textit{potential} temperature, $\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$ a salinity anomaly, and $\alpha$ ($\beta\,$) the thermal (haline) expansion coefficient. Both $\alpha$ and $\beta$ depend on \textit{potential} temperature, salinity which are averaged at $w$-points prior to the computation.
-
-When a linear equation of state is used (\np{neos}=1 or 2, \eqref{Eq_tra_bn2} reduces to:
+polynomial expression is used (with the pressure in decibar approximated by 
+the depth in meters): 
+\begin{equation} \label{Eq_tra_bn2}
+N^2 = \frac{g}{e_{3w}} \; \beta   \ 
+	   \left(  \alpha / \beta \ \delta_{k+1/2}[T]     - \delta_{k+1/2}[S]   \right) 
+\end{equation} 
+where $\alpha$ ($\beta$) is the thermal (haline) expansion coefficient. 
+They are a function of  
+$\overline{T}^{\,k+1/2},\widetilde{S}=\overline{S}^{\,k+1/2} - 35.$, 
+and  $z_w$, with $T$ the \textit{potential} temperature and 
+$\widetilde{S}$ a salinity anomaly. 
+Note that both $\alpha$ and $\beta$ depend on \textit{potential} 
+temperature and salinity which are averaged at $w$-points prior 
+to the computation instead of being computed at $T$-points and 
+then averaged to $w$-points.
+
+When a linear equation of state is used (\np{neos}=1 or 2, 
+\eqref{Eq_tra_bn2} reduces to:
 \begin{equation} \label{Eq_tra_bn2_linear}
 N^2 =	\frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
 \end{equation} 
-where $\alpha$ and $\beta $ are the constant coefficients used to defined the linear 
-equation of state \eqref{Eq_tra_eos_linear}.
+where $\alpha$ and $\beta $ are the constant coefficients used to 
+defined the linear equation of state \eqref{Eq_tra_eos_linear}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Specific Heat
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Specific Heat (\textit{rcp}, \mdl{phycst})}
+\subsection   [Specific Heat (\textit{phycst})]
+			{Specific Heat (\mdl{phycst})}
 \label{TRA_adv_ldf}
 
-The specific heat of sea water, $C_p$, is a function of temperature, salinity and pressure \citep{UNESCO1983}. It is only used in the model to convert surface heat fluxes into surface temperature increase, thus the pressure dependence is neglected. The dependence on $T$ and $S$ is weak. For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. Its computer name is \textit{rcp} and its value is set in \mdl{phycst} module.
+The specific heat of sea water, $C_p$, is a function of temperature, salinity 
+and pressure \citep{UNESCO1983}. It is only used in the model to convert 
+surface heat fluxes into surface temperature increase and so the pressure 
+dependence is neglected. The dependence on $T$ and $S$ is weak. 
+For example, with $S=35~psu$, $C_p$ increases from $3989$ to $4002$ 
+when $T$ varies from -2~\degres C to 31~\degres C. Therefore, $C_p$ has 
+been chosen as a constant: $C_p=4.10^3~J\,Kg^{-1}\,\degres K^{-1}$. 
+Its value is set in \mdl{phycst} module. 
+
+%%%
+\gmcomment{ STEVEN:  consistency, no other computer variable names are 
+supplied, so why this one}
+%%%
 
 % -------------------------------------------------------------------------------------------------------------
 %        Freezing Point of Seawater
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Freezing Point of Seawater (\mdl{ocfzpt})}
+\subsection   [Freezing Point of Seawater (\textit{ocfzpt})]
+			{Freezing Point of Seawater (\mdl{ocfzpt})}
 \label{TRA_fzp}
 
