Index: /branches/DEV_r1826_DOC/DOC/NEMO_book.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/NEMO_book.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/NEMO_book.tex	(revision 1831)
@@ -70,4 +70,5 @@
 \usepackage{amsmath}
 \usepackage{graphicx}
+\allowdisplaybreaks[1]				% allow page breaks in the middle of equations
 %%%%
 
@@ -134,4 +135,5 @@
 \newcommand{\gmcomment}[1]{}   	% command to allow "commented out" portions of 
 %                                    				% text that span line breaks
+\newcommand{\alpbet} {\left(\alpha / \beta \right)} 	% alpha/beta  for slp computation
 
 %--------------------------------------------namlist---------------------------------------------------------
@@ -165,24 +167,24 @@
 
 \title{ 
-\includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_CNRS.pdf} 
+\psset{unit=1.1in,linewidth=4pt} 	%parameters of the units for pstricks
+\rput(-1.4,2){ \includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_CNRS.pdf}           }
 \quad
-\includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_MERCATOR.pdf} 
+\rput(-.15,2){ \includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_MERCATOR.pdf} }
 \quad
-\includegraphics[width=0.1\textwidth]{./TexFiles/Figures/logo_UKMO.pdf} 
+\rput(.85,2){ \includegraphics[width=0.1\textwidth]{./TexFiles/Figures/logo_UKMO.pdf}            }
 \quad
-\includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_NERC.pdf} \\
-\vspace{3.0cm}
-\includegraphics[width=0.3\textwidth]{./TexFiles/Figures/NEMO_logo_Black.pdf} \\
-\vspace{1.0cm}
+\rput(1.85,2){ \includegraphics[width=0.2\textwidth]{./TexFiles/Figures/logo_NERC.pdf}             } \\
+\vspace{0.1cm}
+\includegraphics[width=0.9\textwidth]{./TexFiles/Figures/NEMO_logo_Black.pdf} \\
+\vspace{1.4cm}
 \rule{345pt}{1.5pt} \\
 \vspace{0.45cm}
  {\Huge NEMO ocean engine} 
 \rule{345pt}{1.5pt} \\ 
-%{\small NEMO v2.3} \\
-{\small  -- version 3.1 --}   }
+{\small  -- version 3.2 --}   }
 %{ -- Draft --}   }
 %\date{\today}
 \date{
-May 2008  \\
+April 2010  \\
 \textit{\small Note du P\^ole de mod\'{e}lisation de l'Institut Pierre-Simon Laplace No 27 }\\
 \vspace{0.45cm}
@@ -199,7 +201,9 @@
 %%        }  }%
 
-\author{ \Large Gurvan Madec \\
+\author{ 
+\Large Gurvan Madec, and the NEMO team  \\
  \texttt{\small gurvan.madec@locean-ipsl.umpc.fr} \\
-{\small Laboratoire d'Oc\'{e}anographie  et du Climat: Exp\'{e}rimentation et Approches Num\'{e}riques}
+ \texttt{\small nemo\_st@hermes.locean-ipsl.umpc.fr} \\
+%{\small Laboratoire d'Oc\'{e}anographie  et du Climat: Exp\'{e}rimentation et Approches Num\'{e}riques }
 }
 
@@ -250,6 +254,4 @@
 \include{./TexFiles/Chapters/Chap_Model_Basics}
 
-%\include{./TexFiles/Chapters/Chap_Model_Basics_zstar}	% attent to describe z*-s*
-
 \include{./TexFiles/Chapters/Chap_DOM}
 
@@ -278,5 +280,5 @@
 \include{./TexFiles/Chapters/Annex_B}
 \include{./TexFiles/Chapters/Annex_C}
-%\include{./TexFiles/Chapters/Annex_D}
+\include{./TexFiles/Chapters/Annex_D}
 %\include{./TexFiles/Chapters/Annex_E}
 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Biblio/Biblio.bib
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Biblio/Biblio.bib	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Biblio/Biblio.bib	(revision 1831)
@@ -64,4 +64,8 @@
 @STRING{Ocean = {Oceanology}}
 
+@STRING{OD = {Ocean Dynamics}}
+
+@STRING{OM = {Ocean Modelling}}
+
 @STRING{OS = {Ocean Science}}
 
@@ -76,5 +80,5 @@
 @STRING{PRSL = {Proceedings of the Royal Society of London}}
 
-@STRING{QJRMS = {Quart J Roy Meteor Soc}}
+@STRING{QJRMS = {Quart. J. Roy. Meteor. Soc.}}
 
 @STRING{Recherche = {La Recherche}}
@@ -90,5 +94,5 @@
   title = {Re-scaled height coordinates for accurate representation of free-surface
 	flows in ocean circulation models},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2004},
   volume = {7},
@@ -104,10 +108,10 @@
   year = {1966},
   volume = {I},
-  pages = {119-149},
+  pages = {119--149},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@ARTICLE{Arakawa1990,
+@ARTICLE{Arakawa_Hsu_MWR90,
   author = {A. Arakawa and Y.-J. G. Hsu},
   title = {Energy Conserving and Potential-Enstrophy Dissipating Schemes for
@@ -116,7 +120,5 @@
   year = {1990},
   volume = {118},
-  pages = {1960--1969
-	
-	},
+  pages = {1960--1969},
   number = {10},
   abstract = {To incorporate potential enstrophy dissipation into discrete shallow
@@ -134,5 +136,5 @@
 }
 
-@ARTICLE{Arakawa1981,
+@ARTICLE{Arakawa_Lamb_MWR81,
   author = {Arakawa, Akio and Lamb, Vivian R.},
   title = {A Potential Enstrophy and Energy Conserving Scheme for the Shallow
@@ -141,7 +143,5 @@
   year = {1981},
   volume = {109},
-  pages = {18--36
-	
-	},
+  pages = {18--36},
   number = {1},
   abstract = {To improve the simulation of nonlinear aspects of the flow over steep
@@ -171,5 +171,5 @@
   year = {2006},
   volume = { 36},
-  pages = {1502-1522}
+  pages = {1502--1522}
 }
 
@@ -180,5 +180,5 @@
   year = {1972},
   volume = {100},
-  pages = {487-490},
+  pages = {487--490},
   number = {6},
   abstract = {A simple filter for controlling high-frequency computational and physical
@@ -198,4 +198,65 @@
 }
 
+@ARTICLE{Atmadipoera_al_DSR09,
+  author = {A. Atmadipoera and R. Molcard and G. Madec and S.Wijffels and J.
+	Sprintall and A. Koch-Larrouy and I. Jaya and A. Supangat},
+  title = {Characteristics and Variability of the Indonesian Throughflow Water
+	at the Outflow Straits},
+  journal = DSR,
+  year = {2009},
+  volume = {in press},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Aumont_al_GBC99,
+  author = {O. Aumont and P. Monfray and J. C. Orr and G. Madec and E. Maier-Reimer},
+  title = {Nutrient trapping in the equatorial Pacific: The ocean circulation
+	solution},
+  journal = GBC,
+  year = {1999},
+  volume = {13},
+  pages = {351--369},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Aumont_al_CD98,
+  author = {O. Aumont and J.C. Orr and D. Jamous and P. Monfray and O. Marti
+	and G. Madec},
+  title = {A degradation approach to accelerate simulations to steady state
+	in a 3-D tracer transport model of the global ocean},
+  journal = CD,
+  year = {1998},
+  volume = {14},
+  pages = {101--116},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Axell_JGR02,
+  author = {L. B. Axell},
+  title = {Wind-driven internal waves and Langmuir circulations in a numerical
+	ocean model of the southern Baltic Sea},
+  journal = JGR,
+  year = {2002},
+  volume = {107},
+  doi = {doi:10.1029/2001JC000922},
+  owner = {gm},
+  timestamp = {2009.01.16}
+}
+
+@ARTICLE{Ayina_al_JC06,
+  author = {L.-H. Ayina and A. Bentamy and A. Munes-Mestaz and G. Madec},
+  title = {The Impact of Satellite Winds and Latent Heat Fluxes in a Numerical
+	Simulation of the Tropical Pacific Ocean},
+  journal = JC,
+  year = {2006},
+  volume = {19},
+  pages = {5889–-5902},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Barnier_al_OD06,
   author = {B. Barnier and G. Madec and T. Penduff and J.-M. Molines and A.-M.
@@ -206,7 +267,9 @@
   title = {Impact of partial steps and momentum advection schemes in a global
 	ocean circulation model at eddy-permitting resolution.},
-  journal = {Ocean Dyn.},
+  journal = OD,
   year = {2006},
-  pages = {doi: 10.1007/s10236-006-0082-1.},
+  volume = {56},
+  pages = {543--567},
+  doi = {10.1007/s10236-006-0082-1},
   owner = {gm},
   timestamp = {2008.01.25}
@@ -229,8 +292,25 @@
   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.},
+  journal = DSR,
   year = {1998},
   volume = {45},
-  pages = {543-572}
+  pages = {543--572}
+}
+
+@ARTICLE{Barthelet_al_CRAS98,
+  author = {P. Barthelet and S. Bony and P. Braconnot and A. Braum and D. Cariolle
+	and E. Cohen-Solal and J.-L. Dufresne and P. Delecluse and M. D\'{e}qu\'{e}
+	and L. Fairhead and M.-A. Filiberti and M. Forichon and J.-Y. Grandpeix
+	and E. Guilyardi and M.-N. Houssais and M. Imbard and H. Le Treut
+	and C. Lévy and Z.X. Li and G. Madec and P. Marquet and O. Marti
+	and S. Planton and L. Terray and O. Thual and S. Valcke},
+  title = {Global coupled simulations of climate change due to increased atmospheric
+	CO2 concentration. C. R. Acad. Sci Paris, 326, 677-684.},
+  journal = {C. R. Acad. Sci Paris},
+  year = {1998},
+  volume = {326},
+  pages = {677--684},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -253,5 +333,5 @@
   year = {1998},
   volume = {27},
-  pages = {581-591},
+  pages = {581--591},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -262,10 +342,8 @@
   title = {Numerical Simulation of Flow around a Tall Isolated Seamount. Part
 	I - Problem Formulation and Model Accuracy},
-  journal = {Journal of Physical Oceanography},
+  journal = JPO,
   year = {1993},
   volume = {23},
-  pages = {1736--1753
-	
-	},
+  pages = {1736--1753},
   number = {8},
   abstract = {A sigma coordinate ocean circulation model is employed to study flow
@@ -304,4 +382,58 @@
 }
 
+@ARTICLE{Bernie_al_CD08,
+  author = {D. Bernie and E. Guilyardi and G. Madec and J. M. Slingo and S. J.
+	Woolnough},
+  title = {Impact of resolving the diurnal cycle in an ocean–atmosphere GCM.
+	Part 2: A diurnally coupled CGCM},
+  journal = CD,
+  year = {2008},
+  volume = {31, 7},
+  pages = {909--925},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1007/s00382-008-0429-z}
+}
+
+@ARTICLE{Bernie_al_CD07,
+  author = {D. Bernie and E. Guilyardi and G. Madec and J. M. Slingo and S. J.
+	Woolnough},
+  title = {Impact of resolving the diurnal cycle in an ocean–atmosphere GCM.
+	Part 1: a diurnally forced OGCM},
+  journal = CD,
+  year = {2007},
+  volume = {29, 6},
+  pages = {575--590},
+  owner = {gm},
+  timestamp = {2009.08.16}
+}
+
+@ARTICLE{Bessiere_al_GRL08,
+  author = {L. Bessi\'{e}res and G. Madec and F. Lyard},
+  title = {Global Tidal Residual Mean Circulation: Does it Affect a Climate
+	OGCM?},
+  journal = GRL,
+  year = {2008},
+  volume = {35},
+  pages = {L03609},
+  doi = {10.1029/2007GL032644},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Biastoch_al_JC08,
+  author = {A. Biastoch and C. W. Böning and J. Getzlaff and J.-M. Molines and
+	G. Madec},
+  title = {Causes of interannual – decadal variability in the meridional overturning
+	circulation of the mid-latitude North Atlantic Ocean},
+  journal = JC,
+  year = {2008},
+  volume = {21, 24},
+  pages = {6599-6615},
+  doi = {10.1175/2008JCLI2404.1},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Blanke_al_JPO99,
   author = {B. Blanke and M. Arhan and G. Madec and S. Roche},
@@ -323,5 +455,5 @@
   year = {1993},
   volume = {23},
-  pages = {1363-1388}
+  pages = {1363--1388}
 }
 
@@ -332,5 +464,5 @@
   year = {1997},
   volume = {10},
-  pages = {1473-1486},
+  pages = {1473--1486},
   abstract = {One open question in El NinoSouthern Oscillation (ENSO) simulation
 	and predictability is the role of random
@@ -403,7 +535,29 @@
   year = {1997},
   volume = {27, 6},
-  pages = {1038-1053},
+  pages = {1038--1053},
   owner = {gm},
   timestamp = {2008.05.27}
+}
+
+@ARTICLE{Blanke_al_JPO01,
+  author = {B. Blanke and S. Speich and G. Madec and K. Döös},
+  title = {A global Diagnostic of interocean mass transfers},
+  journal = JPO,
+  year = {2001},
+  volume = {31, 6},
+  pages = {1623--1632},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Blanke_al_GRL02,
+  author = {B. Blanke and S. Speich and G. Madec and R. Maug\'{e}},
+  title = {A global diagnostic of interior ocean ventilation},
+  journal = GRL,
+  year = {2002},
+  volume = {29, 8},
+  pages = {1081--1084},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -412,8 +566,20 @@
   title = {Revisiting open boundary conditions from the point of view of characteristic
 	variables},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2005},
   volume = {9},
-  pages = {231-252}
+  pages = {231--252}
+}
+
+@ARTICLE{Bopp_al_GBC01,
+  author = {L. Bopp and P. Monfray and O. Aumont and J.-L. Dufresne and H. Le
+	Treut and G. Madec and L. Terray and J.C. Orr},
+  title = {Potential impact of climate change on marine export production},
+  journal = GBC,
+  year = {2001},
+  volume = {15, 1},
+  pages = {81--101},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -425,5 +591,5 @@
   year = {1989},
   volume = {117},
-  pages = {1872-1890},
+  pages = {1872--1890},
   number = {8},
   abstract = {The possibility of extending existing techniques for turbulence parameterization
@@ -442,4 +608,43 @@
 }
 
+@ARTICLE{Boulanger_al_GRL01,
+  author = {J.-P. Boulanger and E. Durand and J.-P. Duvel and C. Menkes and P.
+	Delecluse and M. Imbard and M. Lengaigne and G.Madec and S. Masson},
+  title = {Role of non-linear oceanic processes in the response to westerly
+	wind events: new implications for the 1997 El Niño onset},
+  journal = GRL,
+  year = {2001},
+  volume = {28, 8},
+  pages = {1603--1606},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{de_Boyer_Montegut_al_JGR04,
+  author = {C. de Boyer Mont\'{e}gut and G. Madec and A.S. Fischer and A. Lazar
+	and D. Iudicone},
+  title = {Mixed layer depth over the global ocean: An examination of profile
+	data and a profile-based climatology},
+  journal = JGR,
+  year = {2004},
+  volume = {109},
+  pages = {C12003},
+  doi = {10.1029/2004JC002378},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{de_Boyer_Montegut_al_JC07,
+  author = {C. de Boyer Mont\'{e}gut and J. Vialard and F. Durand and G. Madec},
+  title = {Simulated seasonal and interannual variability of mixed layer heat
+	budget in the northern Indian Ocean},
+  journal = JC,
+  year = {2007},
+  volume = {20 (13)},
+  pages = {3249--3268},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Brown1978,
   author = {J. A. Brown and K. A. Campana},
@@ -448,5 +653,5 @@
   year = {1978},
   volume = {106},
-  pages = {1125-1136},
+  pages = {1125--1136},
   number = {8},
   month = aug,
@@ -495,7 +700,22 @@
   year = {1973},
   volume = {20},
-  pages = {401-408},
+  pages = {401--408},
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@ARTICLE{Burchard_OM02,
+  author = {Hans Burchard},
+  title = {Energy-conserving discretisation of turbulent shear and buoyancy
+	production},
+  journal = OM,
+  year = {2002},
+  volume = {4},
+  pages = {347--361},
+  number = {3-4},
+  doi = {DOI: 10.1016/S1463-5003(02)00009-4},
+  owner = {gm},
+  timestamp = {2008.11.28},
+  url = {http://www.sciencedirect.com/science/article/B6VPS-45GKJ95-2/2/073d76a0fba5defe75fcd65e9b5d1b4f}
 }
 
@@ -503,10 +723,24 @@
   author = {J.-M. Campin and A. Adcroft and C. Hill and J. Marshall},
   title = {Conservation of properties in a free-surface model},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2004},
   volume = {6, 3-4},
-  pages = {221-244},
+  pages = {221--244},
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@ARTICLE{Campin_al_OM08,
+  author = {Jean-Michel Campin and John Marshall and David Ferreira},
+  title = {Sea ice-ocean coupling using a rescaled vertical coordinate z*},
+  journal = {Ocean Modelling},
+  year = {2008},
+  volume = {24},
+  pages = {1 - 14},
+  number = {1-2},
+  doi = {DOI: 10.1016/j.ocemod.2008.05.005},
+  issn = {1463-5003},
+  timestamp = {2010.01.20},
+  url = {http://www.sciencedirect.com/science/article/B6VPS-4SJP7N5-1/2/e098c3abccafb0972b52e0caf6dbf1f3}
 }
 
@@ -518,7 +752,24 @@
   year = {1999},
   volume = {51},
-  pages = {412-430},
+  pages = {412--430},
   owner = {gm},
   timestamp = {2008.01.20}
+}
+
+@ARTICLE{Covey_al_CD00,
+  author = {C. Covey and A. Abe-Ouchi and G.J. Boer and B.A. Boville and U. Cubasch
+	and L. Fairhead and G.M. Flato and H. Gordon and E. Guilyardi and
+	X. Jiang and T.C. Johns and H. Le Treut and G. Madec and G.A. Meehl
+	and R. Miller and A. Noda and S. B. Power and E. Roeckner and G.
+	Russell and E.K. Schneider and R.J. Stouffer and L. Terray and J.-S.
+	von Storch},
+  title = {The seasonal cycle in coupled ocean-atmosphere general circulation
+	models},
+  journal = CD,
+  year = {2000},
+  volume = {16},
+  pages = {775--787},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -526,10 +777,22 @@
   author = {M. Cox},
   title = {Isopycnal diffusion in a z-coordinate ocean model},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {1987},
   volume = {74},
-  pages = {1-9},
+  pages = {1--9},
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Cravatte_al_OM07,
+  author = {Cravatte, S. and G. Madec and T. Izumo and C. Menkes and A. Bozec},
+  title = {Progress in the 3-D circulation of the eastern equatorial Pacific
+	in a climate ocean model},
+  journal = OM,
+  year = {2007},
+  volume = {17, 1},
+  pages = {28--48},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -541,7 +804,19 @@
   year = {2000},
   volume = {17},
-  pages = {698-707},
+  pages = {698--707},
   owner = {gm},
   timestamp = {2008.01.23}
+}
+
+@ARTICLE{Dandonneau_al_S04,
+  author = {Y. Dandonneau and C. Menkes and T. Gorgues and G. Madec},
+  title = {Reply to Peter Killworth, 2004 : « Comment on the Oceanic Rossby
+	Waves acting as a “Hay Rake” for ecosystem by-products »},
+  journal = {Science},
+  year = {2004},
+  volume = {304},
+  pages = {390},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -552,5 +827,5 @@
   year = {2008},
   volume = {34},
-  pages = {8-13},
+  pages = {8--13},
   owner = {gm},
   timestamp = {2008.02.03}
@@ -565,7 +840,26 @@
 	Eds. W. R. Holland, S. Joussaume and F. David, Elsevier Science,},
   year = {2000},
-  pages = {237-313},
+  pages = {237--313},
   owner = {gm},
   timestamp = {2008.02.03}
+}
+
+@ARTICLE{Doney_al_GBC04,
+  author = {S.C. Doney and K. Lindsay and K. Caldeira and J.−M. Campin and H.
+	Drange and J.−C. Dutay and M. Follows and Y. Gao and A. Gnanadesikan
+	and N. Gruber and A. Ishida and F. Joos and G. Madec and E. Maier−Reimer
+	and J.C. Marshall and R.J. Matear and P. Monfray and A. Mouchet and
+	R. Najjar and J.C. Orr and G.−K. Plattner and J. Sarmiento and R.
+	Schlitzer and R. Slater and I.J. Totterdell and M.−F. Weirig and
+	Y. Yamanaka and A. Yoo},
+  title = {Evaluating global ocean carbon models: the importance of realistic
+	physics},
+  journal = GBC,
+  year = {2004},
+  volume = {18},
+  pages = {GB3017},
+  doi = {10.1029/2003GB002150},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -576,7 +870,34 @@
   year = {1994},
   volume = {99},
-  pages = {7991-8014},
+  pages = {7991--8014},
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Durand_al_JC07,
+  author = {F. Durand and D. Shankar and C. de Boyer Mont\'{e}gut and S.S.C.
+	Shenoi and B. Blanke and G. Madec},
+  title = {Modeling the barrier-layer formation in the South-Eastern Arabian
+	Sea},
+  journal = JC,
+  year = {2007},
+  volume = {20 (10)},
+  pages = {2109--2120},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Durand_al_GRL04,
+  author = {F. Durand and S. R. Shetye and J. Vialard and D. Shankar and S.S.C.
+	Shenoi and C. Eth\'{e} and G. Madec},
+  title = {Impact of temperature inversions on SST evolution in the South−Eastern
+	Arabian Sea during the pre−summer monsoon season},
+  journal = GRL,
+  year = {2004},
+  volume = {31},
+  pages = {L01305},
+  doi = {10.1029/2003GL018906},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -590,4 +911,34 @@
 }
 
+@ARTICLE{Dutay_al_OM02,
+  author = {J.-C. Dutay and J.L. Bullister and S.C. Doney and J.C. Orr and R.
+	Najjar and K. Caldeira and J.-M. Campin and H. Drange and M. Follows
+	and Y. Gao and N. Gruber and M. W. Hecht and A. Ishida and F. Joos
+	and K. Lindsay and G. Madec and E. Maier-Reimer and J.C. Mashall
+	and R. J. Matear and P. Monfray and G.-K. Plattner and J. Sarmiento
+	and R. Schlitzer and R. Slater and I.J. Totterdell and M.-F. Weirig
+	and Y. Yamanaka and A. Tool},
+  title = {Evaluation of ocean model ventilation with CFC-11: comparison of
+	13 global ocean models},
+  journal = OM,
+  year = {2002},
+  volume = {4},
+  pages = {89--120},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Dutay_al_EFM09,
+  author = {J.-C. Dutay and J. Emile-Geay and D. Iudicone and P. Jean-Baptiste
+	and G. Madec and C. Carouge},
+  title = {Helium Isotopic Constraints on Simulated Ocean Circulations - Implications
+	for abyssal theories},
+  journal = {Environmental Fluid Mechanics},
+  year = {2009},
+  volume = {in revision},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Dutay.J.C2004,
   author = {J. -C. Dutay and P. J. -Baptiste and J. -M. Campin and A. Ishida
@@ -597,7 +948,7 @@
 	
 	with mantle helium-3},
-  journal = {Journal of Marine Systems},
+  journal = JMS,
   year = {2004},
-  pages = {1-22},
+  pages = {1--22},
   abstract = {We compare simulations of the injection of mantle helium-3 into the
 	deep ocean from six global coarse resolution models which participated
@@ -650,4 +1001,18 @@
 }
 
+@ARTICLE{D'Ortenzio_al_GRL05,
+  author = {F. D’Ortenzio and D. Iudicone and C. de Boyer Mont\'{e}gut and P.
+	Testor and D. Antoine and S. Marullo and R. Santoleri and G. Madec},
+  title = {Seasonal variability of the mixed layer depth in the Mediterranean
+	Sea : a new climatology based on analysis of individual profiles},
+  journal = GRL,
+  year = {2005},
+  volume = {32},
+  pages = {L12605},
+  doi = {10.1029/2005GL022463},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Eiseman1980,
   author = {P. R. Eiseman and A. P. Stone},
@@ -656,18 +1021,48 @@
   year = {1980},
   volume = {22},
-  pages = {12-27},
+  pages = {12--27},
   owner = {gm},
   timestamp = {2007.08.03}
 }
 
-@ARTICLE{Emile-Geay_Madec_OSD08,
+@ARTICLE{Emile-Geay_Madec_OS09,
   author = {J. Emile-Geay and G. Madec},
   title = {Geothermal heating, diapycnal mixing and the abyssal circulation},
-  journal = {Ocean Sci. Discuss.},
-  year = {2008},
+  journal = OS,
+  year = {2009},
   volume = {5},
-  pages = {281-325},
+  pages = {281--325},
   owner = {gm},
   timestamp = {2008.07.16}
+}
+
+@ARTICLE{EUROMODEL_OA95,
+  author = {EUROMODEL Group (P.M. Lehucher, L. Beautier, M. Chartier, F. Martel,
+	L. Mortier, P. Brehmer, C. Millot, C. Alberola, M. Benzhora, I. Taupier-Letage,
+	G. Chabert d'Hieres, H. Didelle, P. Gleizon, D. Obaton, M. Crépon,
+	C. Herbaut, G. Madec, S. Speich, J. Nihoul, J. M. Beckers, P. Brasseur,
+	E. Deleersnijder, S. Djenidi, J. Font, A. Castellon, E. Garcia-Ladona,
+	M. J. Lopez-Garcia, M. Manriquez, M. Maso, J. Salat, J. Tintore,
+	S. Alonso, D. Gomis, A. Viudez, M. Astraldi, D. Bacciola, M. Borghini,
+	F. Dell'amico, C. Galli, E. Lazzoni, G. P. Gasparini, S. Sparnocchia,
+	and A. Harzallah, 1995 : Progress from 1989 to 1992 in understanding
+	the circulation of the Western Mediterranean Sea. Oceanologica Acta,
+	18, 2, 255-271.},
+  title = {EUROMODEL Group (P.M. Lehucher, L. Beautier, M. Chartier, F. Martel,
+	L. Mortier, P. Brehmer, C. Millot, C. Alberola, M. Benzhora, I. Taupier-Letage,
+	G. Chabert d'Hieres, H. Didelle, P. Gleizon, D. Obaton, M. Crépon,
+	C. Herbaut, G. Madec, S. Speich, J. Nihoul, J. M. Beckers, P. Brasseur,
+	E. Deleersnijder, S. Djenidi, J. Font, A. Castellon, E. Garcia-Ladona,
+	M. J. Lopez-Garcia, M. Manriquez, M. Maso, J. Salat, J. Tintore,
+	S. Alonso, D. Gomis, A. Viudez, M. Astraldi, D. Bacciola, M. Borghini,
+	F. Dell'amico, C. Galli, E. Lazzoni, G. P. Gasparini, S. Sparnocchia,
+	and A. Harzallah, 1995 : Progress from 1989 to 1992 in understanding
+	the circulation of the Western Mediterranean Sea.},
+  journal = {Oceanologica Acta},
+  year = {1995},
+  volume = {18, 2},
+  pages = {255--271},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -689,5 +1084,5 @@
   year = {1995},
   volume = {25},
-  pages = {1731-1741.},
+  pages = {1731--1741.},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -701,5 +1096,5 @@
   year = {1991},
   volume = {96},
-  pages = {759-774},
+  pages = {759--774},
   month = jan,
   owner = {gm},
@@ -735,5 +1130,5 @@
   year = {1990},
   volume = {20},
-  pages = {150-155},
+  pages = {150--155},
   number = {1},
   abstract = {A subgrid-scale form for mesoscale eddy mixing on isopycnal surfaces
@@ -766,5 +1161,5 @@
   year = {1993},
   volume = {98},
-  pages = {14703-14726},
+  pages = {14703--14726},
   owner = {gm},
   timestamp = {2007.08.03}
@@ -794,17 +1189,81 @@
   year = {1999},
   volume = {104},
-  pages = {13,681-13,695},
+  pages = {13,681--13,695},
   owner = {gm},
   timestamp = {2008.05.27}
 }
 
-@ARTICLE{Griffes2005,
-  author = {S. M. Griffes and A. Gnanadesikan and K. W. Dixon and J. P. Dunne
-	and R. Gerdes and M. J. Harrison and A. Rosati and J. L. Russell
-	and B. L. Samuels and M. J. Spelman and M. Winton and R. Zhang},
+@ARTICLE{Gorgues_al_GRL07,
+  author = {T. Gorgues and C. Menkes and O. Aumont and K. Rodgers and G. Madec
+	and Y. Dandonneau},
+  title = {Indonesian Throughflow control of the eastern equatorial Pacific
+	biogeochemistry},
+  journal = GRL,
+  year = {2007},
+  volume = {34},
+  pages = {L05609},
+  doi = {10.1029/2006GL028210},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Greatbatch_JGR94,
+  author = {R. J. Greatbatch},
+  title = {A note on the representation of steric sea level in models that conserve
+	volume rather than mass},
+  journal = JGR,
+  year = {1994},
+  volume = {99, C6},
+  pages = {12,767--12,771},
+  owner = {gm},
+  timestamp = {2009.10.01}
+}
+
+@BOOK{Griffies_Bk04,
+  title = {Fundamentals of ocean climate models},
+  publisher = {Princeton University Press, 434pp},
+  year = {2004},
+  author = {S.M. Griffies},
+  owner = {gm},
+  timestamp = {2007.08.05}
+}
+
+@ARTICLE{Griffies_JPO98,
+  author = {S.M. Griffies},
+  title = {The Gent-McWilliams skew-flux},
+  journal = JPO,
+  year = {1998},
+  volume = {28},
+  pages = {831--841},
+  owner = {gm},
+  timestamp = {2008.06.28}
+}
+
+@ARTICLE{Griffies_al_OM09,
+  author = {S.M. Griffies and A. Biastoch and C. Boning and F. Bryan and G. Danabasoglu
+	and E. P. Chassignet and M. H. England and R. Gerdes and H. Haak
+	and R. W. Hallberg and W. Hazeleger and J. Jungclaus and W. G. Large
+	and G. Madec and A. Pirani and B. L. Samuels and M. Scheinert and
+	A. Sen Gupta and C. A. Severijns and H. L. Simmons and A.-M. Treguier
+	and M. Winton and S. Yeager and J. Yin},
+  title = {Coordinated Ocean-ice Reference Experiments (COREs)},
+  journal = OM,
+  year = {2009},
+  volume = {26, 1-2},
+  pages = {1--46},
+  doi = {10.1016/j.ocemod.2008.08.007},
+  owner = {gm},
+  timestamp = {2009.08.15}
+}
+
+@ARTICLE{Griffies_al_OS05,
+  author = {S.M. Griffies and A. Gnanadesikan and K.W. Dixon and J.P. Dunne and
+	R. Gerdes and M.J. Harrison and A. Rosati and J.L. Russell and B.L.
+	Samuels and M.J. Spelman and M. Winton and R. Zhang},
   title = {Formulation of an ocean model for global climate simulations},
   journal = OS,
   year = {2005},
-  pages = {165246},
+  volume = {1},
+  pages = {45--79},
   abstract = {This paper summarizes the formulation of the ocean component to the
 	Geophysical
@@ -865,32 +1324,12 @@
 }
 
-@BOOK{Griffies2004,
-  title = {Fundamentals of ocean climate models},
-  publisher = {Princeton University Press, 434pp},
-  year = {2004},
-  author = {S. M. Griffies},
-  owner = {gm},
-  timestamp = {2007.08.05}
-}
-
-@ARTICLE{Griffies_JPO98,
-  author = {S. M. Griffies},
-  title = {The Gent-McWilliams skew-ﬂux},
-  journal = JPO,
-  year = {1998},
-  volume = {28},
-  pages = {831–841},
-  owner = {gm},
-  timestamp = {2008.06.28}
-}
-
-@ARTICLE{Griffies1998,
-  author = {S. M. Griffies and A. Gnanadesikan and R. C. Pacanowski and V. D.
-	Larichev and J. K. Dukowicz and R. D. Smith},
+@ARTICLE{Griffies_al_JPO98,
+  author = {S.M. Griffies and A. Gnanadesikan and R.C. Pacanowski and V.D. Larichev
+	and J.K. Dukowicz and R.D. Smith},
   title = {Isoneutral Diffusion in a z-Coordinate Ocean Model},
   journal = JPO,
   year = {1998},
   volume = {28},
-  pages = {805-830},
+  pages = {805--830},
   number = {5},
   abstract = {This paper considers the requirements that must be satisfied in order
@@ -910,6 +1349,6 @@
 }
 