@@ -859,18 +1199,41 @@
 \end{equation}
 
-\eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of sea water 
-($i.e.$ referenced to the surface $p=0$), thus the pressure dependent terms in \eqref{Eq_tra_eos_fzp} (last term) has been dropped. The \textit{before} and \textit{now} surface freezing point is introduced in the code as $fzptb$ and $fzptn$ 2D arrays together with a  \textit{now} mask (\textit{freezn}) which takes 0 or 1 whether the ocean temperature is above or at the freezing point. Caution: do not confuse \textit{freezn} with the fraction of lead (\textit{frld}) defined in LIM. 
+\eqref{Eq_tra_eos_fzp} is only used to compute the potential freezing point of 
+sea water ($i.e.$ referenced to the surface $p=0$), thus the pressure dependent 
+terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The \textit{before} 
+and \textit{now} surface freezing point is introduced in the code as $fzptb$ and 
+$fzptn$ 2D arrays together with a  \textit{now} mask (\textit{freezn}) which takes 
+the value 0 or 1 depending on whether the ocean temperature is above or at the 
+freezing point. Caution: do not confuse \textit{freezn} with the fraction of lead 
+(\textit{frld}) defined in LIM.  
+
+%%%
+\gmcomment{STEVEN: consistency, not many computer variable names are supplied, so why these    ===>  gm  I agree   this should evolve both here and in the code itself}
+%%%
 
 % ================================================================
 % Horizontal Derivative in zps-coordinate 
 % ================================================================
-\section{Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})}
+\section  [Horizontal Derivative in \textit{zps}-coordinate (\textit{zpshde})]
+		{Horizontal Derivative in \textit{zps}-coordinate (\mdl{zpshde})}
 \label{TRA_zpshde}
 
-With partial bottom cells (\np{ln\_zps}=T), tracers in horizontally adjacent cells generally live at different depths. Horizontal gradients of tracers are needed for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure gradient (\mdl{dynhpg} module). Before taking horizontal gradients between the tracers next to the bottom, a linear interpolation is used to approximate the deeper tracer as if it actually lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). For example on temperature in the i-direction, the needed interpolated temperature, $\widetilde{T}$, is:
+\gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"}
+
+With partial bottom cells (\np{ln\_zps}=.true.), in general, tracers in horizontally 
+adjacent cells live at different depths. Horizontal gradients of tracers are needed 
+for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure 
+gradient (\mdl{dynhpg} module) to be active. 
+\gmcomment{STEVEN from gm : question: not sure of  what -to be active- means}
+Before taking horizontal gradients between the tracers next to the bottom, a linear 
+interpolation in the vertical is used to approximate the deeper tracer as if it actually 
+lived at the depth of the shallower tracer point (Fig.~\ref{Fig_Partial_step_scheme}). 
+For example, for temperature in the $i$-direction the needed interpolated 
+temperature, $\widetilde{T}$, is:
+
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!p] \label{Fig_Partial_step_scheme}  \begin{center}
 \includegraphics[width=0.9\textwidth]{./Figures/Partial_step_scheme.pdf}
-\caption{ Discretisation of horizontal derivative and mean of tracers in z-partial step coordinate (\np{ln\_zps}=T) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal derivative is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the mean by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
+\caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=.true.) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -884,5 +1247,6 @@
             \end{aligned}   \right.
 \end{equation*}
-and the resulting formulation of horizontal derivative and horizontal mean value of $T$ at $U$-point are: 
+and the resulting forms for the horizontal difference and the horizontal average 
+value of $T$ at a $U$-point are: 
 \begin{equation} \label{Eq_zps_hde}
 \begin{aligned}
@@ -901,5 +1265,12 @@
 \end{equation}
 
-The computation of horizontal derivative of tracers as well as of density is performed once for all at each time step in \mdl{zpshde} module and stored in shared arrays to be used when needed. It has to be emphasized that the computation of the interpolated density, $\widetilde{\rho}$, is not identical to the one of $T$ and $S$. Instead of forming a linear approximation of density, we compute $\widetilde{\rho }$ from the interpolated value of $T$ and $S$, and the pressure of at $u$-point (in the equation of state pressure is approximated by depth, see \S\ref{TRA_eos} ) : 
+The computation of horizontal derivative of tracers as well as of density is 
+performed once for all at each time step in \mdl{zpshde} module and stored 
+in shared arrays to be used when needed. It has to be emphasized that the 
+procedure used to compute the interpolated density, $\widetilde{\rho}$, is not 
+the same as that used for $T$ and $S$. Instead of forming a linear approximation 
+of density, we compute $\widetilde{\rho }$ from the interpolated values of $T$ 
+and $S$, and the pressure at a $u$-point (in the equation of state pressure is 
+approximated by depth, see \S\ref{TRA_eos} ) : 
 \begin{equation} \label{Eq_zps_hde_rho}
 \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
@@ -907,6 +1278,19 @@
 \end{equation} 
 