-@ARTICLE{Griffies2001,
-  author = {S. M. Griffies and R. C. Pacanowski and M. Schmidt and V. Balaji},
+@ARTICLE{Griffies_al_MWR01,
+  author = {S.M. Griffies and R.C. Pacanowski and M. Schmidt and V. Balaji},
   title = {Tracer Conservation with an Explicit Free Surface Method for z-Coordinate
 	Ocean Models},
@@ -917,5 +1356,5 @@
   year = {2001},
   volume = {129},
-  pages = {1081-1098},
+  pages = {1081--1098},
   number = {5},
   abstract = {This paper details a free surface method using an explicit time stepping
@@ -939,5 +1378,31 @@
 }
 
-@ARTICLE{Guily2001,
+@ARTICLE{Guilyardi_al_JC04,
+  author = {E. Guilyardi and S. Gualdi and J. M. Slingo and A. Navarra and P.
+	Delecluse and J. Cole and G. Madec and M. Roberts and M. Latif and
+	L. Terray},
+  title = {Representing El Ni\~{n}o in coupled ocean-atmosphere GCMs: the dominant
+	role of the atmospheric component},
+  journal = JC,
+  year = {2004},
+  volume = {17},
+  pages = {4623--4629},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Guilyardi_Madec_CD98,
+  author = {E. Guilyardi and G. Madec},
+  title = {Performance of the OPA-ARPEGE-T21 global ocean-atmosphere coupled
+	model},
+  journal = CD,
+  year = {1997},
+  volume = {13},
+  pages = {149--165},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Guilyardi_al_CD01,
   author = {E. Guilyardi and G. Madec and L. Terray},
   title = {The role of lateral ocean physics in the upper ocean thermal balance
@@ -946,7 +1411,20 @@
   year = {2001},
   volume = {17},
-  pages = {589-599},
+  pages = {589--599},
   number = {8},
   pdf = {/home/ericg/TeX/Papers/Published_pdfs/Guilyardi_al_CD01.pdf}
+}
+
+@ARTICLE{Guilyardi_al_CRAS95,
+  author = {E. Guilyardi and G. Madec and L. Terray and M. D\'{e}qu\'{e} and
+	M. Pontaud and M. Imbard and D. Stephenson and M.-A. Filiberti and
+	D. Cariolle and P. Delecluse and O. Thual},
+  title = {Simulation couplée océan-atmosphère de la variabilité du climat},
+  journal = {C. R. Acad. Sci Paris},
+  year = {1995},
+  volume = {320, s\'{e}rie IIa},
+  pages = {683--690},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -957,5 +1435,5 @@
   year = {1999},
   volume = {Euro-Par'99},
-  pages = {603-607},
+  pages = {603--607},
   owner = {gm},
   timestamp = {2008.05.27}
@@ -969,5 +1447,5 @@
   year = {1999},
   volume = {11, 4},
-  pages = {499-517},
+  pages = {499--517},
   owner = {gm},
   timestamp = {2008.05.27}
@@ -990,7 +1468,5 @@
   year = {1991},
   volume = {21},
-  pages = {610--619
-	
-	},
+  pages = {610--619},
   number = {4},
   abstract = {The error in computing the pressure gradient force near steep topography
@@ -1019,4 +1495,17 @@
 }
 
+@ARTICLE{Hordoir_al_CD08,
+  author = {R. Hordoir and J. Polcher and J.-C. Brun-Cottan and G. Madec},
+  title = {Towards a parametrization of river discharges into ocean general
+	circulation models: a closure through energy conservation},
+  journal = CD,
+  year = {2008},
+  volume = {31, 7--8},
+  pages = {891--908},
+  doi = {10.1007/s00382-008-0416-4},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Hsu1990,
   author = {Hsu, Yueh-Jiuan G. and Arakawa, Akio},
@@ -1026,7 +1515,5 @@
   year = {1990},
   volume = {118},
-  pages = {1933--1959
-	
-	},
+  pages = {1933--1959},
   number = {10},
   abstract = {In constructing a numerical model of the atmosphere, we must choose
@@ -1056,4 +1543,74 @@
 }
 
+@ARTICLE{Huang_JPO93,
+  author = {R.X. Huang},
+  title = {Real freshwater flux as a natural boundary condition for the salinity
+	balance and thermohaline circulation forced by evaporation and precipitation},
+  journal = JPO,
+  year = {1993},
+  volume = {23},
+  pages = {2428--2446},
+  owner = {gm},
+  timestamp = {2009.05.01}
+}
+
+@ARTICLE{Iudicone_al_JPO08b,
+  author = {D. Iudicone and G. Madec and B. Blanke and S. Speich},
+  title = {The role of Southern Ocean surface forcings and mixing in the global
+	conveyor},
+  journal = JPO,
+  year = {2008},
+  volume = {38},
+  pages = {1377--1400},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Iudicone_al_JPO08a,
+  author = {D. Iudicone and G. Madec and T. J. McDougall},
+  title = {Diagnosing water transformations and the key role of light penetration},
+  journal = JPO,
+  year = {2008},
+  volume = {38},
+  pages = {1357--1376},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Iudicone_al_JPO07,
+  author = {D. Iudicone and K. Rodgers and R. Schopp and G. Madec},
+  title = {An Exchange window for the Antarctic Intermediate Water Injection
+	into the South Pacific},
+  journal = JPO,
+  year = {2007},
+  volume = {37},
+  pages = {31--49},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Iudicone_al_JPO08c,
+  author = {D. Iudicone and S. Speich and G. Madec and B. Blanke},
+  title = {The global Conveyor Belt in a Southern Ocean perspective},
+  journal = JPO,
+  year = {2008},
+  volume = {38},
+  pages = {1401--1425},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Izumo_al_CD09,
+  author = {T. Izumo and S.Masson and J. Vialard and C. de Boyer Montegut and
+	S. K. Behera and G. Madec and K. Takahashi and T. Yamagata},
+  title = {Interannual variations of low-frequency Madden-Julian Oscillation
+	in autral summer: Observations},
+  journal = CD,
+  year = {2009},
+  volume = {in press},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{JackMcD1995,
   author = {D. R. Jackett and T. J. McDougall},
@@ -1062,19 +1619,11 @@
   year = {1995},
   volume = {12},
-  pages = {381-389},
+  pages = {381--389},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@BOOK{Jerlov1968,
+@BOOK{Jerlov_Bk68,
   title = {Optical Oceanography},
-  publisher = {194pp},
-  year = {1968},
-  author = {N. G. Jerlov},
-  owner = {gm},
-  timestamp = {2007.08.04}
-}
-
-@BOOK{Jerlov_Bk1968,
   publisher = {Elsevier},
   year = {1968},
@@ -1085,28 +1634,6 @@
 }
 
-@INPROCEEDINGS{Killworth1989,
-  author = {P. D. Killworth},
-  title = {On the parameterization of deep convection in ocean models},
-  booktitle = {Parameterization of small-scale processes},
-  year = {1989},
-  editor = {Hawaiian winter workshop},
-  month = {January 17-20},
-  organization = {University of Hawaii at Manoa},
-  owner = {gm},
-  timestamp = {2007.08.06}
-}
-
-@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,
-	S. M.},
+@ARTICLE{Killworth_al_JPO91,
+  author = {P.D. Killworth and D. Stainforth and D.J. Webb and S.M. Paterson},
   title = {The Development of a Free-Surface Bryan-Cox-Semtner Ocean Model},
   journal = JPO,
@@ -1133,4 +1660,78 @@
 }
 
+@INPROCEEDINGS{Killworth1989,
+  author = {P. D. Killworth},
+  title = {On the parameterization of deep convection in ocean models},
+  booktitle = {Parameterization of small-scale processes},
+  year = {1989},
+  editor = {Hawaiian winter workshop},
+  month = {January 17-20},
+  organization = {University of Hawaii at Manoa},
+  owner = {gm},
+  timestamp = {2007.08.06}
+}
+
+@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{Koch-Larrouy_al_CD09,
+  author = {A. Koch-Larrouy and M. Lengaigne and P. Terray and G. Madec and S.
+	Masson},
+  title = {Tidal mixing in the Indonesian Seas and its effect on the tropical
+	climate system},
+  journal = CD,
+  year = {2009},
+  pages = {in press},
+  owner = {gm},
+  timestamp = {2009.08.16}
+}
+
+@ARTICLE{Koch-Larrouy_al_OD08b,
+  author = {A. Koch-Larrouy and G. Madec and B. Blanke and R. Molcard},
+  title = {Water mass transformation along the Indonesian throughflow in an
+	OGCM},
+  journal = OD,
+  year = {2008},
+  volume = {58, 3-4},
+  pages = {289--309},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1007/s10236-008-0155-4}
+}
+
+@ARTICLE{Koch-Larrouy_al_GRL07,
+  author = {A. Koch-Larrouy and G. Madec and P. Bouruet-Aubertot and T. Gerkema
+	and L. Bessieres and R. Molcard},
+  title = {Tidal mixing in the Indonesian Seas and its effect on the tropical
+	climate system},
+  journal = GRL,
+  year = {2007},
+  volume = {34},
+  pages = {L04604},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1029/2006GL028405}
+}
+
+@ARTICLE{Koch-Larrouy_al_OD08a,
+  author = {A. Koch-Larrouy and G. Madec and D. Iudicone and A. Atmadipoera and
+	R. Molcard},
+  title = {Physical processes contributing to the water mass transformation
+	of the Indonesian Throughflow},
+  journal = OD,
+  year = {2008},
+  volume = {58, 3-4},
+  pages = {275--288},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1007/s10236-008-0154-5}
+}
+
 @ARTICLE{Kolmogorov1942,
   author = {A. N. Kolmogorov},
@@ -1139,10 +1740,10 @@
   year = {1942},
   volume = {6},
-  pages = {56-58},
+  pages = {56--58},
   owner = {gm},
   timestamp = {2007.08.06}
 }
 
-@PHDTHESIS{Levy1996,
+@PHDTHESIS{Levy_PhD96,
   author = {M. L\'{e}vy},
   title = {Mod\'{e}lisation des processus biog\'{e}ochimiques en M\'{e}diterran\'{e}e
@@ -1154,5 +1755,5 @@
 }
 
-@ARTICLE{Levy2001,
+@ARTICLE{Levy_al_GRL01,
   author = {M. L\'{e}vy and A. Estubier and G Madec},
   title = {Choice of an advection scheme for biogeochemical models},
@@ -1164,18 +1765,53 @@
 }
 
-@ARTICLE{Levy1998,
+@ARTICLE{Levy_al_JGR09,
+  author = {M. L\'{e}vy and P. Klein and A.-M. Tr\'{e}guier and D. Iovino and
+	G. Madec and S. Masson and T. Takahashi},
+  title = {Impacts of sub-mesoscale physics on idealized gyres},
+  journal = JGR,
+  year = {2009},
+  volume = {in revision},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Levy_al_JMS99,
   author = {M. L\'{e}vy and L. M\'{e}mery and G. Madec},
   title = {The onset of a bloom after deep winter convection in the Northwestern
-	Mediterranean Sea: mesoscale 
-	
-	process study with a primitive equation model},
+	Mediterranean Sea: mesoscale process study with a primitive equation
+	model},
   journal = JMS,
+  year = {1999},
+  volume = {16/1-2},
+  pages = {7--21},
+  owner = {gm},
+  timestamp = {2007.08.10}
+}
+
+@ARTICLE{Levy_al_DSR98,
+  author = {M. L\'{e}vy and L. M\'{e}mery and G. Madec},
+  title = {The onset of the spring bloom in the MEDOC area: mesoscale spatial
+	variability},
+  journal = DSR,
   year = {1998},
-  volume = {16/1-2},
+  volume = {I, 46},
+  pages = {1137--1160},
   owner = {gm},
   timestamp = {2007.08.10}
 }
 
-@BOOK{LargeYeager2004,
+@ARTICLE{Levy_al_DSR00,
+  author = {M. L\'{e}vy and L. Mémery and G. Madec},
+  title = {Combined effects of mesoscale processes and atmospheric high-frequency
+	variability on the spring bloom in the MEDOC area},
+  journal = DSR,
+  year = {2000},
+  volume = {47},
+  pages = {527--531},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@BOOK{Large_Yeager_Rep04,
   title = {Diurnal to decadal global forcing for ocean and sea-ice models: the
 	data sets and flux climatologies},
@@ -1195,5 +1831,5 @@
   year = {1994},
   volume = {32},
-  pages = {363-404},
+  pages = {363--404},
   doi = {10.1029/94RG01872},
   owner = {gm},
@@ -1201,5 +1837,17 @@
 }
 
-@PHDTHESIS{Lazar1997,
+@ARTICLE{Latif_al_JC06,
+  author = {M. Latif and C. Böning and J. Willebrand and A. Biastoch and J. Dengg
+	and N. Keenlyside and U. Schweckendiek and G. Madec},
+  title = {Is the Thermohaline Circulation Changing?},
+  journal = JC,
+  year = {2006},
+  volume = {19},
+  pages = {4631--4637},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@PHDTHESIS{Lazar_PhD97,
   author = {A. Lazar},
   title = {La branche froide de la circulation thermohaline - sensibilit\'{e}
@@ -1212,5 +1860,5 @@
 }
 
-@ARTICLE{Lazar1999,
+@ARTICLE{Lazar_al_JPO99,
   author = {A. Lazar and G. Madec and P. Delecluse},
   title = {The Deep Interior Downwelling, the Veronis Effect, and Mesoscale
@@ -1219,5 +1867,5 @@
   year = {1999},
   volume = {29},
-  pages = {2945-2961},
+  pages = {2945--2961},
   number = {11},
   abstract = {Numerous numerical simulations of basin-scale ocean circulation display
@@ -1256,4 +1904,39 @@
 }
 
+@ARTICLE{Leclair_Madec_OM09,
+  author = {M. Leclair and G. Madec},
+  title = {A conservative leap-frog time stepping method},
+  journal = OM,
+  year = {2009},
+  volume = {XX},
+  pages = {YYY},
+  owner = {gm},
+  timestamp = {2009.08.15}
+}
+
+@ARTICLE{Lengaigne_al_JC03,
+  author = {M. Lengaigne and J.-P. Boulanger and C. Menkes and G. Madec and P.
+	Delecluse and E. Guilyardi, and J. Slingo},
+  title = {The March 1997 Westerly Wind Event and the onset of the 1997/98 El
+	Niño: Understanding the role of the atmospheric},
+  journal = JC,
+  year = {2003},
+  volume = {16, 20},
+  pages = {3330--3343},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Lengaigne_al_JGR02,
+  author = {M. Lengaigne and J.-P. Boulanger and C. Menkes and S. Masson and
+	G. Madec and P. Delecluse},
+  title = {Ocean response to the March 1997 Westerly Wind Event},
+  journal = JGR,
+  year = {2002},
+  doi = {10.1029/2001JC000841},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
 @ARTICLE{Lengaigne_al_JGR03,
   author = {M. Lengaigne and G. Madec and G. Alory and C. Menkes},
@@ -1263,7 +1946,33 @@
   year = {2003},
   volume = {108 (C11)},
-  pages = {3345, doi:10.1029/2002JC001704},
+  pages = {3345},
+  doi = {10.1029/2002JC001704},
   owner = {gm},
   timestamp = {2008.01.26}
+}
+
+@ARTICLE{Lengaigne_al_GRL09,
+  author = {M. Lengaigne and G. Madec and L. Bopp and C. Menkes and O. Aumont
+	and P. Cadule},
+  title = {Bio-physical feedbacks in the Arctic Ocean using an Earth System
+	model},
+  journal = GRL,
+  year = {2009},
+  volume = {submitted},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Lengaigne_al_CD07,
+  author = {M. Lengaigne and C. Menkes and O. Aumont and T. Gorgues and L. Bopp
+	and J.-M. Andr\'{e} G. Madec},
+  title = {Bio-physical feedbacks on the tropical Pacific climate in a Coupled
+	General Circulation Model},
+  journal = CD,
+  year = {2007},
+  volume = {28},
+  pages = {503--516},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -1274,10 +1983,10 @@
   journal = {Computer Methods in Applied Mechanics and Engineering},
   year = {1991},
-  pages = {17-74},
+  pages = {17--74},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@TECHREPORT{Leonard1988,
+@TECHREPORT{Leonard_Rep88,
   author = {B. P. Leonard},
   title = {Universal limiter for transient interpolation modelling of the advective
@@ -1296,5 +2005,5 @@
   year = {1979},
   volume = {19},
-  pages = {59-98},
+  pages = {59--98},
   month = jun,
   owner = {gm},
@@ -1331,20 +2040,31 @@
 }
 
-@ARTICLE{Lott1990,
+@ARTICLE{Lott_al_OM90,
   author = {F. Lott and G. Madec and J. Verron},
   title = {Topographic experiments in an Ocean General Circulation Model},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {1990},
   volume = {88},
-  pages = {1-4},
+  pages = {1--4},
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Luo_al_JC05,
+  author = {J.-J. Luo and S. Masson and E. Roeckner and G. Madec and T. Yamagata},
+  title = {Reducing climatology bias in an ocean-atmosphere CGCM with improved
+	coupling physics},
+  journal = JC,
+  year = {2005},
+  volume = {18 (13)},
+  pages = {2344--2360},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
 @BOOK{Madec_Bk08,
   title = {NEMO ocean engine},
-  publisher = {Note du P\^ole de mod\'{e}lisation, Institut Pierre- 
-	
-	Simon Laplace (IPSL), France, No 27, ISSN No 1288-1619},
+  publisher = {Note du P\^ole de mod\'{e}lisation, Institut Pierre-Simon Laplace
+	(IPSL), France, No 27, ISSN No 1288-1619},
   year = {2008},
   author = {G. Madec},
@@ -1353,15 +2073,28 @@
 }
 
-@PHDTHESIS{Madec1990,
+@BOOK{Madec_HDR01,
+  title = {Le Cycle des Masses d'Eau Oc\'{e}aniqueset sa variabilit\'{e} dans
+	le Syst\'{e}me Climatique},
+  year = {2001},
+  author = {G. Madec},
+  pages = {63pp.},
+  series = {Habilitation \'{a} Diriger des Recherches, Universit\'{e} Pierre
+	et Marie Curie},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@PHDTHESIS{Madec_PhD90,
   author = {G. Madec},
   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},
+  month = {2 mai},
   owner = {gm},
   timestamp = {2007.08.10}
 }
 
-@ARTICLE{Madec1991a,
+@ARTICLE{Madec_al_DAO91,
   author = {G. Madec and M. Chartier and M. Cr\'{e}pon},
   title = {Effect of thermohaline forcing variability on deep water formation
@@ -1370,9 +2103,11 @@
   journal = DAO,
   year = {1991},
+  volume = {15},
+  pages = {301--332},
   owner = {gm},
   timestamp = {2007.08.06}
 }
 
-@ARTICLE{Madec1991b,
+@ARTICLE{Madec_al_JPO91,
   author = {G. Madec and M. Chartier and P. Delecluse and M. Cr\'{e}pon},
   title = {A three-dimensional numerical study of deep water formation in the
@@ -1383,13 +2118,15 @@
   year = {1991},
   volume = {21},
+  pages = {1349--1371},
   owner = {gm},
   timestamp = {2007.08.06}
 }
 
-@INBOOK{Madec1991c,
+@INBOOK{Madec_Crepon_Bk91,
   chapter = {Thermohaline-driven deep water formation in the Northwestern Mediterranean
 	Sea},
+  pages = {241--265},
   title = {Deep convection and deep water formation in the oceans},
-  publisher = {Elsevier Oceanographic Series},
+  publisher = {Elsevier Oceanographic Series, P.C. Chu and J.C. Gascard (Eds.)},
   year = {1991},
   author = {G. Madec and M. Cr\'{e}pon},
@@ -1404,5 +2141,5 @@
   year = {1997},
   volume = {26},
-  pages = {12-15},
+  pages = {12--15},
   owner = {gm},
   timestamp = {2007.08.06}
@@ -1416,5 +2153,5 @@
 }
 
-@ARTICLE{MadecImb1996,
+@ARTICLE{Madec_Imbard_CD96,
   author = {G Madec and M Imbard},
   title = {A global ocean mesh to overcome the north pole singularity},
@@ -1422,8 +2159,8 @@
   year = {1996},
   volume = {12},
-  pages = {381-388}
-}
-
-@ARTICLE{Madec1996,
+  pages = {381--388}
+}
+
+@ARTICLE{Madec_al_JPO96,
   author = {G. Madec and F. Lott and P. Delecluse and M. Cr\'{e}pon},
   title = {Large-Scale Preconditioning of Deep-Water Formation in the Northwestern
@@ -1432,5 +2169,5 @@
   year = {1996},
   volume = {26},
-  pages = {1393-1408},
+  pages = {1393--1408},
   number = {8},
   month = aug,
@@ -1467,13 +2204,37 @@
 }
 
-@ARTICLE{Madec1988,
+@ARTICLE{Madec_al_OM88,
   author = {G. Madec and C. Rahier and M. Chartier},
-  title = {A comparison of two-dimensional elliptic solvers for the barotropic
-	streamfunction in a multilevel OGCM},
-  journal = {Ocean Modelling},
+  title = {A comparison of two-dimensional elliptic solvers for the streamfunction
+	in a multilevel OGCM},
+  journal = OM,
   year = {1988},
   volume = {78},
-  owner = {gm},
-  timestamp = {2007.08.10}
+  pages = {1-6},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Maes_al_CD98,
+  author = {C. Maes and P. Delecluse and G. Madec},
+  title = {Impact of westerly wind bursts on the warm pool of the TOGA-COARE
+	domain in an OGCM},
+  journal = CD,
+  year = {1998},
+  volume = {14},
+  pages = {55--70},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Maes_al_MWR97,
+  author = {C. Maes and G. Madec and P. Delecluse},
+  title = {Sensitivity of an Equatorial Pacific OGCM to the lateral diffusion},
+  journal = MWR,
+  year = {1997},
+  volume = {125, 5},
+  pages = {958--971},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -1485,5 +2246,5 @@
   year = {1998},
   volume = {103(C13)},
-  pages = {30,825-30,854},
+  pages = {30,825--30,854},
   owner = {gm},
   timestamp = {2007.08.05}
@@ -1494,8 +2255,23 @@
   title = {Open boundary conditions for long-term integrations of Regional Oceanic
 	Models},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2001},
   volume = {3},
-  pages = {1-20}
+  pages = {1--20}
+}
+
+@ARTICLE{Marsaleix_al_OM08,
+  author = {P. Marsaleix and F. Auclair and J. W. Floor and M. J. Herrmann and
+	C. Estournel and I. Pairaud and C. Ulses},
+  title = {Energy conservation issues in sigma-coordinate free-surface ocean
+	models},
+  journal = OM,
+  year = {2008},
+  volume = {20},
+  pages = {61--89},
+  number = {1},
+  doi = {10.1016/j.ocemod.2007.07.005},
+  owner = {gm},
+  timestamp = {2008.11.28}
 }
 
@@ -1512,5 +2288,5 @@
 }
 
-@PHDTHESIS{MartiTh1992,
+@PHDTHESIS{Marti_PhD92,
   author = {O. Marti},
   title = {Etude de l'oc\'{e}an mondial : mod\'{e}lisation de la circulation
@@ -1522,5 +2298,22 @@
 }
 
-@ARTICLE{Marti1992,
+@ARTICLE{Marti_al_CD09,
+  author = {O. Marti and P. Braconnot and J.-L. Dufresne and J. Bellier and R.
+	Benshila and S. Bony and P. Brockmann and P. Cadule and A. Caubel
+	and F Codron and S. Denvil and L. Fairhead and T. Fichefet and M.-A.
+	Filiberti and M.-A. Foujols and P. Friedlingstein and H. Goosse and
+	J.-Y. Grandpeix and E. Guilyardi and F. Hourdin and G. Krinner and
+	C. L\'{e}vy and G. Madec and J. Mignot and I. Musat and D. Swingedouw
+	and C. Talandier},
+  title = {Key features of the IPSL ocean atmosphere model and its sensitivity
+	to atmospheric resolution},
+  journal = CD,
+  year = {2009},
+  volume = {in press},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Marti_al_JGR92,
   author = {O. Marti and G. Madec and P. Delecluse},
   title = {Comment on "Net diffusivity in ocean general circulation models with
@@ -1529,8 +2322,38 @@
   year = {1992},
   volume = {97},
-  pages = {12763-12766},
+  pages = {12,763--12,766},
   month = aug,
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@INBOOK{Masson_al_Bk08,
+  chapter = {OPA9 - French experiments on the Earth Simulator and Teraflop Workbench
+	tunings},
+  pages = {25-34},
+  title = {In High Performance computing on Vector System 2007, Stuttgart, Germany},
+  publisher = {Springer-Verlag},
+  year = {2008},
+  editor = {Resch M, Roller S, Lammers P, Furui T, Galle M, Bez W},
+  author = {S. Masson and M.-A. Foujols and P. Klein and G. Madec and L. Hua
+	and M. Levy and H. Sasaki and K. Takahashi and F. Svensson},
+  doi = {10.1007/978-3-540-74384-2},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Masson_al_GRL05,
+  author = {S. Masson and J.-J. Luo and G. Madec and J. Vialard and F. Durand
+	and S. Gualdi and E. Guilyardi and S. Behera and P. Delecluse and
+	A. Navarra and T. Yamagata},
+  title = {Impact of barrier layer on winter-spring variability of the South-Eastern
+	Arabian Sea},
+  journal = GRL,
+  year = {2005},
+  volume = {32},
+  pages = {L07703},
+  doi = {10.1029/2004GL021980},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -1538,8 +2361,8 @@
   author = {T. J. McDougall},
   title = {Neutral Surfaces},
-  journal = {Journal of Physical Oceanography},
+  journal = JPO,
   year = {1987},
   volume = {17},
-  pages = {1950-1964},
+  pages = {1950--1964},
   number = {11},
   abstract = {Scalar properties in the ocean are stirred (and subsequently mixed)
@@ -1576,10 +2399,34 @@
   title = {Flux measurements across a finger interface at low values of the
 	stability ratio},
-  journal = {Journal of Marine Research},
+  journal = JMR,
   year = {1984},
   volume = {42},
-  pages = {1-14},
+  pages = {1--14},
   owner = {gm},
   timestamp = {2008.05.20}
+}
+
+@ARTICLE{Mellor_Blumberg_JPO04,
+  author = {G. Mellor and A. Blumberg},
+  title = {Wave Breaking and Ocean Surface Layer Thermal Response},
+  journal = JPO,
+  year = {2004},
+  volume = {34},
+  pages = {693--698},
+  owner = {gm},
+  timestamp = {2009.01.16}
+}
+
+@ARTICLE{Menkes_al_JPO06,
+  author = {C. Menkes and J. Vialard and S C. Kennan and J.-P. Boulanger and
+	G. Madec},
+  title = {A modelling study of the three-dimensional heat budget of Tropical
+	Instability Waves in the Equatorial Pacific},
+  journal = JPO,
+  year = {2006},
+  volume = {36, 5},
+  pages = {847--865},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -1590,5 +2437,5 @@
   year = {1999},
   volume = {29},
-  pages = {1124-1142},
+  pages = {1124--1142},
   number = {6},
   abstract = {A global ocean model is described in which parameterizations of diapycnal
@@ -1617,5 +2464,5 @@
 }
 
-@ARTICLE{Murray1996,
+@ARTICLE{Murray_JCP96,
   author = {R. J. Murray},
   title = {Explicit Generation of Orthogonal Grids for Ocean Models},
@@ -1623,5 +2470,5 @@
   year = {1996},
   volume = {126},
-  pages = {251-273},
+  pages = {251--273},
   number = {2},
   month = {July},
@@ -1630,5 +2477,5 @@
 }
 
-@PHDTHESIS{OlivierPh2001,
+@PHDTHESIS{Olivier_PhD01,
   author = {F. Olivier},
   title = {Etude de l'activit\'{e} biologique et de la circulation oc\'{e}anique
@@ -1640,5 +2487,5 @@
 }
 
-@ARTICLE{PacPhil1981,
+@ARTICLE{Pacanowski_Philander_JPO81,
   author = {R.C. Pacanowski and S.G.H. Philander},
   title = {Parameterization of Vertical Mixing in Numerical Models of Tropical
@@ -1647,5 +2494,5 @@
   year = {1981},
   volume = {11},
-  pages = {1443-1451},
+  pages = {1443--1451},
   number = {11},
   abstract = {Measurements indicate that mixing processes are intense in the surface
@@ -1682,4 +2529,18 @@
 }
 
+@ARTICLE{Park_al_JC09,
+  author = {W. Park and N. Keenlyside and M. Latif and A. Str\¨{o}h and R. Redler
+	and E. Roeckner and G. Madec},
+  title = {Tropical Pacific Climate and its Response to Global Warming in the
+	Kiel Climate Model},
+  journal = JC,
+  year = {2009},
+  volume = {22, 1},
+  pages = {71--92},
+  doi = {10.1175/2008JCLI2261.1},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
 @ARTICLE{Paulson1977,
   author = {C. A. Paulson and J. J. Simpson},
@@ -1688,5 +2549,5 @@
   year = {1977},
   volume = {7},
-  pages = {952-956},
+  pages = {952--956},
   number = {6},
   abstract = {Observations were made of downward solar radiation as a function of
@@ -1715,23 +2576,35 @@
 }
 
-@ARTICLE{Penduff2000,
+@ARTICLE{Penduff_al_OM06,
+  author = {T. Penduff and B. Barnier and J.-M. Molines and G. Madec},
+  title = {On the use of current meter data to assess the realism of ocean model
+	simulations},
+  journal = OM,
+  year = {2006},
+  volume = {11, 3--4},
+  pages = {399--416},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Penduff_al_JGR00,
   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
+  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,
+  pages = {11,279--11,297}
+}
+
+@ARTICLE{Penduff_al_OS07,
   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},
+  journal = OS,
   year = {2007},
-  volume = {?},
-  pages = {in revision}
+  volume = {3},
+  pages = {509--524}
 }
 
@@ -1742,9 +2615,34 @@
   journal = {Transactions of the American Mathematical Society},
   year = {1959},
-  volume = {90(2)},
-  pages = {193-254},
+  volume = {90 (2)},
+  pages = {193--254},
   doi = {doi:10.2307/1993202},
   owner = {gm},
   timestamp = {2007.08.10}
+}
+
+@ARTICLE{Le_Quere_al_GBC00,
+  author = {C. Le Qu\'{e}r\'{e} and J. C. Orr and P. Monfray and O. Aumont and
+	G. Madec},
+  title = {Interannual variability of the global and regional sea-air flux of
+	C02 from 1979 to 1993},
+  journal = GBC,
+  year = {2000},
+  volume = {14},
+  pages = {1247--1266},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Raynaud_al_GRL00,
+  author = {S. Raynaud and S. Speich and E. Guilyardi and G. Madec},
+  title = {Impact of the ocean lateral diffusion on the ENSO-like variability
+	of a global coupled GCM},
+  journal = GRL,
+  year = {2000},
+  volume = {27, 19},
+  pages = {3041--3044},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -1755,5 +2653,5 @@
   year = {1982},
   volume = {13},
-  pages = {1154-1158},
+  pages = {1154--1158},
   owner = {gm},
   timestamp = {2008.02.02}
@@ -1786,9 +2684,33 @@
   title = {The integration of a Low order spectral form of the primitive meteorological
 	equations},
-  journal = {J. Meteo. Soc. Japan},
+  journal = JMSJ,
   year = {1966},
   volume = {44, 2},
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@ARTICLE{Rodgers_al_GRL04,
+  author = {K. Rodgers and O. Aumont and G. Madec and C. Menkes},
+  title = {Radiocarbon as a thermocline proxy for the eastern equatorial Pacific},
+  journal = GRL,
+  year = {2004},
+  volume = {31},
+  pages = {L14314},
+  doi = {10.1029/2004GL019764},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Rodgers_al_GRL03,
+  author = {K. Rodgers and B. Blanke and G. Madec and O. Aumont and P. Ciais
+	and J.-C. Dutay},
+  title = {Extratropical sources of equatorial pacific upwelling in an OGCM},
+  journal = GRL,
+  year = {2003},
+  volume = {30, 2},
+  doi = {10.1029/2002GL016003},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -1802,5 +2724,5 @@
 }
 
-@ARTICLE{Roullet2000,
+@ARTICLE{Roullet_Madec_JGR00,
   author = {G. Roullet and G. Madec},
   title = {salt conservation, free surface, and varying levels: a new formulation
@@ -1809,6 +2731,6 @@
   year = {2000},
   volume = {105},
-  pages = {23,927-23,942},
-  owner = {sandra},
+  pages = {23,927--23,942},
+  owner = {gm},
   pdf = {Roullet_Madec_JGR00.pdf},
   timestamp = {2007.03.22}
@@ -1821,5 +2743,5 @@
   year = {1975},
   volume = {32},
-  pages = {680-689},
+  pages = {680--689},
   number = {4},
   abstract = {Two simple numerical models of the shallow-water equations identical
@@ -1847,5 +2769,5 @@
   year = {1982},
   volume = {87},
-  pages = {394-409},
+  pages = {394--409},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -1859,5 +2781,5 @@
   year = {2005},
   volume = {9, 4},
-  pages = {347-404},
+  pages = {347--404},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -1873,5 +2795,6 @@
   year = {2003},
   volume = {108(C3)},
-  pages = {3090, doi:10.1029/2001JC001047},
+  pages = {3090},
+  doi = {10.1029/2001JC001047},
   owner = {gm},
   timestamp = {2007.08.05}
@@ -1885,5 +2808,5 @@
   year = {1996},
   volume = {124},
-  pages = {1285-1300},
+  pages = {1285--1300},
   number = {6},
   abstract = {Dissipation in numerical ocean models has two purposes: to simulate
@@ -1916,5 +2839,5 @@
 }
 
-@ARTICLE{Simmons2003,
+@ARTICLE{Simmons_al_OM03,
   author = {H. L. Simmons and S. R. Jayne and L. C. St. Laurent and A. J. Weaver},
   title = {Tidally driven mixing in a numerical model of the
@@ -1923,5 +2846,5 @@
   journal = OM,
   year = {2003},
-  pages = {1-19},
+  pages = {1--19},
   abstract = {Astronomical data reveals that approximately 3.5 terawatts (TW) of
 	tidal energy is dissipated in the
@@ -1965,18 +2888,30 @@
 	Climatological comparisons show that the parameterized mixing scheme
 	results in a substantial reduction},
-  owner = {sandra},
+  owner = {gm},
   pdf = {Simmons_mixing_OM2003.pdf},
   timestamp = {2007.03.22}
 }
 
-@ARTICLE{Song1994,
+@ARTICLE{Le_Sommer_al_OM09,
+  author = {J. Le Sommer and T. Penduff and S. Theetten and G. Madec and B. Barnier},
+  title = {How momentum advection schemes influence current-topography interactions
+	at eddy permitting resolution},
+  journal = OM,
+  year = {2009},
+  volume = {29, 1},
+  pages = {1--14},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2008.11.007}
+}
+
+@ARTICLE{Song_Haidvogel_JCP94,
   author = {Y. Song and D. Haidvogel},
-  title = {A Semi-implicit Ocean Circulation Model Using a Generalized Topography-Following
-	Coordinate System
-	
-	Authors:},
+  title = {A semi-implicit ocean circulation model using a generalized topography-following
+	coordinate system},
   journal = JCP,
   year = {1994},
   volume = {115, 1},
+  pages = {228--244},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -1990,5 +2925,5 @@
   year = {1998},
   volume = {126},
-  pages = {3213-3230},
+  pages = {3213--3230},
   number = {12},
   abstract = {A Jacobian formulation of the pressure gradient force for use in models
@@ -2036,5 +2971,5 @@
   year = {1998},
   volume = {126},
-  pages = {3231-3247},
+  pages = {3231--3247},
   number = {12},
   abstract = {A new formulation of the pressure gradient force for use in models
@@ -2071,5 +3006,17 @@
 }
 
-@PHDTHESIS{Speich1992,
+@ARTICLE{Speer_al_Tel00,
+  author = {K. Speer and E. Guilyardi and G. Madec},
+  title = {Southern Ocean transformation in a coupled model with and without
+	eddy mass fluxes},
+  journal = {Tellus},
+  year = {2000},
+  volume = {52A, 5},
+  pages = {554--565},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@PHDTHESIS{Speich_PhD92,
   author = {S. Speich},
   title = {Etude du for\c{c}age de la circulation g\'{e}n\'{e}rale oc\'{e}anique
@@ -2081,10 +3028,22 @@
 }
 
-@ARTICLE{Speich1996,
+@ARTICLE{Speich_al_GRL01,
+  author = {S. Speich and B. Blanke and G. Madec},
+  title = {Warm and cold water paths of an OGCM thermohaline conveyor belt},
+  journal = GRL,
+  year = {2001},
+  volume = {28, 2},
+  pages = {311--314},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Speich_al_JPO96,
   author = {S. Speich and G. Madec and M. Cr\'{e}pon},
   title = {The circulation in the Alboran Sea - a sensitivity study},
   journal = JPO,
   year = {1996},
-  volume = {26},
+  volume = {26, 3},
+  pages = {320--340},
   owner = {gm},
   timestamp = {2007.08.06}
@@ -2094,5 +3053,5 @@
   author = {M. Steele and R. Morley and W. Ermold},
   title = {PHC- A Global Ocean Hydrography with a High-Quality Arctic Ocean},
-  journal = {Journal of Climate},
+  journal = JC,
   year = {2001},
   volume = {14},
@@ -2116,5 +3075,5 @@
 }
 
-@ARTICLE{Stein1992,
+@ARTICLE{Stein_Stein_Nat92,
   author = {C. A. Stein and S. Stein},
   title = {A model for the global variation in oceanic depth and heat flow with
@@ -2123,14 +3082,14 @@
   year = {1992},
   volume = {359},
-  pages = {123-129},
+  pages = {123--129},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@ARTICLE{Thiem2006,
+@ARTICLE{Thiem_Berntsen_OM06,
   author = {O. Thiem and J. Berntsen},
   title = {Internal pressure errors in sigma-coordinate ocean models due to
 	anisotropy},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2006},
   volume = {12, 1-2},
@@ -2146,5 +3105,5 @@
 	
 	sea ice-ocean model.},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2005},
   volume = {8},
@@ -2154,5 +3113,5 @@
 }
 
-@ARTICLE{Treguier1992,
+@ARTICLE{Treguier_JGR92,
   author = {A.M. Tr\'{e}guier},
   title = {Kinetic energy analysis of an eddy resolving, primitive equation
@@ -2164,5 +3123,5 @@
 }
 
-@ARTICLE{Treguier2001,
+@ARTICLE{Treguier_al_JGR01,
   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
@@ -2173,5 +3132,15 @@
   year = {2001},
   volume = {106},
-  pages = {22115-22129}
+  pages = {22,115--22,129}
+}
+
+@ARTICLE{Treguier_al_DSR03,
+  author = {A.-M. Tr\'{e}guier and O. Boedel and B. Barnier and G. Madec},
+  title = {Agulhas eddy fluxes in a 1/6^o Atlantic model},
+  journal = DSR,
+  year = {2003},
+  pages = {251--280},
+  owner = {gm},
+  timestamp = {2009.08.20}
 }
 
@@ -2183,7 +3152,35 @@
   year = {1996},
   volume = {101},
-  pages = {20877-20881},
+  pages = {20,877--20,881},
   owner = {gm},
   timestamp = {2007.08.03}
+}
+
+@ARTICLE{Treguier_al_OS07,
+  author = {A.-M. Tr\'{e}guier and M. H. England and S. R. Rintoul and G. Madec
+	and J. Le Sommer and J.-M. Molines},
+  title = {Southern Ocean overturning across streamlines in an eddying simulation
+	of the Antarctic Circumpolar Current},
+  journal = OS,
+  year = {2007},
+  volume = {4},
+  pages = {653--698},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Treguier_al_OD06,
+  author = {A.-M. Tr\'{e}guier and C. Gourcuff and P. Lherminier and H. Mercier
+	and B. Barnier and G. Madec and J.-M. Molines and T. Penduff and
+	L. Czeschel and C. Böning},
+  title = {Internal and forced variability along a section between Greenland
+	and Portugal in the CLIPPER Atlantic model},
+  journal = OD,
+  year = {2006},
+  volume = {56},
+  pages = {568--580},
+  doi = {i10.1007/s10236-006-0069-y},
+  owner = {gm},
+  timestamp = {2009.08.19}
 }
 
@@ -2195,5 +3192,5 @@
   year = {1997},
   volume = {27},
-  pages = {567-580},
+  pages = {567--580},
   number = {4},
   abstract = {A parameterization of mesoscale eddy fluxes in the ocean should be
@@ -2233,25 +3230,53 @@
 }
 
-@TECHREPORT{valal00,
+@TECHREPORT{Valcke_al_Rep00,
   author = {S. Valcke and L. Terray and A. Piacentini },
   title = {The OASIS Coupled User Guide Version 2.4},
   institution = {CERFACS},
   year = {2000},
-  number = {TR/CMGC/00-10}
-}
-
-@ARTICLE{Vancoppenolle_al_OM08,
+  number = {TR/CMGC/00-10},
+  owner = {gm}
+}
+
+@ARTICLE{Vancoppenolle_al_OM09b,
+  author = {M. Vancoppenolle and T. Fichefet and H. Goosse},
+  title = {Simulating the mass balance and salinity of Arctic and Antarctic
+	sea ice. 2. Importance of sea ice salinity variations},
+  journal = {0M},
+  year = {2009},
+  volume = {27},
+  pages = {54--69},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Vancoppenolle_al_OM09a,
   author = {M. Vancoppenolle and T. Fichefet and H. Goosse and S. Bouillon and
 	G. Madec and M. A. Morales Maqueda},
   title = {Simulating the mass balance and salinity of Arctic and Antarctic
 	sea ice. 1. Model description and validation},
-  journal = {Ocean Modelling},
-  year = {2008},
-  volume = {in press},
+  journal = OM,
+  year = {2009},
+  volume = {27},
+  pages = {33--53},
+  doi = {10.1016/j.ocemod.2008.10.005},
   owner = {gm},
   timestamp = {2008.07.05}
 }
 
-@ARTICLE{Weatherly1984,
+@ARTICLE{Vialard_al_JPO01,
+  author = {J. Vialard and C. Menkes and J.-P. Boulanger and P. Delecluse and
+	E. Guilyardi and M.J. McPhaden and G. Madec},
+  title = {A Model Study of Oceanic Mechanisms Affecting Equatorial Pacific
+	Sea Surface Temperature During the 1997-98 El Niño},
+  journal = JPO,
+  year = {2001},
+  volume = {31, 7},
+  pages = {1649--1675},
+  owner = {gm},
+  timestamp = {2009.08.20}
+}
+
+@ARTICLE{Weatherly_JMR84,
   author = {G. L. Weatherly},
   title = {An estimate of bottom frictional dissipation by Gulf Stream fluctuations},
@@ -2264,5 +3289,5 @@
 }
 
-@ARTICLE{Weaver1997,
+@ARTICLE{Weaver_Eby_JPO97,
   author = {A. J. Weaver and M. Eby},
   title = {On the numerical implementation of advection schemes for use in conjuction
@@ -2277,5 +3302,5 @@
 }
 
-@ARTICLE{Webb1998,
+@ARTICLE{Webb_al_JAOT98,
   author = {D. J. Webb and B. A. de Cuevas and C. S. Richmond},
   title = {Improved Advection Schemes for Ocean Models},
@@ -2310,5 +3335,5 @@
 }
 
-@ARTICLE{Willebrand2001,
+@ARTICLE{Willebrand_al_PO01,
   author = {J. Willebrand and B. Barnier and C. Boning and C. Dieterich and P.
 	D. Killworth and C. Le Provost and Y. Jia and J.-M. Molines and A.
@@ -2316,13 +3341,50 @@
   title = {Circulation characteristics in three eddy-permitting models of the
 	North Atlantic},
-  journal = {Progress in Oceanography},
+  journal = PO,
   year = {2001},
   volume = {48, 2},
-  pages = {123-161},
+  pages = {123--161},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@ARTICLE{Zalesak1979,
+@ARTICLE{Williams_al_DAO09,
+  author = {P.D. Williams and E. Guilyardi and G. Madec and S. Gualdi and E.
+	Scoccimarro},
+  title = {The role of mean ocean salinity on climate},
+  journal = DAO,
+  year = {2009},
+  volume = {in press},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Williams_al_GRL07,
+  author = {P.D. Williams and E. Guilyardi and R. Sutton and J.M. Gregory and
+	G. Madec},
+  title = {A new feedback on climate change from the hydrological cycle},
+  journal = GRL,
+  year = {2007},
+  volume = {34},
+  pages = {L08706},
+  doi = {10.1029/2007GL029275},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Williams_al_CD06,
+  author = {P.D. Williams and E. Guilyardi and R. Sutton and J.M. Gregory and
+	G. Madec},
+  title = {On the climate response of the low-latitude Pacific ocean to changes
+	in the global freshwater cycle},
+  journal = CD,
+  year = {2006},
+  volume = {27},
+  pages = {593--611},
+  owner = {gm},
+  timestamp = {2009.08.19}
+}
+
+@ARTICLE{Zalesak_JCP79,
   author = {S. T. Zalesak},
   title = {Fully multidimensional flux corrected transport algorithms for fluids},
@@ -2334,5 +3396,5 @@
 }
 
-@ARTICLE{Zhang1992,
+@ARTICLE{Zhang_Endoh_JGR92,
   author = {Zhang, R.-H. and Endoh, M.},
   title = {A free surface general circulation model for the tropical Pacific
@@ -2341,5 +3403,5 @@
   year = {1992},
   volume = {97},
-  pages = {11237-11255},
+  pages = {11,237--11,255},
   month = jul,
   owner = {gm}
@@ -2361,5 +3423,5 @@
 2 ExplicitGroup:Previous events\;0\;;
 2 ExplicitGroup:Reviews\;0\;;
-2 ExplicitGroup:Simple models\;0\;Zhang1992\;;
+2 ExplicitGroup:Simple models\;0\;Zhang_Endoh_JGR92\;;
 2 ExplicitGroup:SPL, SC, mean\;0\;;
 2 ExplicitGroup:Teleconnections\;0\;;
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Abstracts_Foreword.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Abstracts_Foreword.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Abstracts_Foreword.tex	(revision 1831)
@@ -45,5 +45,12 @@
 \chapter*{Disclaimer}
 
-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.
+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}
@@ -51,5 +58,7 @@
  \vspace{0.5cm}
 
-Madec, G., 2008: NEMO ocean engine. \textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, No 27, ISSN No 1288-1619.\\
+Madec, G., and the NEMO team, 2008: NEMO ocean engine. 
+\textit{Note du P\^ole de mod\'{e}lisation}, Institut Pierre-Simon Laplace (IPSL), France, 
+No 27, ISSN No 1288-1619.\\
 
 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_A.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_A.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_A.tex	(revision 1831)
@@ -3,16 +3,24 @@
 % Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
 % ================================================================
-\chapter{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 
-($i.e.$ an orthogonal curvilinear coordinate in the horizontal and $s$-coordinate 
-in the vertical), we start from the set of equations 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 
+\newpage
+$\ $\newline    % force a new ligne
+
+% ================================================================
+% Chain rule
+% ================================================================
+\section{Chain rule of $s-$coordinate}
+\label{Apdx_A_continuity}
+
+In order to establish the set of Primitive Equation in curvilinear $s-$coordinates
+($i.e.$ an orthogonal curvilinear coordinate in the horizontal and an Arbitrary Lagrangian 
+Eulerian (ALE) coordinate in the vertical), we start from the set of equations 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 $a = a(i,j,z,t)$. 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 :
+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 
@@ -21,5 +29,5 @@
 \end{equation}
 
-The chain rule to establish the model equations in the curvilinear $s$-coordinate 
+The chain rule to establish the model equations in the curvilinear $s-$coordinate 
 system is:
 \begin{equation} \label{Apdx_A_s_chain_rule}
@@ -42,6 +50,6 @@
 \end{equation}
 
-In particular applying the time derivative chain rule to $z$ provides the 
-expression for $w_s$,  the vertical velocity of the $s-$surfaces:
+In particular applying the time derivative chain rule to $z$ provides the expression 
+for $w_s$,  the vertical velocity of the $s-$surfaces referenced to a fix z-coordinate:
 \begin{equation} \label{Apdx_A_w_in_s}
 w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s 
@@ -50,15 +58,18 @@
 \end{equation}
 
+
 % ================================================================
 % continuity equation
 % ================================================================
-\section{Continuity Equation}
-\label{Apdx_B_continuity}
+\section{Continuity Equation in $s-$coordinate}
+\label{Apdx_A_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*}
+the velocity relative to the ($i$,$j$,$z$) coordinate system is transformed as follows
+in order to obtain its expression in the curvilinear $s-$coordinate system:
+
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
 \nabla \cdot {\rm {\bf U}} 
 &= \frac{1}{e_1 \,e_2 }  \left[ \left. {\frac{\partial (e_2 \,u)}{\partial i}} \right|_z 
@@ -89,20 +100,20 @@
 						-  \sigma _1 \frac{\partial u}{\partial s}
 						-  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\
-\\
-\end{align*}
-
-Noting that $\frac{1}{e_1 }\left. {\frac{\partial e_3 }{\partial i}} 
-\right|_s =\frac{1}{e_1 }\left. {\frac{\partial ^2z}{\partial i\,\partial 
-s}} \right|_s =\frac{\partial }{\partial s}\left( {\frac{1}{e_1 }\left. 
-{\frac{\partial z}{\partial i}} \right|_s } \right)=\frac{\partial \sigma _1 
-}{\partial s}$ and $\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} 
-\right|_s =\frac{\partial \sigma _2 }{\partial s}$, it becomes:
-
-\begin{align*}
+%
+\intertext{Noting that $
+  \frac{1}{e_1} \left.{ \frac{\partial e_3}{\partial i}} \right|_s 
+=\frac{1}{e_1} \left.{ \frac{\partial^2 z}{\partial i\,\partial s}} \right|_s 
+=\frac{\partial}{\partial s} \left( {\frac{1}{e_1 } \left.{ \frac{\partial z}{\partial i} }\right|_s } \right)
+=\frac{\partial \sigma _1}{\partial s}
+$ and $
+\frac{1}{e_2 }\left. {\frac{\partial e_3 }{\partial j}} \right|_s 
+=\frac{\partial \sigma _2}{\partial s}
+$, it becomes:}
+%
 \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
+& \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] \\ 
 \\
@@ -111,15 +122,18 @@
 		+\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*} 
- 
+\end{array} } 		
+\end{align*}
+\end{subequations}
+
 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:
+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 for 
-the divergence of the velocity in the curvilinear $s$-coordinate system:
-\begin{align*} \label{Apdx_A_A4}
+the divergence of the velocity in the curvilinear $s-$coordinate system:
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
 \nabla \cdot {\rm {\bf U}} 
 &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
@@ -147,9 +161,11 @@
 + \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
 + \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\
+\end{array} } 		
 \end{align*}
+\end{subequations}
 
 As a result, the continuity equation \eqref{Eq_PE_continuity} in the 
-$s$-coordinates becomes:
-\begin{equation} \label{Apdx_A_A5}
+$s-$coordinates is:
+\begin{equation} \label{Apdx_A_sco_Continuity}
 \frac{1}{e_3 } \frac{\partial e_3}{\partial t} 
 + \frac{1}{e_1 \,e_2 \,e_3 }\left[ 
@@ -158,111 +174,338 @@
  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0   
 \end{equation}
+A additional term has appeared that take into account the contribution of the time variation 
+of the vertical coordinate to the volume budget.
+
 
 % ================================================================
 % momentum equation
 % ================================================================
-\section{Momentum Equation}
-\label{Apdx_B_momentum}
+\section{Momentum Equation in $s-$coordinate}
+\label{Apdx_A_momentum}
+
+Here we only consider the first component of the momentum equation, 
+the generalization to the second one being straightforward.
+
+$\ $\newline    % force a new ligne
+
+$\bullet$ \textbf{Total derivative in vector invariant form}
 
 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 component). 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}      \\
-\\
-&\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 \\ 
-&\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} }    \\
+equation in the vector invariant form. Its total $z-$coordinate time derivative, 
+$\left. \frac{D u}{D t} \right|_z$ can be transformed as follows in order to obtain 
+its expression in the curvilinear $s-$coordinate system:
+
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
+\left. \frac{D u}{D t} \right|_z 
+&= \left. {\frac{\partial u }{\partial t}} \right|_z 
+   - \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} \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z 
+   - \left. \zeta \right|_z v 
+  +  \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}      \\
+%
+\intertext{introducing the chain rule (\ref{Apdx_A_s_chain_rule}) }
+%
+&= \left. {\frac{\partial u }{\partial t}} \right|_z       
+   - \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  \\ 
+& \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} }    \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z       
+  + \left. \zeta \right|_s \;v
+  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
+&\qquad \qquad \qquad \quad
+  + \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}      \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z       
+  + \left. \zeta \right|_s \;v
+  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s      \\
+&\qquad \qquad \qquad \quad
+ + \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] \\
+\\
+&= \left. {\frac{\partial u }{\partial t}} \right|_z       
+  + \left. \zeta \right|_s \;v
+  + \frac{1}{2\,e_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}   \\
+%
+\intertext{Introducing $\omega$, the dia-a-surface velocity given by (\ref{Apdx_A_w_s}) }
+%
+&= \left. {\frac{\partial u }{\partial t}} \right|_z       
+  + \left. \zeta \right|_s \;v
+  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s  
+  + \frac{1}{e_3 } \left( \omega - w_s \right) \frac{\partial u}{\partial s}   \\
+\end{array} } 		
 \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{subequations}
+%
+Applying the time derivative chain rule (first equation of (\ref{Apdx_A_s_chain_rule}))
+to $u$ and using (\ref{Apdx_A_w_in_s}) provides the expression of the last term 
+of the right hand side,
+\begin{equation*} {\begin{array}{*{20}l} 
+w_s  \;\frac{\partial u}{\partial s} 
+	= \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s}
+	= \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad , 
+\end{array} } 		
+\end{equation*}
+leads to the $s-$coordinate formulation of the total $z-$coordinate time derivative, 
+$i.e.$ the total $s-$coordinate time derivative :
+\begin{align} \label{Apdx_A_sco_Dt_vect}
+\left. \frac{D u}{D t} \right|_s 
+  = \left. {\frac{\partial u }{\partial t}} \right|_s       
+  + \left. \zeta \right|_s \;v
+  + \frac{1}{2\,e_1}\left. {\frac{\partial (u^2+v^2)}{\partial i}} \right|_s  
+  + \frac{1}{e_3 } \omega \;\frac{\partial u}{\partial s}   
+\end{align}
+Therefore, the vector invariant form of the total time derivative has exactly the same 
+mathematical form in $z-$ and $s-$coordinates. This is not the case for the flux form
+as shown in next paragraph.
+
+$\ $\newline    % force a new ligne
+
+$\bullet$ \textbf{Total derivative in flux form}
+
+Let us start from the total time derivative in the curvilinear $s-$coordinate system 
+we have just establish. Following the procedure used to establish (\ref{Eq_PE_flux_form}), 
+it can be transformed into :
+%\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
+\left. \frac{D u}{D t} \right|_s  &= \left. {\frac{\partial u }{\partial t}} \right|_s  
+           					    & -  \zeta \;v 
+           					+ \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+                                                 + \frac{1}{e_3} \omega \;\frac{\partial u}{\partial s}          \\
+\\
+  &= \left. {\frac{\partial u }{\partial t}} \right|_s  
+          &+\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( {\omega\,u} \right)}{\partial s}                                \\ 
+\\
+        &&- \,u \left[     \frac{1}{e_1 e_2 } \left(    \frac{\partial(e_2 u)}{\partial i}
+				  	  	                 + \frac{\partial(e_1 v)}{\partial j}    \right)
+                          + \frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]      \\
+\\
+        &&- \frac{v}{e_1 e_2 }\left(    v	\;\frac{\partial e_2 }{\partial i}
+				 	           -u	\;\frac{\partial e_1 }{\partial j} 	\right)                             \\
+\end{array} } 		
 \end{align*}
-
-Therefore, the non-linear terms of the momentum equation have the same 
-form 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_grad_p}
+%
+Introducing the vertical scale factor inside the horizontal derivative of the first two terms 
+($i.e.$ the horizontal divergence), it becomes :
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
+%\begin{align*} {\begin{array}{*{20}l} 
+%{\begin{array}{*{20}l} 
+\left. \frac{D u}{D t} \right|_s  
+   &= \left. {\frac{\partial u }{\partial t}} \right|_s  
+   &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u^2 )}{\partial i}
+			                  		  + \frac{\partial( e_1 e_3 \,u v )}{\partial j}     
+	     							   -  e_2 u u \frac{\partial e_3}{\partial i}
+							  -  e_1 u v \frac{\partial e_3 }{\partial j}    \right)
+	    + \frac{1}{e_3} \frac{\partial \left( {\omega\,u} \right)}{\partial s}                                  \\
+\\
+           && - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
+			   							 + \frac{\partial(e_1 e_3 \, v)}{\partial j}  
+	                							 -  e_2 u \;\frac{\partial e_3 }{\partial i}
+				 	  	                      -  e_1 v \;\frac{\partial e_3 }{\partial j}   \right)
+             -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                      \\
+\\
+            && - \frac{v}{e_1 e_2 }\left( 	v  \;\frac{\partial e_2 }{\partial i}
+				     	              -u  \;\frac{\partial e_1 }{\partial j} 	\right)                      \\
+\\
+   &= \left. {\frac{\partial u }{\partial t}} \right|_s  
+   &+ \frac{1}{e_1\,e_2\,e_3}  \left(  \frac{\partial( e_2 e_3 \,u\,u )}{\partial i}
+			                  		  + \frac{\partial( e_1 e_3 \,u\,v )}{\partial j}    \right)
+     + \frac{1}{e_3 } \frac{\partial \left( {\omega\,u} \right)}{\partial s}                               \\
+\\
+&& - \,u \left[  \frac{1}{e_1 e_2 e_3} \left(   \frac{\partial(e_2 e_3 \, u)}{\partial i} 
+									+ \frac{\partial(e_1 e_3 \, v)}{\partial j}  \right)
+        -\frac{1}{e_3}        \frac{\partial \omega}{\partial s}                       \right]                  
+     - \frac{v}{e_1 e_2 }\left( 	v   \;\frac{\partial e_2 }{\partial i}
+				            	      -u   \;\frac{\partial e_1 }{\partial j} 	\right)                  \\
+%
+\intertext {Introducing a more compact form for the divergence of the momentum fluxes, 
+and using (\ref{Apdx_A_sco_Continuity}), the $s-$coordinate continuity equation, 
+it becomes : }
+%
+   &= \left. {\frac{\partial u }{\partial t}} \right|_s  
+   &+ \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
+     + \,u \frac{1}{e_3 } \frac{\partial e_3}{\partial t}    
+      - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
+				             -u  \;\frac{\partial e_1 }{\partial j} 	\right) \\
+\end{array} } 		
+\end{align*}
+\end{subequations}
+which leads to the $s-$coordinate flux formulation of the total $s-$coordinate time derivative, 
+$i.e.$ the total $s-$coordinate time derivative in flux form :
+\begin{flalign}\label{Apdx_A_sco_Dt_flux}
+\left. \frac{D u}{D t} \right|_s   = \frac{1}{e_3}  \left. \frac{\partial ( e_3\,u)}{\partial t} \right|_s  
+           + \left.  \nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right)    \right|_s
+           - \frac{v}{e_1 e_2 }\left(    v  \;\frac{\partial e_2 }{\partial i}
+				 	          -u  \;\frac{\partial e_1 }{\partial j}            \right)
+\end{flalign}
+which is the total time derivative expressed in the curvilinear $s-$coordinate system.
+It has the same form as in the $z-$coordinate but for the vertical scale factor 
+that has appeared inside the time derivative which comes from the modification 
+of (\ref{Apdx_A_sco_Continuity}), the continuity equation.
+
+$\ $\newline    % force a new ligne
+
+$\bullet$ \textbf{horizontal pressure gradient}
+
+The horizontal pressure gradient term can be transformed as follows:
+\begin{equation*}
 \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] \\
+ -\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] \\
 & =-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma _1 }{\rho _o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
 &=-\frac{1}{\rho _o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho _o }\sigma _1
 \end{split}
+\end{equation*}
+Applying similar manipulation to the second component and replacing 
+$\sigma _1$ and $\sigma _2$ by their expression \eqref{Apdx_A_s_slope}, it comes:
+\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 
+                                                  + g\;\rho  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right) \\
+%
+ -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
+&=-\frac{1}{\rho _o \,e_2 } \left(    \left.               {\frac{\partial p}{\partial j}} \right|_s 
+                                                   + g\;\rho \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right) \\
+\end{split}
 \end{equation}
 
 An additional term appears in (\ref{Apdx_A_grad_p}) which accounts for the 
-tilt of model levels.
-
-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 the momentum equations in the 
-$s-$coordinate :
+tilt of $s-$surfaces with respect to geopotential $z-$surfaces.
+
+As in $z$-coordinate, the horizontal pressure gradient can be split in two parts
+following \citet{Marsaleix_al_OM08}. Let defined a density anomaly, $d$, by $d=(\rho - \rho_o)/ \rho_o$,
+and a hydrostatic pressure anomaly, $p_h'$, by $p_h'= g \; \int_z^\eta d \; e_3 \; dk$. 
+The pressure is then given by:
+\begin{equation*} 
+\begin{split}
+p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\
+   &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk    
+\end{split}
+\end{equation*}
+Therefore, $p$ and $p_h'$ are linked through:
+\begin{equation} \label{Apdx_A_pressure}
+   p = \rho_o \; p_h' + g \, ( z + \eta )
+\end{equation}
+and the hydrostatic pressure balance expressed in terms of $p_h'$ and $d$ is:
+\begin{equation*} 
+\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\end{equation*}
+
+Substituing \eqref{Apdx_A_pressure} in \eqref{Apdx_A_grad_p} and using the definition of 
+the density anomaly it comes the expression in two parts:
+\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}{e_1 } \left(     \left.              {\frac{\partial p_h'}{\partial i}} \right|_s 
+                                       + g\; d  \;\left. {\frac{\partial z}{\partial i}} \right|_s    \right)  - \frac{g}{e_1 } \frac{\partial \eta}{\partial i} \\
+%
+ -\frac{1}{\rho _o \, e_2 }\left. {\frac{\partial p}{\partial j}} \right|_z
+&=-\frac{1}{e_2 } \left(    \left.               {\frac{\partial p_h'}{\partial j}} \right|_s 
+                                        + g\; d \;\left. {\frac{\partial z}{\partial j}} \right|_s   \right)  - \frac{g}{e_2 } \frac{\partial \eta}{\partial j}\\
+\end{split}
+\end{equation}
+This formulation of the pressure gradient is characterised by the appearance of a term depending on the 
+the sea surface height only (last term on the right hand side of expression \eqref{Apdx_A_grad_p}).
+This term will be abusively named \textit{surface pressure gradient} whereas the first term will be named 
+\textit{hydrostatic pressure gradient} by analogy to the $z$-coordinate formulation. 
+In fact, the the true surface pressure gradient is $1/\rho_o \nabla (\rho \eta)$, and 
+$\eta$ is implicitly included in the computation of $p_h'$ through the upper bound of 
+the vertical integration.
+ 
+
+$\ $\newline    % force a new ligne
+
+$\bullet$ \textbf{The other terms of the momentum equation}
+
+The coriolis and forcing terms as well as the the vertical physics remain unchanged 
+as they involve neither time nor space derivatives. The form of the lateral physics is 
+discussed in appendix~\ref{Apdx_B}.
+
+
+$\ $\newline    % force a new ligne
+
+$\bullet$ \textbf{Full momentum equation}
+
+To sum up, in a curvilinear $s$-coordinate system, the vector invariant momentum equation 
+solved by the model has the same mathematical expression as the one in a curvilinear 
+$z-$coordinate, but the pressure gradient term :
 \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}=
+ \frac{\partial u}{\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 
+        -   \frac{1}{e_1 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)  
+        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
 	+   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}=
+\frac{\partial v}{\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   
+        -   \frac{1}{e_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)  
+        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
 	+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
 \end{multline}
 \end{subequations}
-
-It has the same form as in the $z-$coordinate but for the vertical scale factor 
-that has appeared inside the time derivative. The form of the vertical physics 
-and forcing terms remains unchanged. The form of the lateral physics is 
-discussed in appendix~\ref{Apdx_B}.  
+whereas the flux form momentum equation differ from it by the formulation of both
+the time derivative and the pressure gradient term  :
+\begin{subequations} \label{Apdx_A_dyn_flux}
+\begin{multline} \label{Apdx_A_PE_dyn_flux_u}
+ \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t} =
+	\nabla \cdot \left(   {{\rm {\bf U}}\,u}   \right) 
+	+   \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 } \left(    \frac{\partial p_h'}{\partial i} + g\; d  \; \frac{\partial z}{\partial i}    \right)  
+        -   \frac{g}{e_1 } \frac{\partial \eta}{\partial i}
+	+   D_u^{\vect{U}}  +   F_u^{\vect{U}}
+\end{multline}
+\begin{multline} \label{Apdx_A_dyn_flux_v}
+ \frac{1}{e_3}\frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+	-  \nabla \cdot \left(   {{\rm {\bf U}}\,v}   \right) 
+	+   \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_2 } \left(    \frac{\partial p_h'}{\partial j} + g\; d  \; \frac{\partial z}{\partial j}    \right)  
+        -   \frac{g}{e_2 } \frac{\partial \eta}{\partial j}
+	+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+\end{multline}
+\end{subequations}
+Both formulation share the same hydrostatic pressure balance expressed in terms of
+hydrostatic pressure and density anmalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
+\begin{equation} \label{Apdx_A_dyn_zph}
+\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\end{equation}
+
+It is important to realize that the change in coordinate system has only concerned
+the position on the vertical. It has not affected (\textbf{i},\textbf{j},\textbf{k}), the 
+orthogonal curvilinear set of unit vector. ($u$,$v$) are always horizontal velocities
+so that their evolution is driven by \emph{horizontal} forces, in particular 
+the pressure gradient. By contrast, $\omega$ is not $w$, the third component of the velocity,
+but the dia-surface velocity component, $i.e.$ the velocity relative to the moving 
+$s-$surfaces and normal to them. 
+
 