-This is a much better approximation as the variation of $\rho$ with depth (and thus pressure) is highly non-linear with a true equation of state and thus is badly approximated with a linear interpolation. This approximation is used to compute both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral surfaces (\S\ref{LDF_slp})
-
-\textit{Notes}: in almost all the advection schemes presented in this Chapter, both mean and derivative operators appear. Yet, it has been chosen not to use \eqref{Eq_zps_hde} in those schemes.: contrary to diffusion and pressure gradient computation, no correction for partial steps is applied for advection.The main motivation was to preserve the domain averaged mean variance of the field advected when using $2^{nd}$ order centred scheme. Sensitivity of the advection schemes to the way horizontal means are performed in the vicinity of partial cells should be further investigated in a near future.
-
+This is a much better approximation as the variation of $\rho$ with depth (and 
+thus pressure) is highly non-linear with a true equation of state and thus is badly 
+approximated with a linear interpolation. This approximation is used to compute 
+both the horizontal pressure gradient (\S\ref{DYN_hpg}) and the slopes of neutral 
+surfaces (\S\ref{LDF_slp})
+
+Note that in almost all the advection schemes presented in this Chapter, both 
+averaging and differencing operators appear. Yet \eqref{Eq_zps_hde} has not 
+been used in these schemes: in contrast to diffusion and pressure gradient 
+computations, no correction for partial steps is applied for advection. The main 
+motivation is to preserve the domain averaged mean variance of the advected 
+field when using the $2^{nd}$ order centred scheme. Sensitivity of the advection 
+schemes to the way horizontal averages are performed in the vicinity of partial 
+cells should be further investigated in the near future.
+%%%
+\gmcomment{gm :   this last remark has to be done}
+%%%
Index: trunk/DOC/BETA/Chapters/Chap_ZDF.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Chap_ZDF.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Chap_ZDF.tex	(revision 817)
@@ -197,5 +197,6 @@
 %       Non-Penetrative Convective Adjustment 
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=T) }
+\subsection   [Non-Penetrative Convective Adjustment (\np{ln\_tranpc}) ]
+			{Non-Penetrative Convective Adjustment (\np{ln\_tranpc}=.true.) }
 \label{ZDF_npc}
 
@@ -254,5 +255,6 @@
 %       Enhanced Vertical Diffusion 
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Enhanced Vertical Diffusion (\np{ln\_zdfevd}=T)}
+\subsection   [Enhanced Vertical Diffusion (\np{ln\_zdfevd})]
+			{Enhanced Vertical Diffusion (\np{ln\_zdfevd}=.true.)}
 \label{ZDF_evd}
 
@@ -296,5 +298,6 @@
 % Double Diffusion Mixing
 % ================================================================
-\section{Double Diffusion Mixing (\key{zdfddm})}
+\section  [Double Diffusion Mixing (\textit{zdfddm} - \key{zdfddm})]
+		{Double Diffusion Mixing (\mdl{zdfddm} module - \key{zdfddm})}
 \label{ZDF_ddm}
 
@@ -355,5 +358,6 @@
 % Bottom Friction
 % ================================================================
-\section{Bottom Friction}
+\section  [Bottom Friction (\textit{zdfbfr})]
+		{Bottom Friction (\mdl{zdfbfr} module)}
 \label{ZDF_bfr}
 