 % ================================================================
@@ -270,5 +513,5 @@
 % ================================================================
 \section{Tracer Equation}
-\label{Apdx_B_tracer}
+\label{Apdx_A_tracer}
 
 The tracer equation is obtained using the same calculation as for the continuity 
@@ -277,12 +520,13 @@
 \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} \; \;
+ 	= -\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]       \\
+ 	+  \frac{1}{e_3}  \frac{\partial }{\partial k} \left(                   Tw  \right)  
+	 +  D^{T} +F^{T}
 \end{multline}
 
 
 The expression for the advection term is a straight consequence of (A.4), the 
-expression of the 3D divergence in the $s$-coordinates established above. 
-
+expression of the 3D divergence in the $s-$coordinates established above. 
+
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_B.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_B.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_B.tex	(revision 1831)
@@ -6,4 +6,8 @@
 \minitoc
 
+
+\newpage
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Horizontal/Vertical 2nd Order Tracer Diffusive Operators
@@ -15,15 +19,15 @@
 In the $z$-coordinate, the horizontal/vertical second order tracer diffusion 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.
+\begin{eqnarray} \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}
+\end{eqnarray}
 
 In the $s$-coordinate, we defined the slopes of $s$-surfaces, $\sigma_1$ and 
-$\sigma_2$ by (!!!A.1!!!) and the vertical/horizontal ratio of diffusion coefficient 
-by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
+$\sigma_2$ by \eqref{Apdx_A_s_slope} and the vertical/horizontal ratio of diffusion 
+coefficient by $\epsilon = A^{vT} / A^{lT}$. The diffusion operator is given by:
 
 \begin{equation} \label{Apdx_B2}
@@ -38,74 +42,74 @@
 \end{equation}
 or in expanded form:
-\begin{multline} \label{Apdx_B3}
-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]} 
-\end{multline}
-
-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 derivative 
-$\partial _i \partial _j $ appears in \eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) 
-planes are independent. The derivation can then be demonstrated 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 
-}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
- \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 -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{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;} \right] }\\ 
-\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. \\ 
-\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 
-i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
- \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
--e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
-\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\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 s}} \right)\;\;\;} \right]} \\ 
-\end{multline*}
-using the same remark as just above, 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.\;\;\; \\ 
-+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
--e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
-\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
-\end{multline*}
-
-Since the horizontal scale factors do not depend on the vertical coordinate, 
+\begin{subequations}
+\begin{align*} {\begin{array}{*{20}l} 
+D^T=& \frac{1}{e_1\,e_2\,e_3 }\;\left[ {\ \ \ \ 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.  \\
+&\qquad \quad \ \ \ +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 \\
+&\qquad \quad \ \ \ + 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. 
+ \left. {\left. {+\left( {\varepsilon +\sigma _1^2+\sigma _2 ^2} \right)\;\frac{1}{e_3 }\;\frac{\partial T}{\partial s}} \right)\;\;} \right] 
+\end{array} } 		
+\end{align*}
+\end{subequations}
+
+Equation \eqref{Apdx_B2} 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 derivative $\partial _i \partial _j $ appears in 
+\eqref{Apdx_B1}, the ($i$,$z$) and ($j$,$z$) planes are independent. 
+The derivation can then be demonstrated for the ($i$,$z$)~$\to$~($j$,$s$) 
+transformation without any loss of generality:
+
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
+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)     \\
+\\
+\allowdisplaybreaks
+&=\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. \\ 
+& \qquad \qquad \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]}  \qquad \qquad \qquad \\ 
+\\
+\allowdisplaybreaks
+&=\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}A^{lT}\;\frac{\partial e_3 }{\partial i}} \right|_s \left. {\frac{\partial T}{\partial i}} \right|_s } \right. \\ 
+&  \qquad \qquad \quad \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 -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) \\
+&  \qquad \qquad \quad \shoveright{ -e_1 \,\sigma _1 \frac{\partial }{\partial s}\left( {-\frac{e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right)\;\,\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 }{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\quad} \right] }\\ 
+\end{array} } 		\\
+ {\begin{array}{*{20}l}
+%
+\allowdisplaybreaks
+\intertext{Noting that $\frac{1}{e_1} \left. \frac{\partial e_3 }{\partial i} \right|_s = \frac{\partial \sigma _1 }{\partial s}$, it becomes:}
+%
+& =\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 \qquad \quad -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) \\ 
+& \qquad \qquad \quad\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] }\\ 
+\\
+&=\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 i}\left( {e_2 \,\sigma _1 A^{lT}\;\frac{\partial T}{\partial s}} \right)} \right|_s } \right. \\ 
+& \qquad \qquad \quad \left. {+\frac{e_2 \,\sigma _1 }{e_3}A^{lT}\;\frac{\partial T}{\partial s} \;\frac{\partial e_3 }{\partial i}}  \right|_s -e_2 A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s \\ 
+& \qquad \qquad \quad-e_2 \,\sigma _1 \frac{\partial}{\partial s}\left( {A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}} \right) \\ 
+& \qquad \qquad \quad\shoveright{ \left. {-\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s} \left( {\frac{\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 s}} \right)\;\;\;} \right]} \\ 
+%
+\allowdisplaybreaks
+\intertext{using the same remark as just above, it becomes:}
+%
+&= \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.\;\;\; \\ 
+& \qquad \qquad \quad+\frac{e_1 \,e_2 \,\sigma _1 }{e_3 }A^{lT}\;\frac{\partial T}{\partial s}\;\frac{\partial \sigma _1 }{\partial s} - \frac {\sigma _1 }{e_3} A^{lT} \;\frac{\partial \left( {e_1 \,e_2 \,\sigma _1 } \right)}{\partial s}\;\frac{\partial T}{\partial s} \\ 
+& \qquad \qquad \quad-e_2 \left( {A^{lT}\;\frac{\partial \sigma _1 }{\partial s}\left. {\frac{\partial T}{\partial i}} \right|_s +\frac{\partial }{\partial s}\left( {\sigma _1 A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right)-\frac{\partial \sigma _1 }{\partial s}\;A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s } \right) \\ 
+& \qquad \qquad \quad\shoveright{\left. {+\frac{\partial }{\partial s}\left( {\frac{e_1 \,e_2 \,\sigma _1 ^2}{e_3 }A^{lT}\;\frac{\partial T}{\partial s}+\frac{e_1 \,e_2}{e_3 }A^{vT}\;\frac{\partial T}{\partial s}} \right)\;\;\;} \right] }\\ 
+%
+\allowdisplaybreaks
+\intertext{Since the horizontal scale factors 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. \\ 
- \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
-\end{multline*}
-
-in other words, the horizontal Laplacian operator in the ($i$,$s$) plane takes the following form :
-
-\begin{equation*}
+the second line reduces to a single vertical derivative, so it becomes:}
+%
+& =\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. \\ 
+& \qquad \qquad \quad \shoveright{ \left. {+\frac{\partial }{\partial s}\left( {-e_2 \,\sigma _1 \,A^{lT}\;\left. {\frac{\partial T}{\partial i}} \right|_s +A^{lT}\frac{e_1 \,e_2 }{e_3 }\;\left( {\varepsilon +\sigma _1 ^2} \right)\frac{\partial T}{\partial s}} \right)\;\;\;} \right]} \\ 
+%
+\allowdisplaybreaks
+\intertext{in other words, the horizontal Laplacian operator in the ($i$,$s$) plane takes the following form :}
+\end{array} } 		\\
+%
 D^T = {\frac{1}{e_1\,e_2\,e_3}}
 \left( {{\begin{array}{*{30}c}
@@ -122,7 +126,7 @@
 {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
 {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
-\end{array}}}
-\right) \left( T \right)} \right]
-\end{equation*}
+\end{array}}}       \right) \left( T \right)} \right]
+\end{align*}
+\end{subequations}
  
 
@@ -169,24 +173,28 @@
 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
+\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{align*}
- \nabla T\;.\left( {{\rm {\bf A}}_{\rm {\bf I}} \nabla T} 
+\begin{subequations} 
+\begin{align*} {\begin{array}{*{20}l} 
+\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 
 \frac{\partial T}{\partial i}\frac{\partial T}{\partial k}+\left( 
 {\frac{\partial T}{\partial j}} \right)^2} \right. \\ 
-&\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]} \\
-&=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*}
+&\qquad \qquad \qquad
+{ \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]} \\
+&=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]      \\
+& \geq 0
+\end{array} } 		
+\end{align*}
+\end{subequations}
 the property becomes obvious. 
 
 The resulting diffusion operator in $z$-coordinate has the following form :
 \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\} \\
+ 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]}
+ {+\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\} \\
 \shoveright{+\frac{1}{e_3 }\frac{\partial }{\partial k}\left[ {A_h \left( {-\frac{a_1 }{e_1 }\frac{\partial T}{\partial i}-\frac{a_2 }{e_2 }\frac{\partial T}{\partial j}+\frac{\left( {a_1 ^2+a_2 ^2} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]} \\ 
 \end{multline*}
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_C.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_C.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_C.tex	(revision 1831)
@@ -7,220 +7,459 @@
 
 %%%  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
+%I'm writting this appendix. It will be available in a forthcoming release of the documentation
 
 %\gmcomment{
 
-% ================================================================
-% Conservation Properties on Ocean Dynamics
-% ================================================================
-\section{Conservation Properties on Ocean Dynamics}
+\newpage
+$\ $\newline    % force a new ligne
+
+% ================================================================
+% Introduction / Notations
+% ================================================================
+\section{Introduction / Notations}
+\label{Apdx_C.0}
+
+Notation used in this appendix in the demonstations :
+
+fluxes at the faces of a $T$-box:
+\begin{equation*}
+U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega     \\
+\end{equation*}
+
+volume of cells at $u$-, $v$-, and $T$-points:
+\begin{equation*}
+b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t}     \\
+\end{equation*}
+
+partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$
+
+$dv=e_1\,e_2\,e_3 \,di\,dj\,dk$  is the volume element, with only $e_3$ that depends on time.
+$D$ and $S$ are the ocean domain volume and surface, respectively. 
+No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$) 
+Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 
+($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth).
+\begin{flalign*}
+ z(k) = \eta - \int\limits_{\tilde{k}=k}^{\tilde{k}=k_s}  e_3(\tilde{k}) \;d\tilde{k}
+        = \eta - \int\limits_k^{k_s}  e_3 \;d\tilde{k} 
+\end{flalign*}
+
+Continuity equation with the above notation:
+\begin{equation*}
+\frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0
+\end{equation*}
+
+A quantity, $Q$ is conserved when its domain averaged time change is zero, that is when:
+\begin{equation*}
+\partial_t \left( \int_D{ Q\;dv } \right) =0
+\end{equation*}
+Noting that the coordinate system used ....  blah blah
+\begin{equation*}
+\partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
+                                                       =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0
+\end{equation*}
+equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$
+like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+\begin{flalign*}
+\partial_t \left(   \int_D{ \frac{1}{2} \,Q^2\;dv }   \right)
+=&  \int_D{ \frac{1}{2} \partial_t \left( \frac{1}{e_3}\left( e_3 \, Q \right)^2 \right) e_1e_2\;di\,dj\,dk } \\
+=&  \int_D {         Q   \;\partial_t\left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }  
+-  \int_D { \frac{1}{2} Q^2 \,\partial_t  (e_3) \;e_1e_2\;di\,dj\,dk } \\
+\end{flalign*}
+that is in a more compact form :
+\begin{flalign} \label{Eq_Q2_flux}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=&                   \int_D { \frac{Q}{e_3}  \partial_t \left( e_3 \, Q \right) dv }  
+  -   \frac{1}{2} \int_D {  \frac{Q^2}{e_3} \partial_t (e_3) \;dv }
+\end{flalign}
+equation of evolution of $Q$ written as the time evolution of $Q$
+like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when :
+\begin{flalign*}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=&  \int_D { \frac{1}{2} \partial_t \left( e_3 \, Q^2 \right) \;e_1e_2\;di\,dj\,dk } \\
+=& \int_D {         Q      \partial_t Q  \;e_1e_2e_3\;di\,dj\,dk }  
++  \int_D { \frac{1}{2} Q^2 \, \partial_t e_3  \;e_1e_2\;di\,dj\,dk } \\
+\end{flalign*}
+that is in a more compact form :
+\begin{flalign} \label{Eq_Q2_vect}
+\partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right)
+=& \int_D {         Q   \,\partial_t Q  \;dv }  
++   \frac{1}{2} \int_D { \frac{1}{e_3} Q^2 \partial_t e_3 \;dv }
+\end{flalign}
+
+
+% ================================================================
+% Continuous Total energy Conservation
+% ================================================================
+\section{Continuous conservation}
 \label{Apdx_C.1}
 
-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; in practice the 
-energy is also preserved. In addition, an option is also offered for the vorticity term 
-discretization which provides 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.
-
-% -------------------------------------------------------------------------------------------------------------
-%       Bottom Boundary Condition on Vertical Velocity Field
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Bottom Boundary Condition on Vertical Velocity Field}
-\label{Apdx_C.1.1}
-
-
-The discrete set of momentum equations used in the rigid-lid approximation 
-automatically satisfies the surface and bottom boundary conditions 
-(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)!!!) weighted 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)&&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv \frac{1} {e_{1T}\,e_{2T}\,e_{3T}} 
-	\biggl\{ \quad
-	\delta_i 
-		&\left[ 
-		e_{2u}\,H_u 
-			\left( 
-			M_u - M_u - \frac{1} {H_u\,e_{2u}} \delta_j 
-				\left[ \partial_t\, \psi \right] 
-			\right) 
-		\right] &&
-	\biggr. \\
-	\biggl. 
-	+ \delta_j 
-		&\left[ 
-		e_{1v}\,H_v 
-			\left( M_v - M_v - \frac{1} {H_v\,e_{1v}} \delta_i 
-				\left[ \partial_i\, \psi \right] 
-			\right) 
-		\right] 
-	\biggr\}&& \\
-\end{flalign*}
-\begin{flalign*}
-\equiv \frac{1} {e_{1T} \,e_{2T} \,e_{3T}} \;
-	\biggl\{ 
-	- \delta_i 
-		\Bigl[ 
-		\delta_j 
-			\left[ \partial_t \psi  \right] 
-		\Bigr]
-	+ \delta_j 
-		\Bigl[ 
-		\delta_i 
-			\left[ \partial_t \psi  \right] 
-		\Bigr]\; 
-	\biggr\}\;
-	\equiv 0
-	&&&\\
-\end{flalign*}
-
-
-The surface boundary condition associated with the rigid lid approximation 
-($w_{surface} =0)$ is imposed in the computation of the vertical velocity (!!! II.2.5!!!!). 
-Therefore, it turns out to be:
+
+The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying 
+vertical coordinate) must be chosen so that the discrete equation of the model satisfy 
+integral constrains on energy and enstrophy. 
+
+
+Let us first establish those constraint in the continuous world.
+The total energy ($i.e.$ kinetic plus potential energies) is conserved :
+\begin{flalign} \label{Eq_Tot_Energy}
+  \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 +  \rho \, g \, z \right) \;dv \right)  = & 0
+\end{flalign}
+under the following assumptions: no dissipation, no forcing 
+(wind, buoyancy flux, atmospheric pressure variations), mass 
+conservation, and closed domain. 
+
+This equation can be transformed to obtain several sub-equalities. 
+The transformation for the advection term depends on whether 
+the vector invariant form or the flux form is used for the momentum equation.
+Using \eqref{Eq_Q2_vect} and introducing \eqref{Apdx_A_dyn_vect} in \eqref{Eq_Tot_Energy} 
+for the former form and
+Using \eqref{Eq_Q2_flux} and introducing \eqref{Apdx_A_dyn_flux} in \eqref{Eq_Tot_Energy} 
+for the latter form  leads to:
+
+\begin{subequations} \label{E_tot}
+
+advection term (vector invariant form):
+\begin{equation} \label{E_tot_vect_vor}
+\int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
+\end{equation}
+%
+\begin{equation} \label{E_tot_vect_adv}
+   \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv 
++ \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv    
+-  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\
+\end{equation}
+
+advection term (flux form):
+\begin{equation} \label{E_tot_flux_metric}
+\int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\; 
+ \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
+\end{equation}
+
+\begin{equation} \label{E_tot_flux_adv}
+   \int\limits_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv 
++   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0  \\
+\end{equation}
+
+coriolis term
+\begin{equation} \label{E_tot_cor}
+\int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
+\end{equation}
+
+pressure gradient:
+\begin{equation} \label{E_tot_pg}
+   - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+   + \int\limits_D g\, \rho \; \partial_t z  \;dv   \\
+\end{equation}
+\end{subequations}
+
+where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.
+
+blah blah....
+$\ $\newline    % force a new ligne
+The prognostic ocean dynamics equation can be summarized as follows:
 \begin{equation*}
-\frac{\partial } {\partial t}w_{bottom} \equiv 0
+\text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
+						{\text{COR} + \text{ADV}                       }
+			+ \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
 \end{equation*}
-As the bottom velocity is initially set to zero, it remains zero all the time. 
-Symmetrically, if $w_{bottom} =0$ is used in the computation of the vertical 
-velocity (upward integral of the horizontal divergence), the same computation 
-leads to $w_{surface} =0$ as soon as the surface vertical velocity is initially 
-set to zero.
-
-% -------------------------------------------------------------------------------------------------------------
-%       Coriolis and advection terms: vector invariant form
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Coriolis and advection terms: vector invariant form}
-\label{Apdx_C_vor_zad}
-
-% -------------------------------------------------------------------------------------------------------------
-%       Vorticity Term
-% -------------------------------------------------------------------------------------------------------------
-\subsubsection{Vorticity Term}
-\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 $=0) but not the 
-total kinetic energy. Indeed, using the symmetry or skew symmetry properties 
-of the operators (Eqs \eqref{DOM_mi_adj} and \eqref{DOM_di_adj}), it can 
-be shown that:
-\begin{equation} \label{Apdx_C_1.1}
-\int_D {\zeta / e_3\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {\zeta \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0
+$\ $\newline    % force a new ligne
+
+Vector invariant form:
+\begin{subequations} \label{E_tot_vect}
+\begin{equation} \label{E_tot_vect_vor}
+\int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0   \\
 \end{equation}
-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 
-	\left( 
-	\zeta \; \textbf{k} \times  \textbf{U}_h  
-	\right)\;
-	dv
-	&&& \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[ 
-		   - \overline {\left( {\zeta / e_{3f}} \right)}^{\,i}\;
-		      \overline{\overline {\left( e_{2u}\,e_{3u}\,u \right)}}^{\,i,j+1/ 2} 
-		 \right]  
-	&&  \\ & \qquad \qquad \qquad \;\;
-	- \delta_{j+1/2} 
-		\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}       && \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\} 
-	&& \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 $, it becomes:
-\begin{align*}
-\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left(  \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_{1T}\,e_{2T}\,e_{3T}\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 
-\qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\
-\end{align*}
-
-Note that the derivation is demonstrated here for the relative potential 
-vorticity but it applies also to the planetary ($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 (ENE scheme) :
+\begin{equation} \label{E_tot_vect_adv}
+   \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv 
++ \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv    
+-  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\
+\end{equation}
+\begin{equation} \label{E_tot_pg}
+   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+   + \int\limits_D g\, \rho \; \partial_t z  \;dv   \\
+\end{equation}
+\end{subequations}
+
+Flux form:
+\begin{subequations} \label{E_tot_flux}
+\begin{equation} \label{E_tot_flux_metric}
+\int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0   \\
+\end{equation}
+\begin{equation} \label{E_tot_flux_adv}
+   \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv 
++   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0  \\
+\end{equation}
+\begin{equation} \label{E_tot_pg}
+   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
+= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+   + \int\limits_D g\, \rho \; \partial_t  z  \;dv   \\
+\end{equation}
+\end{subequations}
+
+
+$\ $\newline    % force a new ligne
+
+
+\eqref{E_tot_pg} is the balance between the conversion KE to PE and PE to KE. 
+Indeed the left hand side of \eqref{E_tot_pg} can be transformed as follows:
+\begin{flalign*}
+\partial_t  \left( \int\limits_D { \rho \, g \, z  \;dv} \right) 
+&= + \int\limits_D \frac{1}{e_3} \partial_t (e_3\,\rho) \;g\;z\;\;dv
+     +  \int\limits_D g\, \rho \; \partial_t z  \;dv   &&&\\
+&= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+     + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\
+&= + \int\limits_D  \rho \,g \left( \textbf {U}_h \cdot \nabla_h z + \omega \frac{1}{e_3} \partial_k z \right)  \;dv
+     + \int\limits_D g\, \rho \; \partial_t z \;dv   &&&\\
+&= + \int\limits_D  \rho \,g \left( \omega + \partial_t z + \textbf {U}_h \cdot \nabla_h z  \right)  \;dv  &&&\\
+&=+  \int\limits_D g\, \rho \; w \; dv   &&&\\
+\end{flalign*}
+where the last equality is obtained by noting that the brackets is exactly the expression of $w$, 
+the vertical velocity referenced to the fixe $z$-coordinate system (see \eqref{Apdx_A_w_s}). 
+ 
+The left hand side of \eqref{E_tot_pg} can be transformed as follows:
+\begin{flalign*}
+- \int\limits_D  \left. \nabla p \right|_z & \cdot \textbf{U}_h \;dv  
+= - \int\limits_D  \left( \nabla_h p + \rho \, g \nabla_h z \right) \cdot \textbf{U}_h \;dv   &&&\\
+\allowdisplaybreaks
+&= - \int\limits_D  \nabla_h  p \cdot \textbf{U}_h \;dv   - \int\limits_D  \rho \, g \, \nabla_h z \cdot \textbf{U}_h \;dv   &&&\\
+\allowdisplaybreaks
+&= +\int\limits_D p \,\nabla_h \cdot \textbf{U}_h \;dv   + \int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\
+\allowdisplaybreaks
+&= -\int\limits_D p \left( \frac{1}{e_3} \partial_t e_3 + \frac{1}{e_3} \partial_k \omega  \right) \;dv
+    +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\
+\allowdisplaybreaks
+&= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv    
+     +\int\limits_D \frac{1}{e_3} \partial_k p\; \omega \;dv
+    +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\
+&= -\int\limits_D \frac{p}{e_3} \partial_t e_3  \;dv    
+     -\int\limits_D \rho \, g \, \omega \;dv
+    +\int\limits_D  \rho \, g \left( \omega - w + \partial_t z \right) \;dv   &&&\\
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \; \;dv    
+     - \int\limits_D  \rho \, g \, w \;dv 
+     + \int\limits_D   \rho \, g \, \partial_t z \;dv   &&&\\
+\allowdisplaybreaks
+\intertext{introducing the hydrostatic balance $\partial_k p=-\rho \,g\,e_3$ in the last term,
+it becomes:}
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv    
+     - \int\limits_D  \rho \, g \, w \;dv 
+     - \int\limits_D  \frac{1}{e_3} \partial_k p\, \partial_t z \;dv   &&&\\
+&= - \int\limits_D \frac{p}{e_3} \partial_t e_3 \;dv    
+     - \int\limits_D  \rho \, g \, w \;dv 
+     + \int\limits_D \,\frac{p}{e_3}\partial_t ( \partial_k z )  dv   &&&\\
+% 
+&= - \int\limits_D  \rho \, g \, w \;dv   &&&\\
+\end{flalign*}
+
+
+%gm comment
+\gmcomment{
+%
+The last equality comes from the following equation,
+\begin{flalign*}
+\int\limits_D p \frac{1}{e_3} \frac{\partial e_3}{\partial t}\; \;dv    
+     = \int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv \quad,  \\ 
+\end{flalign*}
+that can be demonstrated as follows:
+
+\begin{flalign*}
+\int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv   
+&= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv   
+  -  \int\limits_D    \rho \, g \, \frac{\partial}{\partial t} \left(  \int\limits_k^{k_s}  e_3 \;d\tilde{k} \right) \;dv   &&&\\
+&= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t} \;dv   
+  -  \int\limits_D    \rho \, g    \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right) \;dv   &&&\\
+%
+\allowdisplaybreaks
+\intertext{The second term of the right hand side can be transformed by applying the integration by part rule: 
+$\left[ a\,b \right]_{k_b}^{k_s} = \int_{k_b}^{k_s}  a\,\frac{\partial b}{\partial k}       \;dk
+                                               + \int_{k_b}^{k_s}      \frac{\partial a}{\partial k} \,b \;dk $
+to the following function: $a=  \int_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$ 
+and $b=  \int_k^{k_s}  \rho \, e_3 \;d\tilde{k}$
+(note that $\frac{\partial}{\partial k} \left(  \int_k^{k_s}  a \;d\tilde{k}  \right) = - a$ as $k$ is the lower bound of the integral).
+This leads to:  }
+\end{flalign*}
+\begin{flalign*}
+&\left[ \int\limits_{k}^{k_s}  \frac{\partial e_3}{\partial t} \,dk \cdot \int\limits_{k}^{k_s}  \rho \, e_3 \,dk   \right]_{k_b}^{k_s}
+=-\int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)  \rho \,e_3 \;dk
+  -\int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t}  \left(  \int\limits_k^{k_s}  \rho \, e_3 \;d\tilde{k} \right)   dk
+&&&\\
+\allowdisplaybreaks
+\intertext{Noting that $\frac{\partial \eta}{\partial t} 
+		= \frac{\partial}{\partial t}  \left( \int_{k_b}^{k_s}   e_3  \;d\tilde{k}  \right)
+		= \int_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k}$
+and 
+		$p(k) = \int_k^{k_s}  \rho \,g \, e_3 \;d\tilde{k} $, 
+but also that $\frac{\partial \eta}{\partial t}$ does not depends on $k$, it comes:
+}
+& - \int\limits_{k_b}^{k_s}  \rho \, \frac{\partial \eta}{\partial t} \, e_3 \;dk
+= - \int\limits_{k_b}^{k_s} \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)   \, \rho \, g   e_3\;dk
+   - \int\limits_{k_b}^{k_s}  \frac{\partial e_3}{\partial t} \frac{p}{g}         \;dk       &&&\\
+\end{flalign*}
+Mutliplying by $g$ and integrating over the $(i,j)$ domain it becomes:
+\begin{flalign*}
+\int\limits_D  \rho \, g \, \left(  \int\limits_k^{k_s}  \frac{\partial e_3}{\partial t} \;d\tilde{k} \right)    \;dv
+=  \int\limits_D  \rho \, g \, \frac{\partial \eta}{\partial t} dv
+  - \int\limits_D  \frac{p}{e_3}\frac{\partial e_3}{\partial t}         \;dv
+\end{flalign*}
+Using this property, we therefore have:
+\begin{flalign*}
+\int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv   
+&= \int\limits_D    \rho \, g \, \frac{\partial \eta}{\partial t}   \;dv   
+  - \left(  \int\limits_D  \rho \, g \, \frac{\partial \eta}{\partial t} dv
+           - \int\limits_D  \frac{p}{e_3}\frac{\partial e_3}{\partial t}   \;dv  \right)    &&&\\
+%
+&=\int\limits_D \frac{p}{e_3} \frac{\partial (e_3\,\rho)}{\partial t}\; \;dv     
+\end{flalign*}
+% end gm comment
+}
+%
+
+
+% ================================================================
+% Discrete Total energy Conservation : vector invariant form
+% ================================================================
+\section{Discrete total energy conservation : vector invariant form}
+\label{Apdx_C.1}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Total energy conservation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Total energy conservation}
+\label{Apdx_C_KE+PE}
+
+The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by:
+\begin{flalign*}
+\partial_t  \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\
+\end{flalign*}
+which in vector invariant forms, it leads to:
+\begin{equation} \label{KE+PE_vect_discrete}   \begin{split}
+                        \sum\limits_{i,j,k} \biggl\{   u\,                        \partial_t u         \;b_u 
+                                                              + v\,                        \partial_t v          \;b_v  \biggr\}
+  + \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\partial_t e_{3u} \;b_u 
+                                                             +  \frac{v^2}{e_{3v}}\partial_t e_{3v} \;b_v   \biggr\}      \\
+= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_{3t}}\partial_t (e_{3t} \rho) \, g \, z_t \;b_t  \biggr\} 
+     - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}                                
+\end{split} \end{equation}
+
+Substituting the discrete expression of the time derivative of the velocity either in vector invariant,
+leads to the discrete equivalent of the four equations \eqref{E_tot_flux}. 
+
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity term (coriolis + vorticity part of the advection)
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Vorticity term (coriolis + vorticity part of the advection)}
+\label{Apdx_C_vor}
+
+Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or  
+the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 
+Two discretisation of the vorticity term (ENE and EEN) allows the conservation of 
+the kinetic energy.
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity Term with ENE scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with ENE scheme (\np{ln\_dynvor\_ene}=.true.)}
+\label{Apdx_C_vorENE} 
+
+For the ENE scheme, the two components of the vorticity term are given by :
 \begin{equation*}
-	- \zeta \;{\textbf{k}}\times {\textbf {U}}_h 
-\equiv 
-	\left( {{
-	\begin{array} {*{20}c}
+- e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv 
+	\left( {{  \begin{array} {*{20}c}
 		+ \frac{1} {e_{1u}} \; 
-		\overline {\left( \zeta / e_{3f} 	  \right)	
-		\overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j} 
-		\hfill \\
+		\overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j}        \hfill \\
 		- \frac{1} {e_{2v}} \; 
-		\overline {\left( \zeta / e_{3f}  	  \right)
-		\overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i} 
-		\hfill \\
-	\end{array}} } 
-	\right)
+		\overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill \\
+	\end{array}} }    \right)
 \end{equation*}
 