Index: trunk/DOC/BETA/Chapters/Introduction.tex
===================================================================
--- trunk/DOC/BETA/Chapters/Introduction.tex	(revision 812)
+++ trunk/DOC/BETA/Chapters/Introduction.tex	(revision 817)
@@ -6,24 +6,93 @@
 \chapter{Introduction}
 
-The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean related engines, namely OPA for the Ocean dynamics and thermodynamics, LIM for the sea-ice dynamics and thermodynamics, TOP for the biogeochemistry (both transport (TRP) and sources minus sinks (LOBSTER, PISCES). It is intended to be a flexible tool for studying the ocean and its interactions with the others components of the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. This documentation provides information about the physics represented by the OPA ocean model and the rationale for the choice of numerical schemes and the model design. More specific information about running the model on different computers, or how to set up a configuration, are found on the \NEMO web site (www.locean-ipsl.upmc.fr/NEMO). 
+The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean 
+related engines, namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the 
+ocean dynamics and thermodynamics, LIM\footnote{LIM= Louvain)la-neuve Ice 
+Model} for the sea-ice dynamics and thermodynamics, TOP\footnote{TOP = Tracer 
+in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources 
+minus sinks (LOBSTER, PISCES)\footnote{Both LOBSTER and PISCES are not 
+acronyms just name}. It is intended to be a flexible tool for studying the ocean and 
+its interactions with the other components of the earth climate system (atmosphere, 
+sea-ice, biogeochemical tracers, ...) over a wide range of space and time scales. 
+This documentation provides information about the physics represented by the ocean component of \NEMO and the rationale for the choice of numerical schemes and 
+the model design. More specific information about running the model on different 
+computers, or how to set up a configuration, are found on the \NEMO web site 
+(www.locean-ipsl.upmc.fr/NEMO). 
 
-The ocean component of \NEMO has been developed from the OPA8.2 model described in \citet{Madec1998}. This model has been used for a wide range of applications, either regional or global, as a forced ocean model or coupled with the atmosphere. A complete list of references is found on the \NEMO web site. 
+The ocean component of \NEMO has been developed from the OPA model, release 8.2, described in \citet{Madec1998}. This model has been used for a wide range of applications, both regional or global, as a forced ocean model and as a model coupled with the atmosphere. A complete list of references is found on the \NEMO web site. 
 
-This manual is organised in as follows. Chapter~\ref{PE} presents the model basics, $i.e.$ the equations and their assumptions, the vertical coordinates used, and the subgrid scale physics. This part deals with the continuous equations of the model (primitive equations, with potential temperature, salinity and an equation of state). The equations are written in a curvilinear coordinate system, with a choice of vertical coordinates (z, s, and variable volumes). Momentum equations are formulated in the vector invariant form. The model equations are written in dimensional units in the meter, kilogram, second (MKS) international system.
+This manual is organised in as follows. Chapter~\ref{PE} presents the model basics, 
+$i.e.$ the equations and their assumptions, the vertical coordinates used, and the 
+subgrid scale physics. This part deals with the continuous equations of the model 
+(primitive equations, with potential temperature, salinity and an equation of state). The 
+equations are written in a curvilinear coordinate system, with a choice of vertical 
+coordinates ($z$ or $s$, with the rescaled height coordinate formulation \textit{z*}, or  
+\textit{s*}). Momentum equations are formulated in the vector invariant form or in the 
+flux form. Dimensional units in the meter, kilogram, second (MKS) international system 
+are used throughout.
 