 This formulation does not conserve the enstrophy but it does conserve the 
-total kinetic energy. It is also possible to mix the two formulations in order 
-to conserve enstrophy on the relative vorticity term and energy on the 
-Coriolis term.
+total kinetic energy. Indeed, the kinetic energy tendency associated to the 
+vorticity term and averaged over the ocean domain can be transformed as
+follows:
+\begin{flalign*}
+&\int\limits_D -  \left(  e_3 \, q \;\textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv &&  \\
+& \qquad \qquad {\begin{array}{*{20}l} 
+&\equiv \sum\limits_{i,j,k} 	\biggl\{ 	
+	  \frac{1} {e_{1u}} \overline { \,q\ \overline{ V }^{\,i+1/2}} ^{\,j} \, u \; b_u 
+	- \frac{1} {e_{2v}}\overline { \, q\ \overline{ U }^{\,j+1/2}} ^{\,i} \, v \; b_v \; \biggr\}    \\ 
+&\equiv  \sum\limits_{i,j,k} 	\biggl\{ 	
+	  \overline { \,q\ \overline{ V }^{\,i+1/2}}^{\,j} \; U 
+	- \overline { \,q\ \overline{ U }^{\,j+1/2}}^{\,i} \; V  \; \biggr\}     \\
+&\equiv \sum\limits_{i,j,k} q \  \biggl\{  \overline{ V }^{\,i+1/2}\; \overline{ U }^{\,j+1/2}        
+		                                        - \overline{ U }^{\,j+1/2}\; \overline{ V }^{\,i+1/2}         \biggr\}  \quad  \equiv 0
+\end{array} } 		
+\end{flalign*}
+In other words, the domain averaged kinetic energy does not change due to the vorticity term.
+
+
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity Term with EEN scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)}
+\label{Apdx_C_vorEEN} 
+
+With the EEN scheme, the vorticity terms are represented as: 
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ {    \begin{aligned}
+ +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
+                         {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\
+ - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
+                         {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2}   \\
+\end{aligned}   } \right.
+\end{equation} 
+where the indices $i_p$ and $k_p$ take the following value: 
+$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
+and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
+\end{equation}
+
+This formulation does conserve the total kinetic energy. Indeed,
 \begin{flalign*}
 &\int\limits_D - \textbf{U}_h \cdot   \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} &	\biggl\{ 
+	    \left[  \sum_{\substack{i_p,\,k_p}} 
+                         {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p} \right] U^{i+1/2}_{j}    %   &&\\
+	  - \left[  \sum_{\substack{i_p,\,k_p}} 
+                         {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p}  \right] V^{i}_{j+1/2}    \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
+\equiv \sum\limits_{i,j,k} &  \sum_{\substack{i_p,\,k_p}} \biggl\{  \ \  
+                         {^{i+1/2-i_p}_j}\mathbb{Q}^{i_p}_{j_p} \; V^{i+1/2-i_p}_{j+j_p}  \, U^{i+1/2}_{j}     %  &&\\
+                       - {^i_{j+1/2-j_p}}\mathbb{Q}^{i_p}_{j_p} \; U^{i+i_p}_{j+1/2-j_p} \, V^{i}_{j+1/2}     \ \;     \biggr\}     &&  \\
+%
+\allowdisplaybreaks
+\intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:}
+%
+\equiv \sum\limits_{i,j,k} & \biggl\{  \ \  
+                    {^{i+1}_j     }\mathbb{Q}^{-1/2}_{+1/2} \;V^{i+1}_{j+1/2} \; U^{\,i+1/2}_{j}      
+                -  {^i_{j}\quad}\mathbb{Q}^{-1/2}_{+1/2} \; U^{i-1/2}_{j}    \; V^{\,i}_{j+1/2}         &&  \\
+        &       + {^{i+1}_j     }\mathbb{Q}^{-1/2}_{-1/2} \; V^{i+1}_{j-1/2} \; U^{\,i+1/2}_{j}        
+                  - {^i_{j+1}     }\mathbb{Q}^{-1/2}_{-1/2} \; U^{i-1/2}_{j+1} \; V^{\,i}_{j+1/2}        \biggr.     &&  \\
+        &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{+1/2} \; V^{i}_{j+1/2}   \; U^{\,i+1/2}_{j}      
+                  - {^i_{j}\quad}\mathbb{Q}^{+1/2}_{+1/2} \; U^{i+1/2}_{j}   \; V^{\,i}_{j+1/2}          \biggr.        &&  \\
+        &       + {^{i}_j\quad}\mathbb{Q}^{+1/2}_{-1/2} \; V^{i}_{j-1/2}     \; U^{\,i+1/2}_{j}       
+                 -  {^i_{j+1}     }\mathbb{Q}^{+1/2}_{-1/2} \; U^{i+1/2}_{j+1}\; V^{\,i}_{j+1/2}  \ \;     \biggr\}     &&  \\
+%
+\allowdisplaybreaks
+\intertext{The summation is done over all $i$ and $j$ indices, it is therefore possible to introduce 
+a shift of $-1$ either in $i$ or $j$ direction in some of the term of the summation (first term of the 
+first and second lines, second term of the second and fourth lines). By doning so, we can regroup 
+all the terms of the summation by triad at a ($i$,$j$) point. In other words, we regroup all the terms 
+in the neighbourhood  that contain a triad at the same ($i$,$j$) indices. It becomes: }
+\allowdisplaybreaks
+%
+\equiv \sum\limits_{i,j,k} & \biggl\{  \ \  
+             {^{i}_j}\mathbb{Q}^{-1/2}_{+1/2}  \left[  V^{i}_{j+1/2}\, U^{\,i-1/2}_{j}    
+                                                                       -  U^{i-1/2}_{j} \, V^{\,i}_{j+1/2}      \right]    &&  \\
+ &       + {^{i}_j}\mathbb{Q}^{-1/2}_{-1/2}  \left[  V^{i}_{j-1/2} \, U^{\,i-1/2}_{j}        
+                                                                     -    U^{i-1/2}_{j} \, V^{\,i}_{j-1/2}      \right]    \biggr.   &&  \\
+ &      + {^{i}_j}\mathbb{Q}^{+1/2}_{+1/2}  \left[  V^{i}_{j+1/2} \, U^{\,i+1/2}_{j}      
+                                                                     -    U^{i+1/2}_{j} \, V^{\,i}_{j+1/2}     \right]  \biggr.  &&  \\
+ &     + {^{i}_j}\mathbb{Q}^{+1/2}_{-1/2}  \left[   V^{i}_{j-1/2} \, U^{\,i+1/2}_{j}                                                                     
+                                                                    -    U^{i+1/2}_{j-1} \, V^{\,i}_{j-1/2}  \right]  \ \;   \biggr\}   \qquad
+\equiv \ 0   &&
 \end{flalign*}
 
@@ -235,9 +474,10 @@
 balanced by the change of KE due to the horizontal gradient of KE~:
 \begin{equation*}
-      \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
+    \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv
+=  -   \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv
+   +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv }  \\
 \end{equation*}
 Indeed, using successively \eqref{DOM_di_adj} ($i.e.$ the skew symmetry 
-property of the $\delta$ operator) and the incompressibility, then 
+property of the $\delta$ operator) and the continuity equation, then 
 \eqref{DOM_di_adj} again, then the commutativity of operators 
 $\overline {\,\cdot \,}$ and $\delta$, and finally \eqref{DOM_mi_adj} 
@@ -245,55 +485,61 @@
 applied in the horizontal and vertical directions, 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\} 
-	&&& \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} 
-	+ \overline{ v^2}^{\,j} 
-	\right] \;
-	e_{1v}\,e_{2v}\,w 
-	&&& \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] \\
+& - \int_D \textbf{U}_h \cdot \text{KEG}\;dv 	
+= - \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv    &&&\\
+%
+\equiv  & -  \sum\limits_{i,j,k} \frac{1}{2}  \biggl\{	
+	\frac{1} {e_{1u}}  \delta_{i+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  u \ b_u 
+     + \frac{1} {e_{2v}}  \delta_{j+1/2}   \left[   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right]  v \ b_v   \biggr\}  	&&&  \\ 
+%
+\equiv & + \sum\limits_{i,j,k} \frac{1}{2}  \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)\;
+                                                              \biggl\{ \delta_{i} \left[  U   \right] +  \delta_{j} \left[  V  \right]    \biggr\}       &&&  \\ 
+\allowdisplaybreaks
+%
+\equiv   & - \sum\limits_{i,j,k}  \frac{1}{2}
+	\left(       \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right)  \;  
+	\biggl\{   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[  W   \right]    \biggr\}    &&&\\
+\allowdisplaybreaks
+%
+\equiv & +  \sum\limits_{i,j,k} \frac{1}{2} \delta_{k+1/2}   \left[ \overline{ u^2}^{\,i} + \overline{ v^2}^{\,j}   \right] \;  W 
+	       -  \sum\limits_{i,j,k} \frac{1}{2} \left(   \overline {u^2}^{\,i} + \overline {v^2}^{\,j}   \right) \;\partial_t b_t   &&& \\
+\allowdisplaybreaks
+%
+\equiv   & + \sum\limits_{i,j,k} \frac{1} {2} \left(    \overline{\delta_{k+1/2} \left[ u^2 \right]}^{\,i} 
+	                                                                + \overline{\delta_{k+1/2} \left[ v^2 \right]}^{\,j}    \right) \; W     
+	       -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t \overline{b_t}^{\,{i+1/2}}
+	                                              + \frac{v^2}{2}\,\partial_t \overline{b_t}^{\,{j+1/2}}   \right)    &&& \\
+\allowdisplaybreaks
+\intertext{Assuming that $b_u= \overline{b_t}^{\,i+1/2}$ and $b_v= \overline{b_t}^{\,j+1/2}$, or at least that the time
+derivative of these two equations is satisfied, it becomes:}
+%
+\equiv &     \sum\limits_{i,j,k} \frac{1} {2}
+	\biggl\{ \; \overline{W}^{\,i+1/2}\;\delta_{k+1/2} \left[ u^2 \right]    
+	            + \overline{W}^{\,j+1/2}\;\delta_{k+1/2} \left[ v^2 \right]  \;  \biggr\}   
+	       -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u
+	                                             + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\
+\allowdisplaybreaks
 %	
-\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\} 
-	&&\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}  
-	&& \\
-	&\qquad \quad\; + \frac{1} {b_v } \;
-	\overline {\Bigl\{ \overline {e_{1T}\,e_{2T} \,w}^{\,j+1/2} \delta_{k+1/2}
-		 \left[ v \right] 
-		 	\Bigr\} }^{\,k} \;v\;e_{1v}\,e_{2v}\,e_{3v}  \;
-	\biggr\} 
-	&& \\ 
-\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 
+\equiv &     \sum\limits_{i,j,k} 
+	\biggl\{ \; \overline{W}^{\,i+1/2}\; \overline {u}^{\,k+1/2}\; \delta_{k+1/2}[ u ]    
+	            + \overline{W}^{\,j+1/2}\; \overline {v}^{\,k+1/2}\; \delta_{k+1/2}[ v ]  \;  \biggr\} 
+	       -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u
+	                                             + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\
+%
+\allowdisplaybreaks
+\equiv  &  \sum\limits_{i,j,k}  
+	\biggl\{ \; \frac{1} {b_u } \; \overline { \overline{W}^{\,i+1/2}\,\delta_{k+1/2}  \left[ u \right] }^{\,k} \;u\;b_u  
+                    + \frac{1} {b_v } \; \overline { \overline{W}^{\,j+1/2} \delta_{k+1/2}  \left[ v \right]  }^{\,k} \;v\;b_v  \; \biggr\}  
+	       -  \sum\limits_{i,j,k}  \left(  \frac{u^2}{2}\,\partial_t b_u
+	                                             + \frac{v^2}{2}\,\partial_t b_v   \right)    &&& \\
+%
+\intertext{The first term provides the discrete expression for the vertical advection of momentum (ZAD), 
+while the second term corresponds exactly to \eqref{KE+PE_vect_discrete}, therefore:}
+\equiv&                   \int\limits_D \textbf{U}_h \cdot \text{ZAD} \;dv  
+           + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\
+\equiv&                   \int\limits_D \textbf{U}_h \cdot w \partial_k \textbf{U}_h \;dv  
+           + \frac{1}{2} \int_D { {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t  (e_3)  \;dv }    &&&\\
+\end{flalign*}
+
+There is two main points here. First, the satisfaction of this property links the choice of 
 the discrete formulation of the vertical advection and of the horizontal gradient 
 of KE. Choosing one imposes the other. For example KE can also be discretized 
@@ -301,11 +547,9 @@
 expression for the vertical advection:
 \begin{equation*}
-\frac{1} {e_3 }\; w\; \frac{\partial \textbf{U}_h } {\partial k}
+\frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h
 \equiv \left( {{\begin{array} {*{20}c}
-\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;
-\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2} 
+\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;  \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2} 
 \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\
-\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;
-\overline{\overline {e_{1T}\,e_{2T} \,w\;\delta_{k+1/2}
+\frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;   \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2}
 \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\
 \end{array}} } \right)
@@ -315,4 +559,202 @@
 This is the reason why it has not been chosen.
 
+Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint
+arises on the time derivative of the volume at $u$- and $v$-points:
+\begin{flalign*}
+e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2}    \\
+e_{1v}\,e_{2v}\,\partial_t (e_{3v})  =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,j+1/2}
+\end{flalign*}
+which is (over-)satified by defining the vertical scale factor as follows:
+\begin{flalign} \label{e3u-e3v}
+e_{3u} = \frac{1}{e_{1u}\,e_{2u}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,i+1/2}    \\
+e_{3v} = \frac{1}{e_{1v}\,e_{2v}}\;\overline{ e_{1t}^{ }\,e_{2t}^{ }\,e_{3t}^{ } }^{\,j+1/2} 
+\end{flalign}
+
+Blah blah required on the the step representation of bottom topography.....
+
+
+% -------------------------------------------------------------------------------------------------------------
+%       Pressure Gradient Term
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Pressure Gradient Term}
+\label{Apdx_C.1.4}
+
+\gmcomment{
+A pressure gradient has no contribution to the evolution of the vorticity as the 
+curl of a gradient is zero. In the $z$-coordinate, this property 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 advection-diffusion 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  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv 
+= - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv
+  + \int_D g\, \rho \; \partial_t (z)  \;dv
+\end{equation*}
+
+This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 
+Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale 
+factors at $w$-points starting from the surface, the work of pressure forces can be 
+written as:
+\begin{flalign*}
+&- \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv   
+\equiv \sum\limits_{i,j,k} \biggl\{ \;  - \frac{1} {e_{1u}}   \Bigl( 
+\delta_{i+1/2} [p_t] - g\;\overline \rho^{\,i+1/2}\;\delta_{i+1/2} [z_t]     \Bigr)  \; u\;b_u  
+	&&  \\ & \qquad \qquad  \qquad \qquad  \qquad \quad \ \,
+					 	      - \frac{1} {e_{2v}}    \Bigl( 
+\delta_{j+1/2} [p_t] - g\;\overline \rho^{\,j+1/2}\delta_{j+1/2} [z_t]      \Bigr)  \; v\;b_v \;  \biggr\}   && \\ 
+%
+\allowdisplaybreaks
+\intertext{Using successively  \eqref{DOM_di_adj}, $i.e.$ the skew symmetry property of 
+the $\delta$ operator, \eqref{Eq_wzv}, the continuity equation, \eqref{Eq_dynhpg_sco}, 
+the hydrostatic equation in the $s$-coordinate, and $\delta_{k+1/2} \left[ z_t \right] \equiv e_{3w} $,
+which comes from the definition of $z_t$, it becomes: }
+\allowdisplaybreaks
+%
+\equiv& +  \sum\limits_{i,j,k}   g  \biggl\{ 
+		\overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]    
+	+  	\overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]     
+	+\Bigl(  \delta_i[U] + \delta_j [V]  \Bigr)\;\frac{p_t}{g} \biggr\}  &&\\
+%
+\equiv& +  \sum\limits_{i,j,k}   g   \biggl\{ 
+		\overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]    
+	+  	\overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]     
+	 -       \left(   \frac{b_t}{e_{3t}} \partial_t (e_{3t})  +  \delta_k \left[ W \right]    \right) \frac{p_t}{g}    \biggr\}   &&&\\ 
+%
+\equiv& +  \sum\limits_{i,j,k}  g   \biggl\{ 
+		\overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]    
+	+  	\overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]     
+	+	\frac{W}{g}\;\delta_{k+1/2} [p_t] 
+	 -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\
+%
+\equiv& +  \sum\limits_{i,j,k}  g   \biggl\{ 
+		\overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]    
+	+  	\overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]     
+	- 	W\;e_{3w} \overline \rho^{\,k+1/2}         
+	 -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\
+%
+\equiv& +  \sum\limits_{i,j,k}    g   \biggl\{ 
+		\overline\rho^{\,i+1/2}\,U\,\delta_{i+1/2}[z_t]    
+	+  	\overline\rho^{\,j+1/2}\,V\,\delta_{j+1/2}[z_t]     
+	+ 	W\; \overline \rho^{\,k+1/2}\;\delta_{k+1/2} [z_t]   
+	 -        \frac{p_t}{g}\,\partial_t b_t    \biggr\}    &&&\\
+%
+\allowdisplaybreaks
+%
+\equiv& - \sum\limits_{i,j,k}   g \; z_t      \biggl\{ 
+		\delta_i 	\left[ U\;	\overline \rho^{\,i+1/2} 	\right]
+	+ 	\delta_j 	\left[ V\;	\overline \rho^{\,j+1/2} 	\right]
+	+ 	\delta_k 	\left[ W\;	\overline \rho^{\,k+1/2} 	\right]       \biggr\}    
+             - \sum\limits_{i,j,k}       \biggl\{ p_t\;\partial_t b_t    \biggr\}   &&&\\	
+%
+\equiv& + \sum\limits_{i,j,k}   g \; z_t    \biggl\{      \partial_t ( e_{3t} \,\rho)    \biggr\}  \; b_t    
+             -  \sum\limits_{i,j,k}                 \biggl\{  p_t\;\partial_t b_t                     \biggr\}              &&&\\	
+%
+\end{flalign*}
+The first term is exactly the first term of the right-hand-side of \eqref{KE+PE_vect_discrete}.
+It remains to demonstrate that the last term, which is obviously a discrete analogue of 
+$\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \eqref{KE+PE_vect_discrete}.
+In other words, the following property must be satisfied:
+\begin{flalign*}
+           \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t b_t                  \biggr\}        
+\equiv  \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}                                
+\end{flalign*}
+
+Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. 
+The right-hand-side of the above equation can be transformed as follows:
+
+\begin{flalign*}
+               \sum\limits_{i,j,k}  \biggl\{ \rho \,g\,\partial_t (z_t) \,b_t  \biggr\}  
+&\equiv   - \sum\limits_{i,j,k}  \biggl\{ \delta_k [p_w]\,\partial_t (z_t) \,e_{1t}\,e_{2t}  \biggr\}        &&&\\
+%
+&\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \delta_{k+1/2} [\partial_t (z_t)] \,e_{1t}\,e_{2t}  \biggr\}  
+  \equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (e_{3w}) \,e_{1t}\,e_{2t}  \biggr\}        &&&\\
+&\equiv  + \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\}  
+ %
+% & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w   \right)   \biggr\}           &&&\\
+% & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_k [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\
+% & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\}          
+\end{flalign*}
+therefore, the balance to be satisfied is:
+\begin{flalign*}
+           \sum\limits_{i,j,k}  \biggl\{  p_t\;\partial_t (b_t) \biggr\}  \equiv  \sum\limits_{i,j,k}  \biggl\{  p_w\, \partial_t (b_w) \biggr\}  
+\end{flalign*}
+which is a purely vertical balance:
+\begin{flalign*}
+           \sum\limits_{k}  \biggl\{  p_t\;\partial_t (e_{3t}) \biggr\}  \equiv  \sum\limits_{k}  \biggl\{  p_w\, \partial_t (e_{3w}) \biggr\}  
+\end{flalign*}
+Defining $p_w = \overline{p_t}^{\,k+1/2}$
+
+%gm comment
+\gmcomment{
+\begin{flalign*}
+ \sum\limits_{i,j,k} \biggl\{   p_t\;\partial_t b_t   \biggr\}                                &&&\\
+ %
+ & \equiv     \sum\limits_{i,j,k} \biggl\{  \frac{1}{e_{3t}} \delta_k [p_w]\;\partial_t (z_t) \,b_w    \biggr\}           &&&\\
+ & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t \left(    \delta_{k+1/2} [z_t]   \right)  e_{1w}\,e_{2w}   \biggr\}           &&&\\
+ & \equiv     \sum\limits_{i,j,k} \biggl\{   p_w\;\partial_t b_w   \biggr\}          
+\end{flalign*}
+
+
+\begin{flalign*}
+\int\limits_D   \rho \, g \, \frac{\partial z }{\partial t} \;dv
+\equiv&  \sum\limits_{i,j,k}   \biggl\{  \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} p   \biggr\} \; b_t   &&&\\
+\equiv&  \sum\limits_{i,j,k}   \biggl\{      \biggr\} \; b_t   &&&\\
+\end{flalign*}
+
+%
+\begin{flalign*}
+\equiv& - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+   + \int\limits_D g\, \rho \; \frac{\partial z}{\partial t}  \;dv     &&& \\
+\end{flalign*}
+%
+}
+%end gm comment
+
+
+Note that this property strongly constrains the discrete expression of both 
+the depth of $T-$points and of the term added to the pressure gradient in the
+$s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation 
+of state is rarely used.
+
+
+
+
+
+
+
+% ================================================================
+% Discrete Total energy Conservation : flux form
+% ================================================================
+\section{Discrete total energy conservation : flux form}
+\label{Apdx_C.1}
+
+% -------------------------------------------------------------------------------------------------------------
+%       Total energy conservation
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Total energy conservation}
+\label{Apdx_C_KE+PE}
+
+The discrete form of the total energy conservation, \eqref{Eq_Tot_Energy}, is given by:
+\begin{flalign*}
+\partial_t \left(  \sum\limits_{i,j,k} \biggl\{ \frac{u^2}{2} \,b_u + \frac{v^2}{2}\, b_v +  \rho \, g \, z_t \,b_t  \biggr\} \right) &=0  \\
+\end{flalign*}
+which in flux form, it leads to:
+\begin{flalign*}
+                        \sum\limits_{i,j,k} \biggl\{  \frac{u    }{e_{3u}}\,\frac{\partial (e_{3u}u)}{\partial t} \,b_u 
+                                                             +  \frac{v    }{e_{3v}}\,\frac{\partial (e_{3v}v)}{\partial t} \,b_v  \biggr\}
+&  -  \frac{1}{2} \sum\limits_{i,j,k} \biggl\{  \frac{u^2}{e_{3u}}\frac{\partial    e_{3u}    }{\partial t} \,b_u 
+                                                             +  \frac{v^2}{e_{3v}}\frac{\partial    e_{3v}    }{\partial t} \,b_v   \biggr\}      \\
+&= - \sum\limits_{i,j,k} \biggl\{ \frac{1}{e_3t}\frac{\partial e_{3t} \rho}{\partial t} \, g \, z_t \,b_t  \biggr\} 
+     - \sum\limits_{i,j,k} \biggl\{ \rho \,g\,\frac{\partial z_t}{\partial t} \,b_t  \biggr\}                                     \\
+\end{flalign*}
+
+Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form,
+leads to the discrete equivalent of the 
+
+
 % -------------------------------------------------------------------------------------------------------------
 %       Coriolis and advection terms: flux form
@@ -331,13 +773,11 @@
 Coriolis parameter is discretised at an f-point. It is given by: 
 \begin{equation*}
-f+\frac{1} {e_1 e_2 } 
-\left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
+f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
 \equiv \;
-f+\frac{1} {e_{1f}\,e_{2f}} 
-\left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 
--\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right)
+f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right] 
+                                               -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right)
 \end{equation*}
 
-The ENE scheme is then applied to obtain the vorticity term in flux form. 
+Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 
 It therefore conserves the total KE. The derivation is the same as for the 
 vorticity term in the vector invariant form (\S\ref{Apdx_C_vor}).
@@ -355,69 +795,65 @@
 the horizontal kinetic energy, that is :
 
-\begin{equation} \label{Apdx_C_I.3.10}
-\int_D \textbf{U}_h \cdot 
-\left( {{\begin{array} {*{20}c}
+\begin{equation} \label{Apdx_C_ADV_KE_flux}
+ -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c}
 \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
-\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\
-\end{array}} } \right)\;dv =\;0
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv 
+-   \frac{1}{2} \int_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } =\;0
 \end{equation}
 
-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    &&&\\
-%
-\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} 
+Let us first consider the first term of the scalar product ($i.e.$ 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   \\
+%
+\equiv& - \sum\limits_{i,j,k} \biggl\{    \frac{1} {b_u}    \biggl(   
+      \delta_{i+1/2}  \left[   \overline {U}^{\,i}      \;\overline u^{\,i}          \right]   
+   + \delta_j           \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right]   
+   + \delta_k          \left[   \overline {W}^{\,i+1/2}\;\overline u^{\,k+1/2} \right]  \biggr)   \;   \biggr\} \, b_u \;u &&&  \\ 
+%
+\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]
-		&&& \\ & \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\} \; u    &&& \displaybreak[0] \\
-%
-\equiv& - \sum\limits_{i,j,k}
+      \delta_{i+1/2} \left[   \overline {U}^{\,i}\;  \overline u^{\,i}  \right]
+   + \delta_j          \left[   \overline {V}^{\,i+1/2}\;\overline u^{\,j+1/2}   \right]
+   + \delta_k         \left[   \overline {W}^{\,i+12}\;\overline u^{\,k+1/2}  \right]
+   	\; \biggr\} \; u     \\
+%
+\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} 
-		\left[ u \right] 
-		&&& \\ & \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]
-	 + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2} 	\delta_{k+1/2} 	\left[ u^2 \right] 
-	\biggr\} 
-	&& \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]
-	 + e_{1w}\,e_{2w}\,w\;	\delta_{k+1/2}  	   \left[ \overline {u^2}^{\,i} \right] 
-	\biggr\} 
-	&& \displaybreak[0] \\
-%
-\equiv& \sum\limits_{i,j,k}
-\overline {u^2}^{\,i} 
-	\biggl\{ 
-	   \delta_{i+1/2} 	\left[ e_{2u}\,e_{3u}\,u  \right]
-	+ \delta_{j+1/2} 	\left[ e_{1u}\,e_{3u}\,v  \right]
-	+ \delta_{k+1/2} 	\left[ e_{1w}\,e_{2w}\,w \right] 
-	\biggr\}  \;\;  \equiv 0
-	&&& \\
-\end{flalign*}
+	   \overline {U}^{\,i}\;	\overline u^{\,i} 	\delta_i \left[ u \right] 
+        + \overline {V}^{\,i+1/2}\;	\overline u^{\,j+1/2} 	\delta_{j+1/2} \left[ u \right] 
+        + \overline {W}^{\,i+1/2}\;	\overline u^{\,k+1/2}	\delta_{k+1/2} 	\left[ u \right]     \biggr\}     && \\
+%
+\equiv& + \frac{1}{2} \sum\limits_{i,j,k}    \biggl\{ 
+	    \overline{U}^{\,i} 		\delta_i 		\left[ u^2 \right] 
+	 + \overline{V}^{\,i+1/2} 	\delta_{j+/2} 	\left[ u^2 \right]
+	 + \overline{W}^{\,i+1/2} 	\delta_{k+1/2} 	\left[ u^2 \right]      \biggr\} && \\
+%
+\equiv& -  \sum\limits_{i,j,k}    \frac{1}{2}   \bigg\{ 
+	    U  \; \delta_{i+1/2}    \left[ \overline {u^2}^{\,i} \right]
+         + V  \; \delta_{j+1/2}    \left[ \overline {u^2}^{\,i} \right]
+	 + W \; \delta_{k+1/2}   \left[ \overline {u^2}^{\,i} \right]     \biggr\}    &&& \\
+%
+\equiv& - \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i}     \biggl\{ 
+	   \delta_{i+1/2} 	\left[ U  \right]
+	+ \delta_{j+1/2} 	\left[ V  \right]
+	+ \delta_{k+1/2} 	\left[ W \right]     \biggr\}    &&& \\
+%
+\equiv& + \sum\limits_{i,j,k}  \frac{1}{2}  \overline {u^2}^{\,i} 
+	\biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}    &&& \\
+\end{flalign*}
+Applying similar manipulation applied to the second term of the scalar product 
+leads to :
+\begin{equation*}
+ -  \int_D \textbf{U}_h \cdot     \left(                 {{\begin{array} {*{20}c}
+\nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+\nabla \cdot \left( \textbf{U}\,v \right) \hfill \\       \end{array}} }           \right)   \;dv     
+\equiv + \sum\limits_{i,j,k}  \frac{1}{2}  \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)
+	\biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}    
+\end{equation*}
+which is the discrete form of 
+$ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $. 
+\eqref{Apdx_C_ADV_KE_flux} is thus satisfied.
+
 
 When the UBS scheme is used to evaluate the flux form momentum advection, 
@@ -426,181 +862,152 @@
 ($i.e.$ the scheme is diffusive). 
 