-The following chapters deal with the discrete equations. Chapter~\ref{DOM} presents the space and time domain. The model is discretised on a staggered grid (Arakawa C grid) with masking of land areas. Vertical discretisation uses $z$-coordinates (including partial step), $s$- (terrain-following) coordinate (fixed volume thickness and linear free surface), or $s*$-coordinate (variable volume thickness and nonlinear free surface). The following chapters describe the discretisation of the  prognostic  equations (momentum and tracers). Explicit, split-explicit or implicit free surface formulations are implemented as well as arid-lid approximation. A number of numerical schemes are available for momentum advection, for the computation of the pressure gradients, as well as for the advection of tracers (second or higher order advection schemes, including positive ones).
+The following chapters deal with the discrete equations. Chapter~\ref{DOM} presents the 
+space and time domain. The model is discretised on a staggered grid (Arakawa C grid) 
+with masking of land areas and uses a Leap-frog environment for time-stepping. Vertical 
+discretisation used depends on both how the bottom topography is represented and 
+whether the free surface is linear or not. Full step or partial step $z$-coordinate or 
+$s$- (terrain-following) coordinate is used with linear free surface (level position are then 
+fixed in time). In non-linear free surface, the corresponding rescaled height coordinate 
+formulation (\textit{z*} or \textit{s*}) is used (the level position then vary in time as a 
+function of the sea surface heigh). The following two chapters (\ref{TRA} and \ref{DYN}) 
+describe the discretisation of the prognostic equations for the active tracers and the 
+momentum. Explicit, split-explicit and implicit free surface formulations are implemented 
+as well as rigid-lid case. A number of numerical schemes are available for momentum 
+advection, for the computation of the pressure gradients, as well as for the advection of 
+tracers (second or higher order advection schemes, including positive ones).
 
-Other model characteristics are the lateral boundary conditions (chapter~\ref{LBC}).  Global configurations of the model make use of the ORCA tripolar grid, with special north fold boundary condition. Free-slip or no-slip boundary conditions are allowed at land boundaries. Closed basin geometries as well as periodic domains and open boundary conditions are possible. 
+Surface boundary conditions (chapter~\ref{SBC}) can be implemented as prescribed
+fluxes, or bulk formulations for the surface fluxes (wind stress, heat, freshwater). The 
+model allows penetration of solar radiation  There is an optional geothermal heating at 
+the ocean bottom. Within the \NEMO system the ocean model is interactively coupled 
+with a sea ice model (LIM) and with biogeochemistry models (PISCES, LOBSTER). 
+Interactive coupling to Atmospheric models is possible via the OASIS coupler 
+\citep{OASIS2006}. 
 
-Surface boundary conditions (chapter~\ref{SBC}) can be implemented as prescribed fluxes, or bulk formulations for the surface fluxes (wind stress, heat, freshwater). The model allows penetration of solar radiation  There is an optional geothermal heating at the ocean bottom. Within the \NEMO system the ocean model is interactively coupled with a sea ice model (LIM) and with biogeochemistry models (PISCES, LOBSTER). Interactive coupling to Atmospheric models is possible via the OASIS coupler \citep{OASIS2006}. 
+Other model characteristics are the lateral boundary conditions (chapter~\ref{LBC}).  
+Global configurations of the model make use of the ORCA tripolar grid, with special north 
+fold boundary condition. Free-slip or no-slip boundary conditions are allowed at land 
+boundaries. Closed basin geometries as well as periodic domains and open boundary 
+conditions are possible. 
 
-Physical parameterisations are described in chapters~\ref{LDF} and \ref{ZDF}. The model includes an implicit treatment of vertical viscosity and diffusivity. The lateral Laplacian and biharmonic viscosity and diffusion can be rotated following a geopotential or neutral direction. There is an optional eddy induced velocity \citep{Gent1990} with a space and time variable coefficient \citet{Treguier1997}. The model has vertical harmonic viscosity and diffusion with a space and time variable coefficient, with options to compute the coefficients with \citet{Blanke1993}, \citet{Large1994}, or \citet{PacPhil1981} mixing schemes.
+Physical parameterisations are described in chapters~\ref{LDF} and \ref{ZDF}. The 
+model includes an implicit treatment of vertical viscosity and diffusivity. The lateral 
+Laplacian and biharmonic viscosity and diffusion can be rotated following a geopotential 
+or neutral direction. There is an optional eddy induced velocity \citep{Gent1990} with a 
+space and time variable coefficient \citet{Treguier1997}. The model has vertical harmonic 
+viscosity and diffusion with a space and time variable coefficient, with options to compute 
+the coefficients with \citet{Blanke1993}, \citet{Large1994}, or \citet{PacPhil1981} mixing 
+schemes.
 