-% -------------------------------------------------------------------------------------------------------------
-%       Hydrostatic Pressure Gradient Term
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Hydrostatic Pressure Gradient Term}
-\label{Apdx_C.1.4}
-
-
-A pressure gradient has no contribution to the evolution of the vorticity as the 
-curl of a gradient is zero. In the $z$-coordinate, this property 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 advection-diffusion 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 
-= \int_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
-\end{equation*}
-
-This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 
-Indeed, defining the depth of a $T$-point, $z_T$, as the sum of the vertical scale 
-factors at $w$-points starting from the surface, the work of 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 the $s$-coordinate, it becomes: 
+
+
+
+
+
+
+
+
+
+% ================================================================
+% Discrete Enstrophy Conservation
+% ================================================================
+\section{Discrete enstrophy conservation}
+\label{Apdx_C.1}
+
+
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity Term with ENS scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with ENS scheme  (\np{ln\_dynvor\_ens}=.true.)}
+\label{Apdx_C_vorENS} 
+
+In the ENS scheme, the vorticity term is descretized as follows:
+\begin{equation} \label{Eq_dynvor_ens}
+\left\{   \begin{aligned}
++\frac{1}{e_{1u}} & \overline{q}^{\,i}  & {\overline{ \overline{\left( e_{1v}\,e_{3v}\;  v \right) } } }^{\,i, j+1/2}    \\
+- \frac{1}{e_{2v}} & \overline{q}^{\,j}  & {\overline{ \overline{\left( e_{2u}\,e_{3u}\; u \right) } } }^{\,i+1/2, j}  
+\end{aligned}  \right.
+\end{equation} 
+
+The scheme does not allow but the conservation of the total kinetic energy but the conservation 
+of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). 
+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 {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\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 
+\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*} 
-\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]     
-&& \\  & \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] 
-	&&&\\ & \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] 
-	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
-	+	 e_{1w} e_{2w} \;w\;\delta_{k+1/2} \left[ p_h \right] 
-	\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] 	
-	&&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\;\,
-	- 	e_{1w} e_{2w} \;w\;e_{3w} \overline \rho^{\,k+1/2} 
-	\biggr\}   &&&\\ 
-\end{flalign*}
-noting that by definition of $z_T$, $\delta_{k+1/2} \left[ z_T \right] \equiv - e_{3w} $, thus:
-\begin{multline*}
-\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_{1w} e_{2w} \;w\;	\overline \rho^{\,k+1/2}\;\delta_{k+1/2} \left[ z_T \right] 
-	\biggr\} } \\ 
-\end{multline*}
-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]
-	+ 	\delta_j 	\left[ e_{1v}\,e_{3v}\,v\;	\overline \rho^{\,j+1/2} 	\right]
-	+ 	\delta_k 	\left[ e_{1w} e_{2w}\,w\;	\overline \rho^{\,k+1/2} 	\right] 
+&\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times 
+	\left(  e_3 \, q \; \textbf{k} \times  \textbf{U}_h   \right)\;   dv       \\
+%
+& \qquad {\begin{array}{*{20}l} 
+&\equiv \sum\limits_{i,j,k} 
+q \ \left\{  \delta_{i+1/2}  \left[ - \,\overline {q}^{\,i}\;  \overline{\overline  U }^{\,i,j+1/ 2} \right]  
+             - \delta_{j+1/2} \left[	 \overline {q}^{\,j}\;  \overline{\overline  V }^{\,i+1/2, j} \right]     \right\}    \\ 
+%
+&\equiv \sum\limits_{i,j,k} 
+	\left\{   \delta_i [q] \; \overline{q}^{\,i} \; \overline{  \overline U  }^{\,i,j+1/2} 
+		+ \delta_j [q] \; \overline{q}^{\,j} \; \overline{\overline V }^{\,i+1/2,j}        \right\}       &&  \\ 
+%
+&\equiv \,\frac{1} {2} \sum\limits_{i,j,k}  
+	\left\{         \delta_i  \left[ q^2 \right] \; \overline{\overline U }^{\,i,j+1/2} 
+		      + \delta_j  \left[ q^2 \right] \; \overline{\overline V }^{\,i+1/2,j}      \right\}    &&  \\ 
+%
+&\equiv - \frac{1} {2} \sum\limits_{i,j,k} 	q^2 \;
+	\left\{    \delta_{i+1/2}   \left[   \overline{\overline{ U }}^{\,i,j+1/2}    \right]  
+	         + \delta_{j+1/2}  \left[   \overline{\overline{ V }}^{\,i+1/2,j}     \right]    \right\}    && \\ 
+\end{array} } 		
+%
+\allowdisplaybreaks
+\intertext{ 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: }
+\allowdisplaybreaks
+%
+& \qquad {\begin{array}{*{20}l} 
+&\equiv \sum\limits_{i,j,k} - \frac{1} {2} q^2 \; \overline{\overline{ e_{1t}\,e_{2t}\,e_{3t}^{}\, \chi}}^{\,i+1/2,j+1/2} 
+\quad \equiv 0 &&
+\end{array} } 		
+\end{flalign*}
+The later equality is obtain only when the flow is horizontally non-divergent, $i.e.$ $\chi$=$0$. 
+
+
+% -------------------------------------------------------------------------------------------------------------
+%       Vorticity Term with EEN scheme
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Vorticity Term with EEN scheme (\np{ln\_dynvor\_een}=.true.)}
+\label{Apdx_C_vorEEN} 
+
+With the EEN scheme, the vorticity terms are represented as: 
+\begin{equation} \label{Eq_dynvor_een}
+\left\{ {    \begin{aligned}
+ +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
+                         {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v} e_{3v} \ v  \right)^{i+i_p-1/2}_{j+j_p}   \\
+ - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
+                         {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u} e_{3u} \ u  \right)^{i+i_p}_{j+j_p-1/2}   \\
+\end{aligned}   } \right.
+\end{equation} 
+where the indices $i_p$ and $k_p$ take the following value: 
+$i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$,
+and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
+\end{equation}
+
+
+This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$). 
+
+Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 
+similar manipulation can be done for the 3 others. The discrete form of the right hand 
+side of \eqref{Apdx_C_1.1} applied to this triad only can be transformed as follow:
+
+\begin{flalign*} 
+&\int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \\
+%
+\equiv& \sum\limits_{i,j,k} 
+ {q} \    \biggl\{ \;\;
+	\delta_{i+1/2} \left[   -\, {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}}    \right]  
+      - \delta_{j+1/2} \left[       {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}    \right] 
+	\;\;\biggr\}        &&  \\ 
+%
+\equiv& \sum\limits_{i,j,k} 
+	\biggl\{   \delta_i [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; U^{i+1/2}_{j}}
+		   + \delta_j [q] \  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}  \; V^{i}_{j+1/2}}   \biggr\}
+		&& \\ 
+%
+... & &&\\
+&Demonstation \ to \ be \ done... &&\\
+... & &&\\
+%
+\equiv& \frac{1} {2} \sum\limits_{i,j,k}  
+	\biggl\{	\delta_i    \Bigl[    \left(  {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2   \Bigr]\;
+			\overline{\overline {U}}^{\,i,j+1/2} 
+		      + \delta_j 	\Bigl[    \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2     \Bigr]\; 
+			\overline{\overline {V}}^{\,i+1/2,j} 
 	\biggr\} 
-	&&& \\
-%
-\equiv& -\int_D \nabla \cdot \left( \rho \, \textbf{U} \right)\;g\;z\;\;dv 	&&& \\
-\end{flalign*}
-
-Note that this property strongly constrains the discrete expression of both 
-the depth of $T-$points and of the term added to the pressure gradient in the
-$s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation 
-of state is rarely used.
-
-% -------------------------------------------------------------------------------------------------------------
-%       Surface Pressure Gradient Term
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Surface Pressure Gradient Term}
-\label{Apdx_C.1.5}
-
-
-The surface pressure gradient has no contribution to the evolution of the vorticity. 
-This property is trivially satisfied locally since the equation verified by $\psi$ has 
-been derived from the discrete formulation of the momentum equation and of the curl. 
-But it has to be noted that since the elliptic equation satisfied by $\psi$ is solved 
-numerically by an iterative solver (preconditioned conjugate gradient or successive 
-over relaxation), the property is only satisfied at the precision requested for the 
-solver used.
-
-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 
-requested for 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%	&&& \\
-%
-&\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\} 
-	&& \\ 
-%
-\intertext{using the relation between \textit{$\psi $} and the vertical 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] 
-	&& \\ &  \qquad \;\;\,
-		+ \left( 
-		- M_v + \frac{1} {H_v \,e_{1v}}  \delta_i \left[ \partial_t \psi \right] 
-		\right)\;
-		e_{2v} \,\delta_i \left[ \partial_t \psi \right] 	\;
-	\biggr\} 
-	&& \\ 
-%
-\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] 
-	&& \\ &  \qquad \;\;\,
-	+ \delta_{i+1/2} 
-		\left[ \frac{e_{2v}} {H_v \,e_{2v}}  \delta_i \left[ \partial_t \psi \right] 
-		\right] 
-	+ \delta_{j+1/2} 
-	 	 \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 \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 to compute the streamfunction, 
-otherwise the surface pressure forces will do work. Nevertheless, since 
-the elliptic equation satisfied by $\psi $ is solved numerically by an iterative 
-solver, the property is only satisfied at the precision requested for the solver.
+	&&  \\ 
+%
+\equiv& - \frac{1} {2} \sum\limits_{i,j,k} 	\left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2\;
+	\biggl\{    \delta_{i+1/2} 
+	      \left[   \overline{\overline {U}}^{\,i,j+1/2}    \right]  
+	            + \delta_{j+1/2}
+		\left[   \overline{\overline {V}}^{\,i+1/2,j}     \right]  
+	\biggr\}    && \\ 
+%
+\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left( {{^i_j}\mathbb{Q}^{+1/2}_{+1/2}} \right)^2
+                                                            \; \overline{\overline{ b_t^{}\, \chi}}^{\,i+1/2,\,j+1/2}  &&\\
+%
+\ \ \equiv& \ 0     &&\\
+\end{flalign*}
+
+
+
+
 
 % ================================================================
@@ -626,59 +1033,64 @@
 \label{Apdx_C.2.1}
 
+conservation of a tracer, $T$:
+\begin{equation*}
+\frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right) 
+=  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0
+\end{equation*}
+
+conservation of its variance:
+\begin{flalign*} 
+\frac{\partial }{\partial t} \left( \int_D {\frac{1}{2} T^2\;dv} \right)
+=&  \int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }  
+-   \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }
+\end{flalign*}
+
+
 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}\; 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\} 
- 	 && \\
+the scheme are written in flux form. Indeed,  let $T$ be the tracer and $\tau_u$, $\tau_v$, 
+and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is), 
+the conservation of the tracer content due to the advection tendency is obtained as follows: 
+\begin{flalign*}
+&\int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv } = - \int_D \nabla \cdot \left( T \textbf{U} \right)\;dv    &&&\\
+&\equiv - \sum\limits_{i,j,k}    \biggl\{
+	 \frac{1} {b_t}  \left(  \delta_i    \left[   U \;\tau_u   \right]
+	                             + \delta_j    \left[   V  \;\tau_v   \right] \right) 
+	+ \frac{1} {e_{3t}} \delta_k \left[ w\;\tau_w \right]    \biggl\}  b_t   &&&\\
+%
+&\equiv - \sum\limits_{i,j,k}     \left\{
+	    \delta_i  \left[   U \;\tau_u   \right]
+         + \delta_j  \left[   V  \;\tau_v   \right]
+ 	 + \delta_k \left[ W \;\tau_w \right] \right\}   && \\
 &\equiv 0 &&&
 \end{flalign*}
 
-The conservation of the variance of tracer can be achieved only with the CEN2 scheme. 
+The conservation of the variance of tracer due to the advection tendency 
+can be achieved only with the CEN2 scheme, $i.e.$ when 
+$\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 
 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\;
+&\int_D { \frac{1}{e_3} Q      \frac{\partial \left( e_3 \, T \right) }{\partial t} \;dv }
+= - \int\limits_D \tau\;\nabla \cdot \left( T\; \textbf{U} \right)\;dv &&&\\
+\equiv& - \sum\limits_{i,j,k} T\;
 	\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\} 
-	&& \\
-\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]
-&&&\\& \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\} 
-	&&\\
-\equiv&  -\frac{1} {2}  \sum\limits_{i,j,k}
-	\Bigl\{
-	   e_{2u}\,e_{3u} \;  u\;\delta_{i+1/2} \left[ T^2 \right]
-	+ e_{1v}\,e_{3v} \;  v\;\delta_{j+1/2}  \left[ T^2 \right]
-	+ e_{1T}\,e_{2T} \;w\;\delta_{k+1/2} \left[ T^2 \right]
-	\Bigr\} 
-	&& \\ 
-\equiv& \frac{1} {2}  \sum\limits_{i,j,k} T^2
-	\Bigl\{
-	   \delta_i  \left[ e_{2u}\,e_{3u}\,u \right]
-	+ \delta_j  \left[ e_{1v}\,e_{3v}\,v \right]
-	+ \delta_k \left[ e_{1T}\,e_{2T}\,w \right]
-	\Bigr\} 
-\quad \equiv 0 &&&
-\end{flalign*}
-
+	   \delta_i  \left[ U  \overline T^{\,i+1/2}  \right]
+	+ \delta_j  \left[ V  \overline T^{\,j+1/2}  \right]
+	+ \delta_k \left[ W \overline T^{\,k+1/2} \right]          \right\} && \\
+\equiv& + \sum\limits_{i,j,k} 
+	\left\{     U  \overline T^{\,i+1/2} \,\delta_{i+1/2}  \left[ T \right] 
+                 +  V  \overline T^{\,j+1/2} \;\delta_{j+1/2}  \left[ T \right]
+                 +  W \overline T^{\,k+1/2}\;\delta_{k+1/2} \left[ T \right]     \right\}      &&\\
+\equiv&  + \frac{1} {2}  \sum\limits_{i,j,k}
+	\Bigl\{   U  \;\delta_{i+1/2} \left[ T^2 \right]
+                 + V  \;\delta_{j+1/2}  \left[ T^2 \right]
+                 + W \;\delta_{k+1/2} \left[ T^2 \right]   \Bigr\}     && \\ 
+\equiv& - \frac{1} {2}  \sum\limits_{i,j,k} T^2
+	\Bigl\{    \delta_i  \left[ U  \right]
+                  + \delta_j  \left[ V  \right]
+                  + \delta_k \left[ W \right]     \Bigr\}      &&&  \\
+\equiv& + \frac{1} {2}  \sum\limits_{i,j,k} T^2
+	\Bigl\{   \frac{1}{e_{3t}} \frac{\partial e_{3t}\,T }{\partial t}     \Bigr\}      &&& \\
+\end{flalign*}
+which is the discrete form of $ \frac{1}{2} \int_D {  T^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv }$.
 
 % ================================================================
@@ -714,16 +1126,11 @@
 \begin{flalign*}
 &\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  = 0
+	\Bigl[    \nabla_h  \left( A^{\,lm}\;\chi  \right)
+	          - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)    \Bigr]\;dv  = 0
 \end{flalign*}
 %%%%%%%%%%  recheck here....  (gm)
 \begin{flalign*}
 = \int \limits_D  -\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
-	\Bigl[ \nabla_h \times 
-		\left( A^{\,lm}\;\zeta \; \textbf{k} \right) 
-	\Bigr]\;dv &&& \\ 
+	\Bigl[ \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)  \Bigr]\;dv &&& \\ 
 \end{flalign*}
 \begin{flalign*}
@@ -802,5 +1209,5 @@
 \\  %%%
 \equiv& \sum\limits_{i,j,k} 
-	- A_T^{\,lm}  \,\chi^2   \;e_{1T}\,e_{2T}\,e_{3T}
+	- 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       \\
@@ -819,43 +1226,17 @@
 \begin{flalign*}
 &\int\limits_D  \zeta \; \textbf{k} \cdot \nabla \times 
-	\left[ 
-	  \nabla_h 
-		\left( A^{\,lm}\;\chi  \right)
-	-\nabla_h \times 
-		\left( A^{\,lm}\;\zeta \; \textbf{k} \right)
-	\right]\;dv &&&\\
+	\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 \zeta \textbf{k} \cdot \nabla \times 
- 	\left[ 
-	\nabla_h \times 
-		\left( \zeta \; \textbf{k} \right)
-	\right]\;dv &&&\displaybreak[0]\\
+ 	\left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\
 &\equiv A^{\,lm} \sum\limits_{i,j,k}  \zeta \;e_{3f} 
-	\left\{
-	  \delta_{i+1/2} 
-	\left[ 
-	\frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i 
-		\left[ e_{3f} \zeta  \right]
-	\right]
-	+ \delta_{j+1/2} 
-	\left[
-	\frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j 
-		\left[ e_{3f} \zeta  \right]
-	\right]
-	\right\} 
-	&&&\\ 
+	\left\{     \delta_{i+1/2} \left[  \frac{e_{2v}} {e_{1v}\,e_{3v}} \delta_i \left[ e_{3f} \zeta  \right]   \right]
+	          + \delta_{j+1/2} \left[  \frac{e_{1u}} {e_{2u}\,e_{3u}} \delta_j \left[ e_{3f} \zeta  \right]   \right]      \right\}   &&&\\ 
 %
 \intertext{Using \eqref{DOM_di_adj}, it follows:}
 %
 &\equiv  - A^{\,lm} \sum\limits_{i,j,k} 
-	\left\{
-	  \left( 
-	  \frac{1} {e_{1v}\,e_{3v}}  \delta_i 
-		\left[ e_{3f} \zeta  \right] 
-	  \right)^2   e_{1v}\,e_{2v}\,e_{3v}
-	+ \left(
-	  \frac{1} {e_{2u}\,e_{3u}}  \delta_j 
-	  	\left[ e_{3f} \zeta  \right]
-	  \right)^2   e_{1u}\,e_{2u}\,e_{3u}
-	  \right\}      &&&\\
+	\left\{    \left(  \frac{1} {e_{1v}\,e_{3v}}  \delta_i \left[ e_{3f} \zeta  \right]  \right)^2   b_v
+	         + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}      &&&\\
 & \leq \;0       &&&\\ 
 \end{flalign*}
@@ -874,45 +1255,18 @@
 \begin{flalign*}
 & \int\limits_D  \nabla_h \cdot 
-	\Bigl[ 
-	  \nabla_h 
-		\left( A^{\,lm}\;\chi \right)
-	- \nabla_h \times 
-		\left( A^{\,lm}\;\zeta \;\textbf{k} \right)
-	\Bigr]
-	dv
-= \int\limits_D  \nabla_h \cdot \nabla_h 
-	\left( A^{\,lm}\;\chi  \right)
-	dv 
-&&&\\
+	\Bigl[     \nabla_h \left( A^{\,lm}\;\chi \right)
+	          - \nabla_h \times \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \Bigr]  dv
+= \int\limits_D  \nabla_h \cdot \nabla_h \left( A^{\,lm}\;\chi  \right)   dv   &&&\\
+%
 &\equiv \sum\limits_{i,j,k} 
-	\left\{ 
-	  \delta_i
-		\left[ 
-		A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} 
-			\left[ \chi \right] 
-		\right]
-	+ \delta_j 
-		\left[ 
-		A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} 
-			\left[ \chi \right] 
-		\right] 
-	\right\}
-	&&&\\ 
+	\left\{   \delta_i \left[ A_u^{\,lm} \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]  \right]
+	        + \delta_j \left[ A_v^{\,lm} \frac{e_{1v}\,e_{3v}} {e_{2v}}  \delta_{j+1/2} \left[ \chi \right]  \right]    \right\}    &&&\\ 
 %
 \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} 
-		\left[ \chi \right]
-	\delta_{i+1/2} 
-		\left[ 1 \right] 
-	+ \frac{e_{1v}\,e_{3v}} {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} 
-		\left[ \chi \right]
-	\delta_{j+1/2} 
-		\left[ 1 \right]
-	\right\} \;
-	\equiv 0 
-	&&& \\ 
+	- \left\{   \frac{e_{2u}\,e_{3u}} {e_{1u}}  A_u^{\,lm} \delta_{i+1/2} \left[ \chi \right] \delta_{i+1/2} \left[ 1 \right] 
+	          + \frac{e_{1v}\,e_{3v}}  {e_{2v}}  A_v^{\,lm} \delta_{j+1/2} \left[ \chi \right] \delta_{j+1/2} \left[ 1 \right]    \right\} 
+	\qquad \equiv 0     &&& \\ 
 \end{flalign*}
 
@@ -929,17 +1283,16 @@
  = 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 
+&\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}    &&&\\ 
+	\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\} \;    &&&\\
+	\left\{    \left(  \frac{1} {e_{1u}}  \delta_{i+1/2}  \left[ \chi \right]  \right)^2  b_u
+                 + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\} \;    &&&\\
+%
 &\leq 0              &&&\\
 \end{flalign*}
@@ -956,22 +1309,13 @@
 with the conservation of momentum and the dissipation of horizontal kinetic energy:
 \begin{align*}
- \int\limits_D  
-	 \frac{1} {e_3 }\; \frac{\partial } {\partial k}
- 	\left( 
-	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
-	\right)\;
-	dv \qquad \quad &= \vec{\textbf{0}}
-	\\
+ \int\limits_D   \frac{1} {e_3 }\; \frac{\partial } {\partial k} 
+       \left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;  dv 
+	\qquad \quad &= \vec{\textbf{0}}    \\
 %
 \intertext{and}
 %
 \int\limits_D 
-	\textbf{U}_h \cdot 
-	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-	\left(
-	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}
-	\right)\;
-	dv \quad &\leq 0
-	\\
+	\textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+	\left(   \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\; dv    \quad &\leq 0     \\
 \end{align*}
 The first property is obvious. The second results from:
@@ -979,47 +1323,19 @@
 \begin{flalign*}
 \int\limits_D 
-	\textbf{U}_h \cdot 
-	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-	\left( 
-	\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-	\right)\;dv
-	&&&\\
+	\textbf{U}_h \cdot   \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+	\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)\;dv    &&&\\
 \end{flalign*}
 \begin{flalign*}
 &\equiv \sum\limits_{i,j,k} 
 	\left( 
-	  u\; \delta_k 
-		\left[ 
-		\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
-			\left[ u \right] 
-		\right]\;
-		e_{1u}\,e_{2u} 
-	+ v\;\delta_k 
-		\left[
-		\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
-			\left[ v \right] 
-		\right]\;
-		e_{1v}\,e_{2v} 
-	\right)
-	&&&\\ 
+	   u\; \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2}  \left[ u \right]  \right]\;  e_{1u}\,e_{2u} 
+	+ v\; \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}   \left[ v \right]  \right]\;  e_{1v}\,e_{2v} \right)   &&&\\ 
 %
 \intertext{since the horizontal scale factor does not depend on $k$, it follows:}
 %
 &\equiv - \sum\limits_{i,j,k} 
-	\left( 
-	   \frac{A_u^{\,vm}} {e_{3uw}}
-		\left( 
-		\delta_{k+1/2} 
-			\left[ u \right] 
-		\right)^2\;
-		e_{1u}\,e_{2u} 
-	+ \frac{A_v^{\,vm}} {e_{3vw}} 
-		\left( \delta_{k+1/2} 
-			\left[ v \right] 
-		\right)^2\;
-		e_{1v}\,e_{2v}
-	\right)
-	 \quad \leq 0
-	 &&&\\
+	\left(  \frac{A_u^{\,vm}} {e_{3uw}} \left( \delta_{k+1/2} \left[ u \right] \right)^2\; e_{1u}\,e_{2u} 
+	      + \frac{A_v^{\,vm}} {e_{3vw}}  \left( \delta_{k+1/2} \left[ v \right] \right)^2\; e_{1v}\,e_{2v}  \right)
+\quad \leq 0   &&&\\
 \end{flalign*}
 
@@ -1028,12 +1344,7 @@
 \int \limits_D 
 	\frac{1} {e_3 } \textbf{k} \cdot \nabla \times 
-		\left( 
-		\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-			\left( 
-			\frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k} 
-			\right)
-		\right)\;
-		dv
-		&&&\\ 
+		\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}  \left( 
+		        \frac{A^{\,vm}} {e_3}\; \frac{\partial \textbf{U}_h } {\partial k}  
+		  \right)  \right)\; dv   &&&\\ 
 \end{flalign*}
 \begin{flalign*}
@@ -1041,24 +1352,10 @@
 	\bigg\{    \biggr.   \quad
 	\delta_{i+1/2} 
-		&\left(
-		\frac{e_{2v}} {e_{3v}} \delta_k 
-			\left[
-			\frac{1} {e_{3vw}} \delta_{k+1/2} 
-				\left[ v \right] 
-			\right]
-		\right)
-	 &&\\
+		&\left(   \frac{e_{2v}} {e_{3v}} \delta_k  \left[  \frac{1} {e_{3vw}} \delta_{k+1/2} \left[ v \right]  \right]  \right)   &&\\
 	\biggl. 
 	- \delta_{j+1/2} 
-		&\left(
-		\frac{e_{1u}} {e_{3u}} \delta_k 
-			\left[
-			\frac{1} {e_{3uw}}\delta_{k+1/2} 
-				\left[ u \right]
-			\right]
-		\right)
+		&\left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[  \frac{1} {e_{3uw}}\delta_{k+1/2} \left[ u \right]  \right]   \right)
 	\biggr\} \;
-	e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0
-	&& \\
+	e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   && \\
 \end{flalign*}
 If the vertical diffusion coefficient is uniform over the whole domain, the 
@@ -1066,12 +1363,6 @@
 \begin{flalign*}
 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
-	\left( 
-	\frac{1} {e_3}\; \frac{\partial } {\partial k}
-		\left( 
-		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-		\right) 
-	\right)\;
-	dv = 0
-	&&&\\
+	\left(   \frac{1} {e_3}\; \frac{\partial } {\partial k}
+		\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}   \right)   \right)\; dv = 0   &&&\\
 \end{flalign*}
 This property is only satisfied in $z$-coordinates:
@@ -1079,12 +1370,6 @@
 \begin{flalign*}
 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times 
-	\left( 
-	\frac{1} {e_3}\; \frac{\partial } {\partial k}
-		\left( 
-		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-		\right)
-	\right)\;
-	dv
-	&&& \\ 
+	\left(  \frac{1} {e_3}\; \frac{\partial } {\partial k}
+		\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right)   \right)\; dv   &&& \\ 
 \end{flalign*}
 \begin{flalign*}
@@ -1092,23 +1377,8 @@
 	\biggl\{ 	\biggr.	\quad
 	\delta_{i+1/2} 
-		&\left( 
-		   \frac{e_{2v}} {e_{3v}} \delta_k 
-			\left[ 
-			\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
-				\left[ v \right] 
-			\right]
-		   \right)
-		   &&\\
+		&\left(   \frac{e_{2v}} {e_{3v}} \delta_k \left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2}[v]  \right]   \right)   &&\\
 	- \delta_{j+1/2} 
 		&\biggl.
-		\left( 	
-		   \frac{e_{1u}} {e_{3u}} \delta_k 
-			\left[ 
-			\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
-				\left[ u \right]
-			 \right] 
-		   \right) 
-	\biggr\} 
-	&&\\ 
+		\left(   \frac{e_{1u}} {e_{3u}} \delta_k \left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u]  \right]    \right) \biggr\}   &&\\ 
 \end{flalign*}
 \begin{flalign*}
@@ -1116,48 +1386,21 @@
 	\biggl\{		\biggr.	\quad
 	\frac{1} {e_{3v}} \delta_k 
-		&\left[ 
-		\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
-			\left[ \delta_{i+1/2} 
-				\left[ e_{2v}\,v \right]
-			 \right]
-		\right]
-		&&\\ 
+		&\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} \left[ \delta_{i+1/2} \left[ e_{2v}\,v \right] \right]   \right]    &&\\ 
  	 \biggl. 
 	- \frac{1} {e_{3u}} \delta_k 
-		&\left[ 
-		\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
-			\left[ \delta_{j+1/2} 
-				\left[ e_{1u}\,u \right]
-			 \right]
-		\right]
-	\biggr\} 
-	&&\\ 
+		&\left[  \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right]  \right]  \biggr\}  &&\\ 
 \end{flalign*}
 Using the fact that the vertical diffusion coefficients are uniform, and that in 
 $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so 
-that: $e_{3f} =e_{3u} =e_{3v} =e_{3T} $ and $e_{3w} =e_{3uw} =e_{3vw} $, 
+that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, 
 it follows:
 \begin{flalign*}
 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k 
-	\left[ 
-	\frac{1} {e_{3w}} \delta_{k+1/2} 
-		\Bigl[ 
-		\delta_{i+1/2} 
-			\left[ e_{2v}\,v \right]
-		- \delta_{j+1/ 2} 
-			\left[ e_{1u}\,u \right]
-		 \Bigr] 
-	\right] 
-	&&&\\
+	\left[   \frac{1} {e_{3w}} \delta_{k+1/2} \Bigl[   \delta_{i+1/2} \left[ e_{2v}\,v \right]
+		                          - \delta_{j+1/ 2} \left[ e_{1u}\,u \right]   \Bigr]    \right]    &&&\\
 \end{flalign*}
 \begin{flalign*}
 \equiv - A^{\,vm} \sum\limits_{i,j,k} \frac{1} {e_{3w}}
-	\left( 
-	\delta_{k+1/2} 
-		\left[ \zeta  \right]
-	 \right)^2 \;
-	 e_{1f}\,e_{2f} 
-	 \; \leq 0
-	 &&&\\
+	\left( \delta_{k+1/2} \left[ \zeta  \right] \right)^2 \; e_{1f}\,e_{2f}  \; \leq 0    &&&\\
 \end{flalign*}
 Similarly, the horizontal divergence is obviously conserved:
@@ -1165,12 +1408,6 @@
 \begin{flalign*}
 \int\limits_D \nabla \cdot 
-	\left( 
-	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-		\left( 
-		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-		\right) 
-	\right)\;
-	dv = 0
-	&&&\\
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0    &&&\\
 \end{flalign*}
 and the square of the horizontal divergence decreases ($i.e.$ the horizontal 
@@ -1180,49 +1417,24 @@
 \begin{flalign*}
 \int\limits_D \chi \;\nabla \cdot 
-	\left( 
-	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-		\left( 
-		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-		\right) 
-	\right)\;
-	dv = 0
-	&&&\\
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k}  \right) \right)\;  dv = 0  &&&\\
 \end{flalign*}
 This property is only satisfied in the $z$-coordinate:
 \begin{flalign*}
 \int\limits_D \chi \;\nabla \cdot 
-	\left( 
-	\frac{1} {e_3 }\; \frac{\partial } {\partial k}
-		\left( 
-		\frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} 
-		\right) 
-	\right)\;
-	dv
-	&&&\\
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1T}\,e_{2T}}
+\left( \frac{1} {e_3 }\; \frac{\partial } {\partial k}
+		\left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right)  \right)\; dv    &&&\\
+\end{flalign*}
+\begin{flalign*}
+\equiv \sum\limits_{i,j,k} \frac{\chi } {e_{1t}\,e_{2t}}
 	\biggl\{ 	\Biggr.	\quad
 	\delta_{i+1/2} 
-		&\left( 
-		   \frac{e_{2u}} {e_{3u}} \delta_k 
-		   	\left[ 
-			\frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} 
-				\left[ u \right] 
-			\right]
-		 \right)
-		 &&\\ 
+		&\left(   \frac{e_{2u}} {e_{3u}} \delta_k 
+		   	\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} [u] \right] \right)    &&\\ 
  	\Biggl. 
 	+ \delta_{j+1/2} 
-		&\left( 
-		\frac{e_{1v}} {e_{3v}} \delta_k 
-			\left[ 
-			\frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} 
-				\left[ v \right]
-			 \right]
-		\right)
-	\Biggr\} \;
-	e_{1T}\,e_{2T}\,e_{3T} 
-	&&\\ 
+		&\left( \frac{e_{1v}} {e_{3v}} \delta_k 
+			\left[ \frac{A_v^{\,vm}} {e_{3vw}} \delta_{k+1/2} [v] \right]   \right)
+	\Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   &&\\ 
 \end{flalign*}
 
@@ -1232,58 +1444,28 @@
 	\delta_{i+1/2}
 		&\left( 
-		   \delta_k 
-			\left[ 
-			\frac{1} {e_{3uw}} \delta_{k+1/2} 
-				\left[ e_{2u}\,u \right]
-			 \right]
-		   \right) 
-		   && \\ 
+		   \delta_k \left[ 
+			\frac{1} {e_{3uw}} \delta_{k+1/2} \left[ e_{2u}\,u \right] \right]   \right)    && \\ 
 	\biggl. 
 	+ \delta_{j+1/2} 
-		&\left( 
-		   \delta_k 
-			\left[ 
-			\frac{1} {e_{3vw}} \delta_{k+1/2} 
-				\left[ e_{1v}\,v \right]
-			 \right] 
-		   \right)
-	\biggr\} 
-	&& \\ 
+		&\left(    \delta_k \left[ 
+			\frac{1} {e_{3vw}} \delta_{k+1/2} \left[ e_{1v}\,v \right] \right]   \right)   \biggr\}    && \\ 
 \end{flalign*}
 
 \begin{flalign*}
 \equiv -A^{\,vm} \sum\limits_{i,j,k} 
-\frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\;
-	\biggl\{ 
-	\delta_{k+1/2} 
-		\Bigl[
-		   \delta_{i+1/2} 
-			\left[ e_{2u}\,u \right]
-		+ \delta_{j+1/2} 
-			\left[ e_{1v}\,v \right]
-		\Bigr]
-	\biggr\} 
-	&&&\\
+\frac{\delta_{k+1/2} \left[ \chi \right]} {e_{3w}}\; \biggl\{ 
+	\delta_{k+1/2} \Bigl[
+		   \delta_{i+1/2} \left[ e_{2u}\,u \right]
+		+ \delta_{j+1/2} \left[ e_{1v}\,v \right]  \Bigr]    \biggr\}    &&&\\
 \end{flalign*}
 
 \begin{flalign*}
 \equiv -A^{\,vm} \sum\limits_{i,j,k}
- \frac{1} {e_{3w}} 
-\delta_{k+1/2} 
-	\left[ \chi \right]\;
-\delta_{k+1/2} 
-	\left[ e_{1T}\,e_{2T} \;\chi \right]
-&&&\\
+ \frac{1} {e_{3w}} \delta_{k+1/2} \left[ \chi \right]\; \delta_{k+1/2} \left[ e_{1t}\,e_{2t} \;\chi \right]   &&&\\
 \end{flalign*}
 
 \begin{flalign*}
 \equiv -A^{\,vm} \sum\limits_{i,j,k} 
-\frac{e_{1T}\,e_{2T}} {e_{3w}}\;
-	\left( 
-	\delta_{k+1/2} 
-		\left[ \chi \right]
-	\right)^2
-	\quad  \equiv 0
-&&&\\
+\frac{e_{1t}\,e_{2t}} {e_{3w}}\; \left( \delta_{k+1/2} \left[ \chi \right]  \right)^2     \quad  \equiv 0    &&&\\
 \end{flalign*}
 
@@ -1327,5 +1509,5 @@
 	+ \delta_k 
 		\left[ 
-		A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
+		A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 
 			\left[ T \right] 
 		\right]
@@ -1351,5 +1533,5 @@
 		\quad&& \\ 
  \biggl. 
-&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1T}\,e_{2T}} {e_{3T}}\delta_{k+1/2}\left[T\right]\right]
+&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right]
 \biggr\} && 
 \end{flalign*}
@@ -1357,28 +1539,8 @@
 \equiv - \sum\limits_{i,j,k} 
 	\biggl\{ 	\biggr.	\quad
-	& A_u^{\,lT} 
-		\left( 
-		\frac{1} {e_{1u}} \delta_{i+1/2} 
-			\left[ T \right]
-		\right)^2
-		e_{1u}\,e_{2u}\,e_{3u}
-	&& \\
-	& + A_v^{\,lT} 
-		\left( 
-		\frac{1} {e_{2v}} \delta_{j+1/2} 
-			\left[ T \right] 
-		\right)^2
-		e_{1v}\,e_{2v}\,e_{3v}
-	&& \\ 
- 	\biggl. 
-	& + A_w^{\,vT} 
-		\left( 
-		\frac{1} {e_{3w}} \delta_{k+1/2} 
-			\left[ T \right] 
-		\right)^2
-		e_{1w}\,e_{2w}\,e_{3w} 
-	\biggr\} 
-	\quad \leq 0 
-	&& \\ 
+	&    A_u^{\,lT} \left( \frac{1} {e_{1u}} \delta_{i+1/2} \left[ T \right]  \right)^2   e_{1u}\,e_{2u}\,e_{3u}    && \\
+	& + A_v^{\,lT} \left( \frac{1} {e_{2v}} \delta_{j+1/2} \left[ T \right]  \right)^2   e_{1v}\,e_{2v}\,e_{3v}     && \\ \biggl. 
+	& + A_w^{\,vT} \left( \frac{1} {e_{3w}} \delta_{k+1/2} \left[ T \right]   \right)^2    e_{1w}\,e_{2w}\,e_{3w}   \biggr\} 
+	\quad      \leq 0      && \\ 
 \end{flalign*}
 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_D.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_D.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Annex_D.tex	(revision 1831)
@@ -5,4 +5,8 @@
 \label{Apdx_D}
 \minitoc
+
+\newpage
+$\ $\newline    % force a new ligne
+$\ $\newline    % force a new ligne
 
 
@@ -33,6 +37,7 @@
 
 - to use rules for variable names which allow recognition of their type 
-(integer, real, parameter, local or shared variables, etc. ) so that debugging is 
-facilitated.
+(integer, real, parameter, local or shared variables, etc. ). 
+
+This facilitates both the understanding and the debugging of an algorithm.
 
 % ================================================================
@@ -63,4 +68,6 @@
 - A program has not more than 200 to 300 lines.
 
+A template of a module style can be found on the NEMO depository 
+in the following file : NEMO/OPA\_SRC/module\_example.
 % ================================================================
 % Coding conventions
@@ -72,12 +79,26 @@
 features like statement functions, do not use GO TO and EQUIVALENCE statements.
 
-- A continuation line begins with the character {\$} indented by three spaces 
-compared to the previous line.
+- A continuation line begins with the character {\&} indented by three spaces 
+compared to the previous line, while the previous line ended with the character {\&}.
+
+- All the variables must be declared. The code is usually compiled with implicit none.
+ 
+- Never use continuation lines in the declaration of a variable. When searching a 
+variable in the code through a \textit{grep} command, the declaration line will be found.
+
+- In the declaration of a PUBLIC variable, the comment part at the end of the line 
+should start with the two characters "\verb?!:?". the following UNIX command, 
+\verb?grep var_name *90 \ grep \!: ? 
+will display the module name and the line where the var\_name declaration is.
 
 - Always use a three spaces indentation in DO loop, CASE, or IF-ELSEIF-ELSE-ENDIF 
 statements.
 
+- use a space after a comma, except when it appears to separate the indices of an array.
+
 - use call to ctl\_stop routine instead of just a STOP.
 
+
+\newpage
 % ================================================================
 % Naming Conventions
@@ -87,26 +108,30 @@
 
 The purpose of the naming conventions is to use prefix letters to classify 
-model variables. These conventions allow the variable type to be easily known
-and rapidly identified:
+model variables. These conventions allow the variable type to be easily 
+known and rapidly identified. The naming conventions are summarised 
+in the Table below:
+
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]
+\begin{table}[htbp]  \label{Tab_VarName}
 \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 \\  
+\begin{tabular}{|p{45pt}|p{35pt}|p{45pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|p{40pt}|}
+\hline  Type \par / Status &   integer&   real&   logical &   character  & structure &   double \par precision&   complex \\  
 \hline
 public  \par or  \par module variable& 
-\textbf{m n} \par \textit{but not } \par \textbf{nn\_}& 
-\textbf{a b e f g h o} \textbf{q} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_}& 
-\textbf{l} \par \textit{but not} \par \textbf{lp ld ll ln\_}& 
-\textbf{c} \par \textit{but not} \par \textbf{cp cd cl cn\_}& 
-\textbf{d} \par \textit{but not} \par \textbf{dp dd dl dn\_}& 
-\textbf{y} \par \textit{but not} \par \textbf{yp yd yl} \\
+\textbf{m n} \par \textit{but not} \par \textbf{nn\_}& 
+\textbf{a b e f g h o q r} \par \textbf{t} \textit{to} \textbf{x} \par but not \par \textbf{fs rn\_}& 
+\textbf{l} \par \textit{but not} \par \textbf{lp ld} \par \textbf{ ll ln\_}& 
+\textbf{c} \par \textit{but not} \par \textbf{cp cd} \par \textbf{cl cn\_}& 
+\textbf{s} \par \textit{but not} \par \textbf{sd sd} \par \textbf{sl sn\_}& 
+\textbf{d} \par \textit{but not} \par \textbf{dp dd} \par \textbf{dl dn\_}& 
+\textbf{y} \par \textit{but not} \par \textbf{yp yd} \par \textbf{yl yn} \\
 \hline
 dummy \par argument& 
 \textbf{k} \par \textit{but not} \par \textbf{kf}& 
-\textbf{p} \par \textit{but not}  \par \textbf{pp pf}& 
+\textbf{p} \par \textit{but not} \par \textbf{pp pf}& 
 \textbf{ld}& 
 \textbf{cd}& 
+\textbf{sd}& 
 \textbf{dd}& 
 \textbf{yd} \\
@@ -117,11 +142,13 @@
 \textbf{ll}& 
 \textbf{cl}& 
-\textbf{cd}& 
+\textbf{sl}& 
+\textbf{dl}& 
 \textbf{yl} \\
 \hline
 loop \par control& 
-\textbf{j} \par \textit{but not } \par \textbf{jp}& 
+\textbf{j} \par \textit{but not} \par \textbf{jp}& 
 & 
 & 
+&
 & 
 & 
@@ -133,4 +160,5 @@
 \textbf{lp}& 
 \textbf{cp}& 
+\textbf{sp}& 
 \textbf{dp}& 
 \textbf{yp} \\
@@ -142,11 +170,14 @@
 \textbf{ln\_}& 
 \textbf{cn\_}& 
+\textbf{sn\_}& 
 \textbf{dn\_}& 
+\textbf{yn\_}
 \\
 \hline
 CPP \par macro& 
 \textbf{kf}& 
-\textbf{sf} \par & 
+\textbf{fs} \par & 
 & 
+&
 & 
 & 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DOM.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 1831)
@@ -31,4 +31,6 @@
 as well as the DOM (DOMain) directory. 
 
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Fundamentals of the Discretisation
@@ -46,5 +48,5 @@
 \begin{figure}[!tb] \label{Fig_cell}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_cell.pdf}
-\caption{Arrangement of variables. $T$ indicates scalar points where temperature, 
+\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 
@@ -57,5 +59,5 @@
 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 
+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 
@@ -120,41 +122,31 @@
 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 
+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}}
-			+	\frac{1}{e_{2v} }\delta _{j+1/2} \left[ q \right]\;\,{\rm {\bf j}}
-			+	\frac{1}{e_{3w} }\delta _{k+1/2} \left[ q \right]\;\,{\rm {\bf k}}
+\nabla q\equiv 	\frac{1}{e_{1u} } \delta _{i+1/2 } [q] \;\,{\rm {\bf i}}
+		+	\frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,{\rm {\bf j}}
+		+	\frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,{\rm {\bf k}}
 \end{equation}
 \begin{multline} \label{Eq_DOM_lap}
-\Delta q\equiv \frac{1}{e_{1T} {\kern 1pt}e_{2T} {\kern 1pt}e_{3T} }\;\left( 
-{\delta _i \left[ {\frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
-\left[ q \right]} \right]
-+\delta _j \left[ {\frac{e_{1v} e_{3v} }{e_{2v} 
-}\;\delta _{j+1/2} \left[ q \right]} \right]\;} \right)		\\
-+\frac{1}{e_{3T} }\delta _k \left[ {\frac{1}{e_{3w} }\;\delta _{k+1/2} 
-\left[ q \right]} \right]
+\Delta q\equiv \frac{1}{e_{1t}\,e_{2t}\,e_{3t} } 
+       \;\left(          \delta_i  \left[ \frac{e_{2u}\,e_{3u}} {e_{1u}} \;\delta_{i+1/2} [q] \right]
++                        \delta_j  \left[ \frac{e_{1v}\,e_{3v}}  {e_{2v}} \;\delta_{j+1/2} [q] \right] \;  \right)		\\
++\frac{1}{e_{3t}} \delta_k \left[ \frac{1}{e_{3w} }                     \;\delta_{k+1/2} [q] \right]
 \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 $vw$-, $uw$, and $f$-points, and its divergence defined 
-at $T$-points:
-\begin{equation} \label{Eq_DOM_curl}
-\begin{split}
- \nabla \times {\rm {\bf A}}\equiv \frac{1}{e_{2v} {\kern 1pt}e_{3vw} 
-}{\kern 1pt}\,\;\left( {\delta _{j+1/2} \left[ {e_{3w} a_3 } \right]-\delta 
-_{k+1/2} \left[ {e_{2v} a_2 } \right]} \right)	&\;\;{\rm {\bf i}} \\ 
- +\frac{1}{e_{2u} {\kern 1pt}e_{3uw} }\;\left( {\delta _{k+1/2} \left[ {e_{1u} a_1 } 
-\right]-\delta _{i+1/2} \left[ {e_{3w} a_3 } \right]} \right) 	&\;\;{\rm{\bf j}} \\ 
- +\frac{e_{3f} }{e_{1f} {\kern 1pt}e_{2f} }\,{\kern 1pt}\;\left( {\delta 
-_{i+1/2} \left[ {e_{2v} a_2 } \right]-\delta _{j+12} \left[ {e_{1u} a_1 } \right]} 
-\right)	&\;\;{\rm {\bf k}}
- \end{split}
-\end{equation}
+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{eqnarray}  \label{Eq_DOM_curl}
+ \nabla \times {\rm {\bf A}}\equiv &
+      \frac{1}{e_{2v}\,e_{3vw} } \ \left( \delta_{j +1/2} \left[e_{3w}\,a_3 \right] -\delta_{k+1/2} \left[e_{2v} \,a_2 \right] \right)  &\ \rm{\bf i} \\ 
+ +& \frac{1}{e_{2u}\,e_{3uw}} \ \left( \delta_{k+1/2} \left[e_{1u}\,a_1  \right] -\delta_{i +1/2} \left[e_{3w}\,a_3 \right] \right)  &\ \rm{\bf j} \\ 
+ +& \frac{1}{e_{1f} \,e_{2f}    } \ \left( \delta_{i +1/2} \left[e_{2v}\,a_2  \right] -\delta_{j +1/2} \left[e_{1u}\,a_1 \right] \right)  &\ \rm{\bf k}
+ \end{eqnarray}
 \begin{equation} \label{Eq_DOM_div}
-\nabla \cdot {\rm {\bf A}}=\frac{1}{e_{1T} e_{2T} e_{3T} }\left( {\delta 
-_i \left[ {e_{2u} e_{3u} a_1 } \right]+\delta _j \left[ {e_{1v} e_{3v} a_2 } 
-\right]} \right)+\frac{1}{e_{3T} }\delta _k \left[ {a_3 } \right]
+\nabla \cdot \rm{\bf A}=\frac{1}{e_{1t}\,e_{2t}\,e_{3t}}\left( \delta_i \left[e_{2u}\,e_{3u}\,a_1 \right]
+                                                                                         +\delta_j \left[e_{1v}\,e_{3v}\,a_2 \right] \right)+\frac{1}{e_{3t} }\delta_k \left[a_3 \right]
 \end{equation}
 
@@ -164,7 +156,7 @@
 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]
+\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*}
 
@@ -172,5 +164,5 @@
 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} 
+\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}
@@ -189,5 +181,5 @@
 
 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 
+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 $(u,v,w)$.
 
@@ -230,7 +222,7 @@
 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 
+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 
+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 $k=1$.
@@ -242,8 +234,8 @@
 \label{DOM_Num_Index_hor}
 
-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.
+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.
 
 % -----------------------------------
@@ -257,10 +249,10 @@
 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 ($k=jpk$) 
+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 
+$t$-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 
+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 
@@ -309,4 +301,7 @@
 \S\ref{LBC_mpp}).
 
+
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Domain: Horizontal Grid (mesh) 
@@ -341,5 +336,5 @@
 factors in the horizontal plane as follows:
 \begin{flalign*}
-\lambda_T &\equiv \text{glamt}= \lambda(i)	  & \varphi_T &\equiv \text{gphit} = \varphi(j)\\
+\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)\\
@@ -347,6 +342,6 @@
 \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_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\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)|\\
@@ -359,13 +354,13 @@
 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 
+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 
+specific to the \NEMO model \citep{Marti_al_JGR92}. 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 
+$t$-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 
@@ -380,5 +375,5 @@
 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 
+$t$-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 
@@ -422,5 +417,5 @@
 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 $T$- 
+in order to ensure that the equator corresponds to line passing through $t$- 
 and $u$-points.  
 
@@ -431,6 +426,6 @@
 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 
+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, 
@@ -460,4 +455,6 @@
 the output grid written when $\np{nmsh} \not=0$ is no more equal to the input grid.
 
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Domain: Vertical Grid (domzgr)
@@ -467,5 +464,5 @@
 \label{DOM_zgr}
 %-----------------------------------------nam_zgr & namdom-------------------------------------------
-\namdisplay{nam_zgr} 
+\namdisplay{namzgr} 
 \namdisplay{namdom} 
 %-------------------------------------------------------------------------------------------------------------
@@ -593,9 +590,9 @@
 
 The reference coordinate transformation $z_0 (k)$ defines the arrays $gdept_0$ 
-and $gdepw_0$ for $T$- and $w$-points, respectively. As indicated on 
+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 
+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 
@@ -663,5 +660,5 @@
 \begin{center} \begin{tabular}{c||r|r|r|r}
 \hline
-\textbf{LEVEL}& \textbf{GDEPT}& \textbf{GDEPW}& \textbf{E3T }& \textbf{E3W  } \\ \hline
+\textbf{LEVEL}& \textbf{gdept}& \textbf{gdepw}& \textbf{e3t }& \textbf{e3w  } \\ \hline
 1	&	\textbf{  5.00}	&       0.00 &	\textbf{ 10.00} &	 10.00 \\	\hline
 2	&	\textbf{15.00}	& 	  10.00 &	\textbf{ 10.00} &	 10.00 \\	\hline
@@ -728,6 +725,6 @@
 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 
-partially filled with bathymetry at level jk is the minimum of \np{e3zpsmin} 
-(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{e3zps\_rat}$ (a fraction, 
+partially filled with bathymetry at level jk is the minimum of \np{rn\_e3zps\_min} 
+(thickness in meters, usually $20~m$) or $e_{3t}(jk)*\np{rn\_e3zps\_rat}$ (a fraction, 
 usually 10\%, of the default thickness $e_{3t}(jk)$).
 
@@ -738,8 +735,8 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [$s$-coordinate (\np{ln\_sco})]
-			{$s$-coordinate (\np{ln\_sco}=true)}
+		     {$s$-coordinate (\np{ln\_sco}=true)}
 \label{DOM_sco}
 %------------------------------------------nam_zgr_sco---------------------------------------------------
-\namdisplay{nam_zgr_sco} 
+\namdisplay{namzgr_sco} 
 %--------------------------------------------------------------------------------------------------------------
 In $s$-coordinate (\key{sco} is defined), the depth and thickness of the model 
@@ -752,5 +749,5 @@
 \end{split}
 \end{equation}
-where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $T$-point 
+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 
@@ -760,5 +757,5 @@
 sharp bathymetric gradients.
 
-A new flexible stretching function, modified from \citet{Song1994} is provided as an example:
+A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example:
 \begin{equation} \label{DOM_sco_function}
 \begin{split}
@@ -801,9 +798,9 @@
 Whatever the vertical coordinate used, the model offers the possibility of 
 representing the bottom topography with steps that follow the face of the 
-model cells (step like topography) \citep{Madec1996}. The distribution of 
+model cells (step like topography) \citep{Madec_al_JPO96}. The distribution of 
 the steps in the horizontal is defined in a 2D integer array, mbathy, which 
 gives the number of ocean levels ($i.e.$ those that are not masked) at each 
-$T$-point. mbathy is computed from the meter bathymetry using the definiton of 
-gdept as the number of $T$-points which gdept $\leq$ bathy. Note that in version 
+$t$-point. mbathy is computed from the meter bathymetry using the definiton of 
+gdept as the number of $t$-points which gdept $\leq$ bathy. Note that in version 
 NEMO v2.3, the user still has to provide the "level" bathymetry in a NetCDF
 file when using the full step option (\np{ln\_zco}), rather than the bathymetry 
@@ -817,7 +814,7 @@
 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 
+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. 
 
@@ -849,4 +846,6 @@
 %%%
 
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Time Discretisation
@@ -860,5 +859,5 @@
    x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t,t,t+\Delta t}
 \end{equation} 
-where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the 
+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 
@@ -995,5 +994,5 @@
 
 This is diffusive in time and conditionally stable. For example, the 
-conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies2004}:
+conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
 \begin{equation} \label{Eq_DOM_nxt_euler_stability}
 A^h < \left\{
@@ -1040,5 +1039,5 @@
 approximation of the temperature equation is:
 \begin{equation} \label{Eq_DOM_nxt_imp_zdf}
-\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3T} }\delta 
+\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\Delta t}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta 
 _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
 \right]
@@ -1109,2 +1108,54 @@
 }     	%% end add
 
+
+
+
+
+Implicit time stepping in case of variable volume thickness.
+
+Tracer case (NB for momentum in vector invariant form take care!)
+
+\begin{flalign*}
+&\frac{\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}}{2\Delta t}
+\equiv \text{RHS}+ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
+\right]      \\
+&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}
+\equiv {2\Delta t} \ \text{RHS}+ {2\Delta t} \ \delta _k \left[ {\frac{A_w^{vt} }{e_{3w}^{t+1} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
+\right]      \\
+&\left( e_{3t}\,T \right)_k^{t+1}-\left( e_{3t}\,T \right)_k^{t-1}
+\equiv 2\Delta t \ \text{RHS}
++ 2\Delta t \ \left\{ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} [ T_{k+1}^{t+1} - T_k      ^{t+1} ]
+                          - \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} [ T_k       ^{t+1} - T_{k-1}^{t+1} ]  \right\}     \\
+&\\
+&\left( e_{3t}\,T \right)_k^{t+1}
+-  {2\Delta t} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}                  T_{k+1}^{t+1} 
++ {2\Delta t} \ \left\{  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2} 
+                            +  \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}     \right\}   T_{k    }^{t+1}
+-  {2\Delta t} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                  T_{k-1}^{t+1}      \\
+&\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}    \\
+%
+\end{flalign*}
+
+\begin{flalign*}
+\allowdisplaybreaks
+\intertext{ Tracer case }
+%
+&  \qquad \qquad  \quad   -  {2\Delta t}                  \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}   
+                                                                                                      \qquad \qquad \qquad  \qquad  T_{k+1}^{t+1}   \\
+&+ {2\Delta t} \ \biggl\{  (e_{3t})_{k   }^{t+1}  \bigg. +    \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}  
+                                                                               +   \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2} \bigg. \biggr\}  \ \ \ T_{k   }^{t+1}  &&\\
+& \qquad \qquad  \qquad \qquad \qquad \quad \ \ -  {2\Delta t} \                          \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k-1/2}                          \quad \ \ T_{k-1}^{t+1}     
+\ \equiv \ \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}  \\
+%
+\end{flalign*}
+\begin{flalign*}
+\allowdisplaybreaks
+\intertext{ Tracer content case }
+%
+& -  {2\Delta t} \              & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_{k+1}^{t+1}}  && \  \left( e_{3t}\,T \right)_{k+1}^{t+1}   &\\
+& + {2\Delta t} \ \left[ 1  \right.+ & \frac{(A_w^{vt})_{k+1/2}} {(e_{3w})_{k+1/2}^{t+1}\;(e_{3t})_k^{t+1}} 
+                                    + & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_k^{t+1}}  \left.  \right]  & \left( e_{3t}\,T \right)_{k   }^{t+1}  &\\
+& -  {2\Delta t} \               & \frac{(A_w^{vt})_{k -1/2}} {(e_{3w})_{k -1/2}^{t+1}\;(e_{3t})_{k-1}^{t+1}}     &\  \left( e_{3t}\,T \right)_{k-1}^{t+1}   
+\equiv \left( e_{3t}\,T \right)_k^{t-1} + {2\Delta t} \ \text{RHS}  &
+\end{flalign*}
+
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DYN.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 1831)
@@ -24,10 +24,9 @@
 			+ \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
 \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 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 
+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 
@@ -60,4 +59,103 @@
 
 $\ $\newline    % force a new ligne
+
+% ================================================================
+% Sea Surface Height evolution & Diagnostics variables
+% ================================================================
+\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
+\label{DYN_divcur_wzv}
+
+%--------------------------------------------------------------------------------------------------------------
+%           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 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 a $T$-point. It is given by:
+\begin{equation} \label{Eq_divcur_div}
+\chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+		\left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
+		       +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
+\end{equation} 
+
+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. 
+
+%--------------------------------------------------------------------------------------------------------------
+%           Sea Surface Height evolution
+%--------------------------------------------------------------------------------------------------------------
+\subsection   [Sea surface height evolution and vertical velocity (\textit{sshwzv})]
+			{Horizontal divergence and relative vorticity (\mdl{sshwzv})}
+\label{DYN_sshwzv}
+
+The sea surface height is given by :
+\begin{equation} \label{Eq_dynspg_ssh}
+\begin{aligned}
+\frac{\partial \eta }{\partial t}
+&\equiv    \frac{1}{e_{1t} e_{2t} }\sum\limits_k { \left(  \delta _i \left[ {e_{2u}\,e_{3u}\;u} \right]
+                                                                                  +\delta _j \left[ {e_{1v}\,e_{3v}\;v} \right]  \right) } 
+           -    \frac{\textit{emp}}{\rho _w }   \\
+&\equiv    \sum\limits_k {\chi \ e_{3t}}  -  \frac{\textit{emp}}{\rho _w }
+\end{aligned}
+\end{equation}
+where \textit{emp} is the surface freshwater budget (evaporation minus precipitation), 
+expressed in Kg/m$^2$/s (which is equal to mm/s), 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 \textit{emp} can be written as the evaporation minus 
+precipitation, minus the river runoff. The sea-surface height is evaluated 
+using exactly the same time stepping as the tracer equation \eqref{Eq_tra_nxt}: 
+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). 
+This is of paramount importance. Substituing $T$ by $1$ in the tracer equation and summing
+over the water column must lead to the sea surface height equation otherwise tracer content
+could not be conserved \ref{Griffies_al_MWR01, LeclairMadec2009}.
+
+The vertical velocity is computed by an upward integration of the horizontal 
+divergence from the bottom, taken into account the change of the thickness of the levels :
+
+\begin{equation} \label{Eq_wzv}
+\left\{   \begin{aligned}
+&\left. w \right|_{3/2} \quad= 0   	\\
+&\left. w \right|_{k+1/2}     = \left. w \right|_{k+1/2}  + e_{3t}\;  \left. \chi \right|_k  
+                                         - \frac{ e_{3t}^{t+1} - e_{3t}^{t-1} } {2 \Delta t}
+\end{aligned}   \right.
+\end{equation}
+
+In case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 
+as changes in the the divergence of the barotropic transport is absorbed in the change 
+of the levels thickness.re-oriented downward co
+In case of linear free surface, the time derivative in \eqref{Eq_wzv} cancel out.
+The upper boundary condition applies at a fixed level $z=0$. The top vertical velocity 
+is thus equal to the divergence of the barotropic transport ($i.e.$ the first term in the
+right-hand-side of \eqref{Eq_dynspg_ssh}).
+
+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. 
+Note also that the $k$-axis is re-oriented downward in the \textsc{fortran} code compare 
+to the indexing used in the semi-discrete equations such as \eqref{Eq_wzv} 
+(see  \S\ref{DOM_Num_Index_vertical}). 
+
+
 % ================================================================
 % Coriolis and Advection terms: vector invariant form
@@ -66,5 +164,5 @@
 \label{DYN_adv_cor_vect}
 %-----------------------------------------nam_dynadv----------------------------------------------------
-\namdisplay{nam_dynadv} 
+\namdisplay{namdyn_adv} 
 %-------------------------------------------------------------------------------------------------------------
 
@@ -85,15 +183,17 @@
 \label{DYN_vor}
 %------------------------------------------nam_dynvor----------------------------------------------------
-\namdisplay{nam_dynvor} 
+\namdisplay{namdyn_vor} 
 %-------------------------------------------------------------------------------------------------------------
 
-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. They are 
-all computed in dedicated routines that can be found in the \mdl{dynvor} module.
+Four discretisations of the vorticity term (\textit{ln\_dynvor\_xxx}=.true.) are available: 
+conserving potential enstrophy of horizontally non-divergent flow (ENS scheme) ; 
+conserving horizontal kinetic energy (ENE scheme) ; conserving potential enstrophy for 
+the relative vorticity term and horizontal kinetic energy for the planetary vorticity 
+term (MIX scheme) ; or conserving both the potential enstrophy of horizontally non-divergent 
+flow and horizontal kinetic energy (ENE scheme) (see  Appendix~\ref{Apdx_C_vor_zad}). 
+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. They are all computed in dedicated routines that can be found in 
+the \mdl{dynvor} module.
 
 %-------------------------------------------------------------
@@ -106,12 +206,12 @@
 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:
+flow ($i.e.$ $\chi$=$0$), but does not conserve the total kinetic energy. It is given by:
 \begin{equation} \label{Eq_dynvor_ens}
 \left\{ 
 \begin{aligned}
 {+\frac{1}{e_{1u} } } & {\overline {\left( { \frac{\zeta +f}{e_{3f} }} \right)} }^{\,i} 
-                                & {\overline{\overline {\left( {e_{1v} e_{3v} v} \right)}} }^{\,i, j+1/2}    \\
+                                & {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i, j+1/2}    \\
 {- \frac{1}{e_{2v} } } & {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)} }^{\,j}  
-                                & {\overline{\overline {\left( {e_{2u} e_{3u} u} \right)}} }^{\,i+1/2, j}  
+                                & {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2, j}  
 \end{aligned} 
  \right.
@@ -129,7 +229,7 @@
 \left\{   \begin{aligned}
 {+\frac{1}{e_{1u}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
-                            \;  \overline {\left( {e_{1v} e_{3v} v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
+                            \;  \overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} }    \\
 {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
-                            \;  \overline {\left( {e_{2u} e_{3u} u} \right)} ^{\,j+1/2}} }^{\,i} }
+                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
 \end{aligned}    \right.
 \end{equation} 
@@ -146,16 +246,14 @@
 to the planetary vorticity term.
 \begin{equation} \label{Eq_dynvor_mix}
-\left\{ {
-\begin{aligned}
+\left\{ {     \begin{aligned}
  {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i} 
- \; {\overline{\overline {\left( {e_{1v} \; e_{3v} \ v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
+ \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
  \; {\overline {\left( {\frac{f}{e_{3f} }} \right) 
- \;\overline {\left( {e_{1v} \; e_{3v} \ v} \right)} ^{\,i+1/2}} }^{\,j} } \\
+ \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
 {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
- \; {\overline{\overline {\left( {e_{2u} \; e_{3u} \ u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
+ \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
  \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
- \;\overline {\left( {e_{2u}\; e_{3u} \ u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
-\end{aligned} 
-} \right.
+ \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
+\end{aligned}     } \right.
 \end{equation} 
 
@@ -166,20 +264,28 @@
 \label{DYN_vor_een}
 
-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: 
+In both the ENS and ENE schemes, it is apparent that the combination of $i$ and $j$ 
+averages of the velocity allows for the presence of grid point oscillation structures 
+that will be invisible to the operator. These structures are \textit{computational modes} 
+that will beat least partly damped by the momentum diffusive operator ($i.e.$ the 
+subgrid-scale advection), but not by the resolved advection term. These two schemes
+therefore do not contribute to dump grid point noise in the horizontal velocity field, 
+which results in more noise in vertical velocity field, an undesired feature. This is a well-known 
+characteristics of $C$-grid discretization where $u$ and $v$ are located at different grid point,
+a price to pay to avoid a double averaging on the pressure gradient term as in $B$-grid. 
+To circumvent this, Adcroft (ADD REF HERE) 
+we have proposed to introduce a second velocity ... blahblah.... 
+
+Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....
+
+A very nice solution to that problem was proposed by \citet{Arakawa_Hsu_MWR90}. The idea is
+to get rid of the double averaging by considering triad combinations of vorticity. 
+It is noteworthy that this solution is conceptually quite similar to the one proposed by
+\citep{Griffies_al_JPO98} for the discretization of the iso-neutral diffusive operator.
+
+The \citet{Arakawa_Hsu_MWR90} vorticity advection scheme for a single layer is modified 
+for spherical coordinates as described by \citet{Arakawa_Lamb_MWR81} to obtain the EEN scheme. 
+Let first provides the discrete expression of the potential vorticity, $q$, defined at an $f$-point: 
 \begin{equation} \label{Eq_pot_vor}
-q_f  = \frac{\zeta +f} {e_{3f} }
+q  = \frac{\zeta +f} {e_{3f} }
 \end{equation}
 where the relative vorticity is defined by (\ref{Eq_divcur_cur}), the Coriolis parameter 
@@ -202,9 +308,17 @@
 \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}
-%%%
+$e_{3f}$ 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. This feature is essential for 
+$z$-coordinate with partial step.
+
+
+Then, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at $T$-point as 
+the following triad combinations of the neighbouring potential vorticities defined at f-point 
+(Fig.~\ref{Fig_DYN_een_triad}): 
+\begin{equation} \label{Q_triads}
+_i^j \mathbb{Q}^{i_p}_{j_p}
+= \frac{1}{12} \ \left(   q^{i-i_p}_{j+j_p} + q^{i+j_p}_{j+i_p} + q^{i+i_p}_{j-j_p}  \right)
+\end{equation}
+where the indices $i_p$ and $k_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. 
 
 The vorticity terms are represented as: 
@@ -212,39 +326,20 @@
 \left\{ {
 \begin{aligned}
- +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }  \left[ 
-{{\begin{array}{*{20}c}
-    	{\,\ \ a_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } 
-  	{\,+\,b_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i-1/2}  } \\
- \\
-     {  +\,c_{j-1/2}^{i   }  \left( {e_{1v} e_{3v} \ v} \right)_{j    }^{i+1/2}         } 
- 	{\,+\,d_{j+1/2}^{i   } \left( {e_{1v} e_{3v} \ v} \right)_{j+1}^{i+1/2} } \\
-\end{array} }} \right] \\ 
-\\
--q\,e_3 \,u  		&\equiv -\frac{1}{e_{2v} }  \left[ 
-{{\begin{array}{*{20}c}
- 	{\,\ \ a_{j-1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1}^{i+1/2} } 
-  	{\,+\,b_{j-1/2}^{i+1}  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } \\
- \\
-     	{  +\,c_{j+1/2}^{i+1} \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i+1} } 
-  	{\,+\,d_{j+1/2}^{i   }  \left( {e_{2u} e_{3u} \ u} \right)_{j+1/2}^{i   } } \\
-\end{array} }} \right]
+ +q\,e_3 \, v 	&\equiv +\frac{1}{e_{1u} }   \sum_{\substack{i_p,\,k_p}} 
+                         {^{i+1/2-i_p}_j}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{1v}\,e_{3v} \;v  \right)^{i+1/2-i_p}_{j+j_p}   \\
+ - q\,e_3 \, u     &\equiv -\frac{1}{e_{2v} }    \sum_{\substack{i_p,\,k_p}} 
+                         {^i_{j+1/2-j_p}}  \mathbb{Q}^{i_p}_{j_p}  \left( e_{2u}\,e_{3u} \;u  \right)^{i+i_p}_{j+1/2-j_p}   \\
 \end{aligned} 
 } \right.
 \end{equation} 
-where $a$, $b$, $c$ and $d$ are the following triad combinations of the 
-neighbouring potential vorticities (Fig.~\ref{Fig_DYN_een_triad}): 
-\begin{equation} \label{Eq_een_triads}
-\left\{ 
-\begin{aligned}
- a_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i+1} + q_{j+1 /2}^i + q_{j-1/2}^i  \right)    \\ 
- \\
- b_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j+1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
-\\
- c_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i-1} +q_{j+1/2}^i +q_{j-1/2}^i   \right)     \\ 
-\\
- d_{\,j+1/2}^i & = \frac{1}  {12} \left( q_{j-1/2}^{i+1} +q_{j+1/2}^i +q_{j-1/2}^i  \right)     \\ 
-\end{aligned} 
-\right.
-\end{equation}
+
+This EEN scheme in fact combines the conservation properties of ENS and ENE schemes. 
+It conserves both total energy and potential enstrophy in the limit of horizontally 
+nondivergent flow ($i.e.$ $\chi$=$0$) (see  Appendix~\ref{Apdx_C_vor_zad}). 
+Applied to a realistic ocean configuration, it has been shown that its larger mantis
+leads to a significant reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. 
+Furthermore, used in combination with partial step representation of bottom topography,
+it improves the interaction between current and topography, leading to a larger
+topostrophy of the flow  \citep{Barnier_al_OD06, Penduff_al_OS07}. 
 