-Specific online diagnostics (not documented yet) are available in the model: output of all the tendencies of the momentum and tracers equations, output of tracers tendencies averaged over the time evolving mixed layer.
+Specific online diagnostics (not documented yet) are available in the model: output of all 
+the tendencies of the momentum and tracers equations, output of tracers tendencies 
+averaged over the time evolving mixed layer.
 
-The model is implemented in \textsc{Fortran 90}, with preprocessing (C-pre-processor). It runs under UNIX. It is optimized for vector computers and parallelised by domain  decomposition with MPI. All input and output is done in NetCDF (Network Common Data Format) with a optional direct access format for output. To ensure the clarity and readability of the code it is necessary to follow coding rules. The coding rules for OPA include conventions for naming variables, with different starting letters for different types of variables (real, integer, parameter\ldots) Those rules are presented in a document available on the \NEMO web site..
+The model is implemented in \textsc{Fortran 90}, with preprocessing (C-pre-processor). 
+It runs under UNIX. It is optimized for vector computers and parallelised by domain  
+decomposition with MPI. All input and output is done in NetCDF (Network Common Data 
+Format) with a optional direct access format for output. To ensure the clarity and 
+readability of the code it is necessary to follow coding rules. The coding rules for OPA 
+include conventions for naming variables, with different starting letters for different types 
+of variables (real, integer, parameter\ldots). Those rules are presented in a document 
+available on the \NEMO web site.
 
-The model is organized with a high internal modularity based on physics. In particular, each trend (e.g., a term in the rhs of the prognostic equation) for momentum and tracers is computed in a dedicated module.  To make it easier for the user to find his way around the code, the module names follow the three-letter rule.  Each module name is made of three-letter sequences. For example, \mdl{tradmp} is a module related to the TRAcers equation, computing the DaMPing. The complete list of module names is presented in annex. Furthermore, modules are  organized in a few directories that correspond to their category, as indicated by the first three letters of their name. 
-The manual follows this organization. After the presentation of the continuous equations (Chapter \ref{PE}), the following chapters refer to specific terms of the equations each associated with a group of modules .
+The model is organized with a high internal modularity based on physics. For example, 
+each trend ($i.e.$, a term in the RHS of the prognostic equation) for momentum and 
+tracers is computed in a dedicated module.  To make it easier for the user to find his way 
+around the code, the module names follow a three-letter rule. For example, \mdl{tradmp} 
+is a module related to the TRAcers equation, computing the DaMPing. The complete list 
+of module names is presented in \colorbox{yellow}{annex}. Furthermore, modules are  
+organized in a few directories that correspond to their category, as indicated by the first 
+three letters of their name. 
+
+The manual mirrors the organization of the model. After the presentation of the 
+continuous equations (Chapter \ref{PE}), the following chapters refer to specific terms of 
+the equations each associated with a group of modules .
 
 
@@ -31,5 +100,5 @@
 %\begin{center} \begin{tabular}{|p{143pt}|l|l|}	\hline
 \begin{center} \begin{tabular}{|l|l|l|}	\hline
-Chapter \ref{DOM}	& DOM		& Model DOMain \\		\hline
+Chapter \ref{DOM}	& DOM		& model DOMain \\		\hline
 Chapter \ref{TRA}	& TRA		& TRAcer equations (potential temperature and salinity) \\   \hline
 Chapter \ref{DYN}	& DYN		& DYNamic equations (momentum) \\		\hline
@@ -42,19 +111,23 @@
 \end{table}
 
-In the current release (v2.3), LBC directory (see Chap.~\ref{LBC}) does not yet exist. When created LBC will gather OBC directory (Open Boundary Condition), \mdl{lbclnk}, \mdl{mppini} and \mdl{lib\_mpp} modules. 
+In the current release (v2.3), LBC directory (see Chap.~\ref{LBC}) does not yet exist. 
+When created LBC will contain the OBC directory (Open Boundary Condition), and the
+\mdl{lbclnk}, \mdl{mppini} and \mdl{lib\_mpp} modules. 
 
  \vspace{1cm}   Nota Bene : \vspace{0.25cm}
 
-OPA, like all research tools, is in perpetual evolution. The present document describes the OPA model include in the release 2.3 of NEMO. This release differs significantly from version 8, documented in \citet{Madec1998}. The major modifications are :\\
-(1) transition to full native \textsc{Fortran} 90, deep code restructuring and drastic reduction of CPP keys, \\
-(2) introduction of partial step representation of bottom topography \\
-(3) partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$) with the addition of several options for pressure gradient computation \footnote{Partial support of $s$-coordinate: there is presently no support for neutral physics in $s$-coordinate and for the new options for horizontal pressure gradient computation with true equation of state.}, \\ 
-(4) more choices for the treatment of the free surface: full explicit, split-explicit , filtered and rigid-lid \\
-(5) non linear free surface option (variable level thickness distributed on the whole water column) \\
-(6) additional schemes for vector and flux forms of the momentum  advection \\
-(7) addition of several advection schemes on tracers \\
- (8) implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran} ) \\
-(9) online diagnostics : tracers trend in the mixed layer and vorticity balance \\
-(10) rewriting of the I/O management \\
+OPA, like all research tools, is in perpetual evolution. The present document describes 
+the OPA version include in the release 2.3 of NEMO. This release differs significantly
+from version 8, documented in \citet{Madec1998}. The main modifications are :\\
+(1) transition to full native \textsc{Fortran} 90, deep code restructuring and drastic reduction of CPP keys; \\
+(2) introduction of partial step representation of bottom topography \citep{Barnier_al_OD06}; \\
+(3) partial reactivation of a terrain-following vertical coordinate ($s$- and hybrid $s$-$z$) with the addition of several options for pressure gradient computation \footnote{Partial support of $s$-coordinate: there is presently no support for neutral physics in $s$-coordinate and for the new options for horizontal pressure gradient computation with a non-linear equation of state.}; \\ 
+(4) more choices for the treatment of the free surface: full explicit, split-explicit , filtered and rigid-lid; \\
+(5) non linear free surface option (associated with the rescaled height coordinate  \textit{z*} or  \textit{s*}); \\
+(6) additional schemes for vector and flux forms of the momentum  advection; \\
+(7) additional advection schemes for tracers; \\
+(8) implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008}; \\
+(9) online diagnostics : tracers trend in the mixed layer and vorticity balance; \\
+(10) rewriting of the I/O management; \\
 (11) OASIS 3 and 4 couplers interfacing with atmospheric global circulation models. 
 
@@ -63,5 +136,5 @@
 At the time of this writing, the current release is NEMO 2.3. The new surface module described in this document is not yet part of the current distribution.
 
-
+\vspace{1cm}  
 \colorbox{red}{Red color}: not in the current reference version (v2.3) but expected soon.
  
Index: trunk/DOC/BETA/NEMO_book.tex
===================================================================
--- trunk/DOC/BETA/NEMO_book.tex	(revision 812)
+++ trunk/DOC/BETA/NEMO_book.tex	(revision 817)
@@ -6,4 +6,7 @@
 \documentclass[a4paper,12pt]{book}
 
+%  makeindex NEMO_book     <== to regenerate the index
+%  bibtex         NEMO_book	<== to generate  the bibliography
+
 % ================================================================
 % HEADERS DEFINITION
@@ -15,4 +18,7 @@
 \usepackage{alltt}  		%%  alltt for namelist
 \usepackage{verbatim}  	%%  alltt for namelist
+
+%hyperref  ===>>>  don't work.....
+%\usepackage[a4paper=true,ps2pdf=true,pagebackref=true,breaklinks=true]{hyperref}
 
 %\usepackage{amssymb}
@@ -124,6 +130,7 @@
 \newcommand{\pp} [1] {\textit{#1}\index{Model parameters!#1}}%namelist parameter (pp)
 \newcommand{\key} [1] {\textbf{key\_#1}\index{CPP keys!key\_#1}}	%key_cpp (key)
-\newcommand{\NEMO} {\textit{NEMO}}						%NEMO (nemo)
-\newcommand{\amtcomment}[1]{}   	% command to allow "commented out" portions of 
+\newcommand{\NEMO} {\textit{NEMO }}						%NEMO (nemo)
+\newcommand{\amtcomment}[1]{#1}   	% command to allow "commented out" portions of 
+\newcommand{\gmcomment}[1]{}   	% command to allow "commented out" portions of 
 %                                    				% text that span line breaks
 
@@ -134,5 +141,5 @@
 \newcommand{\namdisplay} [1] { 
 	\begin{alltt}  {{\tiny \verbatiminput{./Namelist/#1}}}  \end{alltt} 
-                               \vspace{-15pt}	}
+                               \vspace{-10pt}	}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -159,5 +166,13 @@
 % FRONT PAGE
 % ================================================================
-\title{ \includegraphics[width=0.4\textwidth]{./Figures/NEMO_logo.pdf} \\
+
+\title{ 
+\includegraphics[width=0.2\textwidth]{./Figures/logo_CNRS.pdf} 
+\includegraphics[width=0.2\textwidth]{./Figures/logo_MERCATOR.pdf} 
+\includegraphics[width=0.1\textwidth]{./Figures/logo_UKMO.pdf} 
+\includegraphics[width=0.2\textwidth]{./Figures/logo_NERC.pdf} \\
+\vspace{3.0cm}
+\includegraphics[width=0.4\textwidth]{./Figures/NEMO_logo.pdf} \\
+\vspace{2.0cm}
 \rule{345pt}{1.5pt} \\
 \vspace{0.45cm}
@@ -168,7 +183,10 @@
 \date{\today}
 
+
+%%\includegraphics[width=0.2\textwidth]{./Figures/logo_NERC.pdf} \\
+
 %%    \hbox{\mbox{%
-%%        \hspace{4pt}%
-%%         \fbox{\includegraphics[width=3em]{./NEMO_logo.pdf}}%
+%%       \hspace{4pt}%
+%%         \fbox{\includegraphics[width=3em]{./Figures/logo_CNRS.pdf}}%
 %%        \hspace{4pt}
 %%        }  }%
@@ -186,5 +204,6 @@
 % ================================================================
 
-%\includeonly{./Chapters/Annex_E,./Chapters/Annex_C}
+%\includeonly{./Chapters/Chap_ZDF}
+%\includeonly{./Chapters/Annex_E}
 %\includeonly{./Chapters/Chap_LBC,./Chapters/Chap_MISC}
 %\includeonly{./Chapters/Chap_Model_Basics_zstar}
@@ -223,5 +242,5 @@
 \include{./Chapters/Chap_Model_Basics}
 
-%\include{./Chapters/Chap_Model_Basics_zstar}			% attent to describe z*-s*
+%\include{./Chapters/Chap_Model_Basics_zstar}	% attent to describe z*-s*
 
 \include{./Chapters/Chap_DOM}
@@ -245,10 +264,10 @@
 % ================================================================
 
-%\appendix
+\appendix
 
 %\include{./Chapters/Chap_Conservation}
-%\include{./Chapters/Annex_A}
-%\include{./Chapters/Annex_B}
-%\include{./Chapters/Annex_C}
+\include{./Chapters/Annex_A}
+\include{./Chapters/Annex_B}
+\include{./Chapters/Annex_C}
 %\include{./Chapters/Annex_D}
 %\include{./Chapters/Annex_E}
@@ -268,5 +287,5 @@
 %%\bibliographystyle{plainnat}
 \bibliographystyle{./ametsoc}			% AMS biblio style (JPO)
-\bibliography{./Biblio/biblio}
+\bibliography{./Biblio/Biblio}
 
 % ================================================================