 %--------------------------------------------------------------------------------------------------------------
@@ -260,8 +355,6 @@
 \begin{equation} \label{Eq_dynkeg}
 \left\{ \begin{aligned}
- -\frac{1}{2 \; e_{1u} } 
- & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
- -\frac{1}{2 \; e_{2v} } 
- & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]    
+ -\frac{1}{2 \; e_{1u} }  & \ \delta _{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
+ -\frac{1}{2 \; e_{2v} }  & \ \delta _{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]    
 \end{aligned} \right.
 \end{equation} 
@@ -280,9 +373,7 @@
 \begin{equation} \label{Eq_dynzad}
 \left\{ 		\begin{aligned}
- -\frac{1}  { e_{1u}\,e_{2u}\,e_{3u} }  &  
-  \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,i+1/2}  \;\delta _{k+1/2} \left[ u \right]  }^{\,k}   } \\
- -\frac{1}  { e_{1v}\,e_{2v}\,e_{3v} }  &
-  \ {\overline {\overline{ e_{1T}\,e_{2T}\,w } ^{\,j+1/2}  \;\delta _{k+1/2} \left[ u \right]  }^{\,k}   }
-\end{aligned} \right.
+-\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k}  \\
+-\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta _{k+1/2} \left[ u \right]\  }^{\,k} 
+\end{aligned}         \right.
 \end{equation} 
 
@@ -293,5 +384,5 @@
 \label{DYN_adv_cor_flux}
 %------------------------------------------nam_dynadv----------------------------------------------------
-\namdisplay{nam_dynadv} 
+\namdisplay{namdyn_adv} 
 %-------------------------------------------------------------------------------------------------------------
 
@@ -315,7 +406,6 @@
 \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)  \\
-   \equiv   f + \frac{1}{e_{1f} e_{2f} } 
-	\left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right]  
-	         -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
+   \equiv   f + \frac{1}{e_{1f} e_{2f} } \left( { \ \overline v ^{i+1/2}\delta _{i+1/2} \left[ {e_{2u} } \right]  
+                                                                 -  \overline u ^{j+1/2}\delta _{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
 \end{multline} 
 
@@ -338,12 +428,12 @@
 \begin{aligned}
 \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
-\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\ u }^{i       }  \ u_T      \right]    
-          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\ v }^{i+1/2}  \ u_F      \right] \right.  \ \;   \\
-\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w} w}^{i+1/2}  \ u_{uw} \right] \right)   \\
+\left(      \delta _{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]    
+          + \delta _{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\
+\left.   + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
 \\
 \frac{1}{e_{1v}\,e_{2v}\,e_{3v}} 
-\left( 	\delta _{i      } \left[  \overline{e_{2u}\,e_{3u } \ u }^{j+1/2} \ v_F       \right] 
-         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u } \ v }^{i       } \ v_T       \right] \right.  \ \, \\
-\left.  + \delta _{k     } \left[  \overline{e_{1w}\,e_{2w} \ w}^{j+1/2} \ v_{vw}  \right] \right) \\
+\left(     \delta _{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right] 
+         + \delta _{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\
+\left.  + \delta _{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
 \end{aligned}
 \right.
@@ -369,11 +459,7 @@
 \begin{equation} \label{Eq_dynadv_cen2}
 \left\{ 		\begin{aligned}
- u_T^{cen2} &=\overline u^{i }		\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 &
- v_F^{cen2} &=\overline v^j		\quad &
- v_{vw}^{cen2} &=\overline v ^{k+1/2}      \\
-\end{aligned} \right.
+ u_T^{cen2} &=\overline u^{i }       \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 & v_F^{cen2} &=\overline v^j		\quad &  v_{vw}^{cen2} &=\overline v ^{k+1/2}  \\
+\end{aligned}      \right.
 \end{equation} 
 
@@ -417,9 +503,9 @@
 (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.
+by \citet{Webb_al_JAOT98} 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}. 
+(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 
 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 
@@ -440,10 +526,6 @@
 \label{DYN_hpg}
 %------------------------------------------nam_dynhpg---------------------------------------------------
-\namdisplay{nam_dynhpg} 
-\namdisplay{namflg} 
+\namdisplay{namdyn_hpg} 
 %-------------------------------------------------------------------------------------------------------------
-%%%
-\gmcomment{Suppress the namflg namelist and incorporate it in the namhpg namelist}
-%%%
 
 The key distinction between the different algorithms used for the hydrostatic 
@@ -454,5 +536,5 @@
 
 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 
+$i.e.$ 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.
@@ -495,5 +577,7 @@
 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 ($z=0)$. 
+level ($z=0$). Note also that in case of variable volume level (\key{vvl} defined), the 
+surface pressure gradient is included in \eqref{Eq_dynhpg_zco_surf} and \eqref{Eq_dynhpg_zco} 
+through the space and time variations of the vertical scale factor $e_{3w}$.
 
 %--------------------------------------------------------------------------------------------------------------
@@ -528,12 +612,12 @@
 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., 
+$\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\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{g\; \overline {\rho}^{i+1/2}}	{\rho_o \, e_{1u}} \;	\delta _{i+1/2} \left[  z_T  \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{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  z_T  \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} 
@@ -551,5 +635,5 @@
 (\np{ln\_dynhpg\_djc}=.true.)
 
-$\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.)
+$\bullet$ Rotated axes scheme (rot) \citep{Thiem_Berntsen_OM06} (\np{ln\_dynhpg\_rot}=.true.)
 
 Note that expression \eqref{Eq_dynhpg_sco} is used when the variable volume 
@@ -617,6 +701,5 @@
 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..
+reproducibility. This option is controlled by  \np{nn\_dynhpg\_rst}, a namelist parameter.
 
 % ================================================================
@@ -627,58 +710,31 @@
 \label{DYN_hpg_spg}
 %-----------------------------------------nam_dynspg----------------------------------------------------
-\namdisplay{nam_dynspg} 
+\namdisplay{namdyn_spg} 
 %------------------------------------------------------------------------------------------------------------
 
-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 (\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. 
-
-%-------------------------------------------------------------
-% Explicit
-%-------------------------------------------------------------
-\subsubsection{Explicit (\key{dynspg\_exp})}
+$\ $\newline      %force an empty line
+
+The form of the surface pressure gradient term depends on how the user wants to handle 
+the fast external gravity waves that are solution of the analytical equation (\S\ref{PE_hor_pg}). 
+Three formulation are available, all controlled by a CPP key (ln\_dynspg\_xxx):
+a explicit formulation which required a small time step ;
+a filtered free surface formulation which allows a larger time step by adding a filtering 
+term in the momentum equation ; 
+and a plit-explicit free surface formulation, described below, which also allows a larger time step.
+
+The extra term introduced in the filtered method is calculated 
+implicitly, so that a solver is used to compute it and that the update of the $next$ 
+velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+
+
+
+%--------------------------------------------------------------------------------------------------------------
+% Explicit free surface formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Explicit free surface (\key{dynspg\_exp})}
 \label{DYN_spg_exp}
 
-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} 
-e_{2T} }\sum\limits_k {\left( {\delta _i \left[ {e_{2u} e_{3u} u} 
-\right]+\delta _j \left[ {e_{1v} e_{3v} v} \right]} \right)} 
-\end{equation}
-where EMP is the surface freshwater budget, expressed in Kg/m$^2$/s 
-(which is equal to mm/s), 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). 
+In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step 
+is chosen to be small enough to describe the external gravity waves (typically a few tens of seconds). 
 
 The surface pressure gradient, also evaluated using a leap-frog scheme, is 
@@ -686,26 +742,32 @@
 \begin{equation} \label{Eq_dynspg_exp}
 \left\{ \begin{aligned}
- - \frac{1}{e_{1u}} \;	\delta _{i+1/2} \left[  \,\eta\,  \right] 	\\
- - \frac{1}{e_{2v}} \;	\delta _{j+1/2} \left[  \,\eta\,  \right]  
+ - \frac{1}{e_{1u}\,\rho_o} \;	\delta _{i+1/2} \left[  \,\rho \,\eta\,  \right] 	\\
+ - \frac{1}{e_{2v}\,\rho_o} \;	\delta _{j+1/2} \left[  \,\rho \,\eta\,  \right]  
 \end{aligned} \right.
 \end{equation} 
 
-Consistent with the linearization, a factor of $\left. \rho \right|_{k=1} / \rho _o$ 
-is omitted in \eqref{Eq_dynspg_exp}. 
-
-%-------------------------------------------------------------
-% Split-explicit time-stepping
-%-------------------------------------------------------------
-\subsubsection{Split-explicit time-stepping (\key{dynspg\_ts})}
+Note that in the non-linear free surface case ($i.e.$ \key{vvl} defined), the surface pressure 
+gradient is already included in the momentum tendency  through the level thickness variation 
+when computing the hydrostatic pressure gradient. Thus, nothing is done in the \mdl{dynspg\_exp} module.
+
+%--------------------------------------------------------------------------------------------------------------
+% Split-explict free surface formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Split-Explicit free surface (\key{dynspg\_ts})}
 \label{DYN_spg_ts}
-%--------------------------------------------namdom----------------------------------------------------
-\namdisplay{namdom} 
-%--------------------------------------------------------------------------------------------------------------
+
+In the split-explicit free surface formulation, also called time-splitting formulation 
+(\key{dynspg\_ts} defined)
+
 
 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}). 
+proposed by \citet{Griffies_Bk04}. The general idea is to solve the free surface 
+equation and the associated barotropic velocity equations with a smaller time 
+step than $\Delta t$, the time step used for the three dimensional prognostic 
+variables (Fig.\ref {Fig_DYN_dynspg_ts}). 
+The size of the small time step, $\Delta_e$ (the external mode or barotropic time step)
+ is provided through \np{nn\_baro} namelist parameter as: 
+$\Delta_e = \Delta / nn\_baro$.
+ 
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -739,71 +801,14 @@
 shown by \citet{Levier2007} in the case of an analytical barotropic Kelvin wave. 
 
-%-------------------------------------------------------------
-% Filtered formulation 
-%-------------------------------------------------------------
-\subsubsection{Filtered formulation (\key{dynspg\_flt})}
-\label{DYN_spg_flt}
-
-The filtered formulation follows the \citet{Roullet2000} implementation. The extra 
+%--------------------------------------------------------------------------------------------------------------
+% Filtered free surface formulation
+%--------------------------------------------------------------------------------------------------------------
+\subsection{Filtered free surface (\key{dynspg\_flt})}
+\label{DYN_spg_fltp}
+
+
+The filtered formulation follows the \citet{Roullet_Madec_JGR00} 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 !}
-
-%-------------------------------------------------------------
-% Non-linear free surface formulation 
-%-------------------------------------------------------------
-\subsection{Non-linear free surface formulation (\key{vvl})}
-\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 the filtered case, the matrix to be inverted has to be recomputed at each 
-time-step.
-
-%--------------------------------------------------------------------------------------------------------------
-%           Rigid-lid formulation
-%--------------------------------------------------------------------------------------------------------------
-\subsection{Rigid-lid formulation (\key{dynspg\_rl})}
-\label{DYN_spg_rl}
-
-With the rigid lid formulation, an elliptic equation has to be solved for 
-the barotropic streamfunction. For consistency this equation is obtained by 
-taking the discrete curl of the discrete vertical sum of the discrete 
-momentum equation: 
-\begin{equation}\label{Eq_dynspg_rl}
-\frac{1}{\rho _o }\nabla _h p_s \equiv \left( {{\begin{array}{*{20}c}
- {\overline M_u +\frac{1}{H\;e_2 } \delta_ j \left[ \partial_t \psi \right]}		 \\
- \\
- {\overline M_v -\frac{1}{H\;e_1 }  \delta_i  \left[ \partial_t \psi \right]} 		 \\
-\end{array} }} \right)
-\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:
-\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]
-+ \delta_{j+1/2} \left[ \frac{e_{1u}}{H_u\;e_{2u}} \delta_{j} \left[  \partial_t \psi \right] \right]
-\\ =
-  \delta_{i+1/2} \left[ e_{2v} M_v  \right]
-- \delta_{j+1/2} \left[ e_{1u} M_u  \right]
-\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 \hf{par\_oce} file are 
-related to this option.
+solvers available in the code are documented in \S\ref{MISC}.
 
 
@@ -815,5 +820,5 @@
 \label{DYN_ldf}
 %------------------------------------------nam_dynldf----------------------------------------------------
-\namdisplay{nam_dynldf} 
+\namdisplay{namdyn_ldf} 
 %-------------------------------------------------------------------------------------------------------------
 
@@ -821,6 +826,6 @@
 (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 
+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 
@@ -850,7 +855,7 @@
 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 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}.
+separation between the vorticity and divergence parts of the momentum diffusion. 
+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}.
 
 %--------------------------------------------------------------------------------------------------------------
@@ -875,6 +880,6 @@
  D_u^{l\textbf{U}} &= \frac{1}{e_{1u} \, e_{2u} \, e_{3u} }	\\
 &  \left\{\quad  {\delta _{i+1/2} \left[ {A_T^{lm}  \left( 
-	 {\frac{e_{2T} \; e_{3T} }{e_{1T} } \,\delta _{i}[u]
-	-e_{2T} \; r_{1T} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
+	 {\frac{e_{2t} \; e_{3t} }{e_{1t} } \,\delta _{i}[u]
+	-e_{2t} \; r_{1t} \,\overline{\overline {\delta _{k+1/2}[u]}}^{\,i,\,k}}
  \right)} \right]} 	\right.
 \\ 
@@ -902,6 +907,6 @@
  \right)} \right]} 	\right.
 \\ 
-& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1T}\,e_{3T} }{e_{2T} 
-}\,\delta _{j} [v] - e_{1T}\, r_{2T} 
+& \qquad +\ \delta_j \left[ {A_T^{lm} \left( {\frac{e_{1t}\,e_{3t} }{e_{2t} 
+}\,\delta _{j} [v] - e_{1t}\, r_{2t} 
 \,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
 \right)} \right] 
@@ -954,7 +959,7 @@
 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{nn\_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. 
+\np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both tracers and dynamics. 
 
 The formulation of the vertical subgrid scale physics is the same whatever 
@@ -1021,94 +1026,34 @@
 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}
-&u^{t+\Delta t} = u^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_u^t  	\\
-\\
+(cf. \S\ref{DOM_nxt}). The scheme is applied to the velocity except when using 
+the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in variable 
+volume level case (\key{vvl} defined), where it has to be applied to the thickness 
+weighted velocity (see \S\ref{Apdx_A_momentum})  
+
+$\bullet$ vector invariant form or linear free surface (\np{ln\_dynhpg\_vec}=.true. ; not \key{vvl}):
+\begin{equation} \label{Eq_dynnxt_vec}
+\left\{   \begin{aligned}
+&u^{t+\Delta t} = u_f^{t-\Delta t} + 2\Delta t  \ \text{RHS}_u^t  	\\
 &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\Delta t} -2u^t+u^{t+\Delta t}} \right]
-\end{split}
+\end{aligned}   \right.
+\end{equation} 
+
+$\bullet$ flux form and nonlinear free surface (\np{ln\_dynhpg\_vec}=.false. ; \key{vvl} defined):
+\begin{equation} \label{Eq_dynnxt_flux}
+\left\{   \begin{aligned}
+&\left(e_{3u}\,u\right)^{t+\Delta t} = \left(e_{3u}\,u\right)_f^{t-\Delta t} + 2\Delta t \; e_{3u} \;\text{RHS}_u^t  	\\
+&\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
+  +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\Delta t} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\Delta t}} \right]
+\end{aligned}   \right.
 \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 \textit{next} velocities 
+initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = 0.1.
+
+Note that with 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.}
-
-% ================================================================
-% Diagnostic variables
-% ================================================================
-\section{Diagnostic variables ($\zeta$, $\chi$, $w$)}
-\label{DYN_divcur_wzv}
-
-%--------------------------------------------------------------------------------------------------------------
-%           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 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 a $T$-point. It is given by:
-\begin{equation} \label{Eq_divcur_div}
-\chi =\frac{1}{e_{1T}\,e_{2T}\,e_{3T} }
-		\left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u} \right]
-			  +\delta _j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
-\end{equation} 
-
-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 (\textit{wzvmod})]
-			{Vertical velocity (\mdl{wzvmod})}
-\label{DYN_wzv}
-
-The vertical velocity is computed by an upward integration of the horizontal 
-divergence from the bottom :
-
-\begin{equation} \label{Eq_wzv}
-\left\{   \begin{aligned}
-&\left. w \right|_{3/2} \quad= 0   	\\
-\\
-&\left. w \right|_{k+1/2}     = \left. w \right|_{k+1/2}  + e_{3t}\;  \left. \chi \right|_k  
-\end{aligned}   \right.
-\end{equation} 
-
-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. 
-
-With the variable volume option, the calculation of the vertical velocity is 
-modified (see \citet{Levier2007}, report available on the \NEMO web site). 
-
-% ================================================================
+and Asselin filtering is done in \mdl{dynnxt}.
+
+
+
+% ================================================================
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LBC.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 1831)
@@ -18,5 +18,5 @@
 \label{LBC_coast}
 %--------------------------------------------nam_lbc-------------------------------------------------------
-\namdisplay{nam_lbc} 
+\namdisplay{namlbc} 
 %--------------------------------------------------------------------------------------------------------------
 
@@ -134,7 +134,7 @@
 spectacular improvements have not been found in the half-degree global ocean 
 (ORCA05), but significant reductions of numerically induced coastal upwellings were 
-found in an eddy resolving simulation of the Alboran Sea \citep{OlivierPh2001}. 
+found in an eddy resolving simulation of the Alboran Sea \citep{Olivier_PhD01}. 
 Nevertheless, since a no-slip boundary condition is not recommended in an eddy 
-permitting or resolving simulation \citep{Penduff2007}, the use of this option is also 
+permitting or resolving simulation \citep{Penduff_al_OS07}, the use of this option is also 
 not recommended.
 
@@ -428,5 +428,5 @@
 of the domain, for example at Gibraltar Straits if one wants to avoid including 
 the Mediterranean in an Atlantic domain. This flexibility has been found necessary 
-for the CLIPPER project \citep{Treguier2001}. Because of the complexity of the 
+for the CLIPPER project \citep{Treguier_al_JGR01}. Because of the complexity of the 
 geometry of ocean basins, it may even be necessary to have more than one 
 ''west'' open boundary, more than one ''north'', etc. This is not possible with 
@@ -529,5 +529,5 @@
 
 When ocean observations are used to generate the boundary data (a hydrographic 
-section for example, as in \citet{Treguier2001}) it happens often that only the velocity 
+section for example, as in \citet{Treguier_al_JGR01}) it happens often that only the velocity 
 normal to the boundary is known, which is the reason why the initial OBC code 
 assumes that only $T$, $S$, and the normal velocity ($u$ or $v$) needs to be 
@@ -548,5 +548,5 @@
 remain in NEMO v2.3. Users should read the code carefully before using it. Finally, 
 in the case of the rigid lid approximation the barotropic streamfunction must be 
-provided, as documented in \citet{Treguier2001}). This option is no longer 
+provided, as documented in \citet{Treguier_al_JGR01}). This option is no longer 
 recommended but remains in NEMO V2.3.
 
@@ -623,5 +623,5 @@
 $s$-coordinate model on an Arakawa C-grid. Although the algorithm has 
 been numerically successful in the CLIPPER Atlantic models, the physics 
-do not work as expected \citep{Treguier2001}. Users are invited to consider 
+do not work as expected \citep{Treguier_al_JGR01}. Users are invited to consider 
 open boundary conditions (OBC hereafter) with some scepticism 
 \citep{Durran2001, Blayo2005}.
@@ -637,5 +637,5 @@
 C_{\phi y} = \frac{ -\phi_{t} }{ ( \phi_{x}^{2} + \phi_{y}^{2}) } \phi_{y}. 
 \end{equation}
-Following \citet{Treguier2001} and \citet{Marchesiello2001} we retain only 
+Following \citet{Treguier_al_JGR01} and \citet{Marchesiello2001} we retain only 
 the normal component of the velocity, $C_{\phi x}$, setting $C_{\phi y} =0$ 
 (but unlike the original Orlanski radiation algorithm we retain $\phi_{y}$ in 
@@ -672,8 +672,8 @@
 numerical stability. 
 
-In  the case of a western boundary located in the Eastern Atlantic, \citet{Penduff2000} 
+In  the case of a western boundary located in the Eastern Atlantic, \citet{Penduff_al_JGR00} 
 have been able to implement the radiation algorithm without any boundary data, 
 using persistence from the previous time step instead. This solution has not worked 
-in other cases \citep{Treguier2001}, so that the use of boundary data is recommended. 
+in other cases \citep{Treguier_al_JGR01}, so that the use of boundary data is recommended. 
 Even in the outflow condition (\ref{Eq_obc_rado}), we have found it desirable to 
 maintain a weak relaxation to climatology. The time step is usually chosen so as to 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LDF.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 1831)
@@ -7,5 +7,8 @@
 \minitoc
 
+
+\newpage
 $\ $\newline    % force a new ligne
+
 
 The lateral physics terms in the momentum and tracer equations have been 
@@ -21,6 +24,6 @@
 
 %-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
-\namdisplay{nam_traldf} 
-\namdisplay{nam_dynldf} 
+\namdisplay{namtra_ldf} 
+\namdisplay{namdyn_ldf} 
 %--------------------------------------------------------------------------------------------------------------
 
@@ -89,5 +92,5 @@
 namelist parameter. 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}. 
+the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 
 %%%
 \gmcomment { not only that! stability reasons: with non uniform grid size, it is common 
@@ -99,5 +102,5 @@
 viscosity operator uses \np{ahm0}~=~$4.10^4 m^2/s$ poleward of 20$^{\circ}$ 
 north and south and decreases linearly to \np{aht0}~=~$2.10^3 m^2/s$ 
-at the equator \citep{Madec1996, Delecluse_Madec_Bk00}. This modification 
+at the equator \citep{Madec_al_JPO96, Delecluse_Madec_Bk00}. This modification 
 can be found in routine \rou{ldf\_dyn\_c2d\_orca} defined in \mdl{ldfdyn\_c2d}. 
 Similar modified horizontal variations can be found with the Antarctic or Arctic 
@@ -158,5 +161,5 @@
 %%%
 \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. }
+Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
 
 A direction for lateral mixing has to be defined when the desired operator does 
@@ -168,6 +171,6 @@
 quantity to be diffused. For a tracer, this leads to the following four slopes : 
 $r_{1u}$, $r_{1w}$, $r_{2v}$, $r_{2w}$ (see \eqref{Eq_tra_ldf_iso}), while 
-for momentum the slopes are  $r_{1T}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 
-$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2T}$, $r_{2vw}$ for $v$. 
+for momentum the slopes are  $r_{1t}$, $r_{1uw}$, $r_{2f}$, $r_{2uw}$ for 
+$u$ and  $r_{1f}$, $r_{1vw}$, $r_{2t}$, $r_{2vw}$ for $v$. 
 
 %gm% add here afigure of the slope in i-direction
@@ -186,15 +189,15 @@
 \begin{aligned}
  r_{1u} &= \frac{e_{3u}}{ \left( e_{1u}\;\overline{\overline{e_{3w}}}^{\,i+1/2,\,k} \right)}
- 			  \;\delta_{i+1/2}[z_T] 
- 		&\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_T] 
+ 			  \;\delta_{i+1/2}[z_t] 
+ 		&\approx \frac{1}{e_{1u}}\; \delta_{i+1/2}[z_t] 
 \\
  r_{2v} &= \frac{e_{3v}}{\left( e_{2v}\;\overline{\overline{e_{3w}}}^{\,j+1/2,\,k} \right)} 
- 			  \;\delta_{j+1/2} [z_T] 
-		&\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_T] 
-\\
- r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_T]}}^{\,i,\,k+1/2}
+ 			  \;\delta_{j+1/2} [z_t] 
+		&\approx \frac{1}{e_{2v}}\; \delta_{j+1/2}[z_t] 
+\\
+ r_{1w} &= \frac{1}{e_{1w}}\;\overline{\overline{\delta_{i+1/2}[z_t]}}^{\,i,\,k+1/2}
  		&\approx \frac{1}{e_{1w}}\; \delta_{i+1/2}[z_{uw}] 
  \\
- r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_T]}}^{\,j,\,k+1/2}
+ r_{2w} &= \frac{1}{e_{2w}}\;\overline{\overline{\delta_{j+1/2}[z_t]}}^{\,j,\,k+1/2}
 		&\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 
  \\
@@ -272,5 +275,5 @@
 Note: The solution for $s$-coordinate passes trough the use of different 
 (and better) expression for the constraint on iso-neutral fluxes. Following 
-\citet{Griffies2004}, instead of specifying directly that there is a zero neutral 
+\citet{Griffies_Bk04}, instead of specifying directly that there is a zero neutral 
 diffusive flux of locally referenced potential density, we stay in the $T$-$S$ 
 plane and consider the balance between the neutral direction diffusive fluxes 
@@ -324,6 +327,6 @@
 a minimum background horizontal diffusion for numerical stability reasons. 
 To overcome this problem, several techniques have been proposed in which 
-the numerical schemes of the ocean model are modified \citep{Weaver1997, 
-Griffies1998}. Here, another strategy has been chosen \citep{Lazar1997}: 
+the numerical schemes of the ocean model are modified \citep{Weaver_Eby_JPO97, 
+Griffies_al_JPO98}. Here, another strategy has been chosen \citep{Lazar_PhD97}: 
 a local filtering of the iso-neutral slopes (made on 9 grid-points) prevents 
 the development of grid point noise generated by the iso-neutral diffusion 
@@ -337,5 +340,5 @@
 
 Nevertheless, this iso-neutral operator does not ensure that variance cannot increase, 
-contrary to the \citet{Griffies1998} operator which has that property. 
+contrary to the \citet{Griffies_al_JPO98} operator which has that property. 
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -355,10 +358,10 @@
 
 
-In addition and also for numerical stability reasons \citep{Cox1987, Griffies2004}, 
+In addition and also for numerical stability reasons \citep{Cox1987, Griffies_Bk04}, 
 the slopes are bounded by $1/100$ everywhere. This limit is decreasing linearly 
 to zero fom $70$ meters depth and the surface (the fact that the eddies "feel" the 
 surface motivates this flattening of isopycnals near the surface).
 
-For numerical stability reasons \citep{Cox1987, Griffies2004}, the slopes must also 
+For numerical stability reasons \citep{Cox1987, Griffies_Bk04}, the slopes must also 
 be bounded by $1/100$ everywhere. This constraint is applied in a piecewise linear 
 fashion, increasing from zero at the surface to $1/100$ at $70$ metres and thereafter 
@@ -398,6 +401,6 @@
 \begin{equation} \label{Eq_ldfslp_dyn}
 \begin{aligned}
-&r_{1T}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
-&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& 	r_{2T}\ &= \overline{r_{2v}}^{\,j} \\
+&r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
+&r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& 	r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
 &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
 &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_MISC.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 1831)
@@ -2,8 +2,11 @@
 % Chapter Ñ Miscellaneous Topics
 % ================================================================
-\chapter{Miscellaneous Topics (xxx)}
+\chapter{Miscellaneous Topics}
 \label{MISC}
 \minitoc
 
+\newpage
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Representation of Unresolved Straits
@@ -12,10 +15,24 @@
 \label{MISC_strait}
 
-% In climate modeling, it is often necessary to allow water masses that are separated by land to exchange properties. This situation arises in models when the grid mesh is too coarse to resolve narrow passageways that in reality provide crucial connections between water masses. For example, coarse grid spacing typically closes off the Mediterranean from the Atlantic at the Straits of Gibraltar. In this case, it is important for climate models to include the effects of salty water entering the Atlantic from the Mediterranean. Likewise, it is important for the Mediterranean to replenish its supply of water from the Atlantic to balance the net evaporation occurring over the Mediterranean region. 
-% Same problem occurs as soon as important straits are not properly resolved by the mesh. For example it arises in ORCA 1/4\r{} in several straits of the Indonesian archipelago (Ombai, Lombok...).
-% We describe here the three methods that can be used in \NEMO to handle such improperly resolved straits. 
-%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets.  
-
-% Note: such changes are so specific to a given configuration that no attempt has been made to set them in a generic way. Nevertheless, an example of how they can be set up is given in the ORCA2 configuration (\key{ORCA_R2}).
+In climate modeling, it often occurs that a crucial connections between water masses
+is broken as the grid mesh is too coarse to resolve narrow straits. For example, coarse 
+grid spacing typically closes off the Mediterranean from the Atlantic at the Strait of 
+Gibraltar. In this case, it is important for climate models to include the effects of salty 
+water entering the Atlantic from the Mediterranean. Likewise, it is important for the 
+Mediterranean to replenish its supply of water from the Atlantic to balance the net 
+evaporation occurring over the Mediterranean region. This problem occurs even in 
+eddy permitting simulations. For example, in ORCA 1/4\r{} several straits of the Indonesian 
+archipelago (Ombai, Lombok...) are much narrow than even a single ocean grid-point.
+
+We describe briefly here the three methods that can be used in \NEMO to handle 
+such improperly resolved straits. The first two consist of opening the strait by hand 
+while ensuring that the mass exchanges through the strait are not too large by 
+either artificially reducing the surface of the strait grid-cells or, locally increasing 
+the lateral friction. In the third one, the strait is closed but exchanges of mass, 
+heat and salt across the land are allowed.
+Note that such modifications are so specific to a given configuration that no attempt 
+has been made to set them in a generic way. However, examples of how 
+they can be set up is given in the ORCA 2\r{} and 0.5\r{} configurations (search for 
+\key{ORCA\_R2} or \key{ORCA\_R05} in the code).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -25,11 +42,18 @@
 \label{MISC_strait_hand}
 
-$\bullet$ reduced scale factor, also called partially open face (Hallberg, personnal communication 2006)
-%also called partially open cell  or partial closed cells
-
-$\bullet$ increase of the viscous boundary layer thickness by local increase of the fmask value at the coast
-
-\colorbox{yellow}{Add a short description of scale factor changes staff and fmask increase}
-
+$\bullet$ reduced scale factor in the cross-strait direction to a value in better agreement 
+with the true mean width of the strait. (Fig.~\ref{Fig_MISC_strait_hand}).
+This technique is sometime called "partially open face" or "partially closed cells".
+The key issue here is only to reduce the faces of $T$-cell ($i.e.$ change the value 
+of the horizontal scale factors at $u$- or $v$-point) but not the volume of the $T$-cell.
+Indeed, reducing the volume of strait $T$-cell can easily produce a numerical 
+instability at that grid point that would require a reduction of the model time step.
+The changes associated with strait management are done in \mdl{domhgr}, 
+just after the definition or reading of the horizontal scale factors. 
+
+$\bullet$ increase of the viscous boundary layer thickness by local increase of the 
+fmask value at the coast (Fig.~\ref{Fig_MISC_strait_hand}). This is done in 
+\mdl{dommsk} together with the setting of the coastal value of fmask 
+(see Section \ref{LBC_coast})
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -51,7 +75,6 @@
 % Cross Land Advection 
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Cross Land Advection (\textit{tracla} module)}
+\subsection{Cross Land Advection (\mdl{tracla})}
 \label{MISC_strait_cla}
-
 %--------------------------------------------namcla--------------------------------------------------------
 \namdisplay{namcla} 
@@ -60,9 +83,13 @@
 \colorbox{yellow}{Add a short description of CLA staff here or in lateral boundary condition chapter?}
 
+%The problem is resolved here by allowing the mixing of tracers and mass/volume between non-adjacent water columns at nominated regions within the model. Momentum is not mixed. The scheme conserves total tracer content, and total volume (the latter in $z*$- or $s*$-coordinate), and maintains compatibility between the tracer and mass/volume budgets.  
+
 % ================================================================
 % Closed seas
 % ================================================================
-\section{Closed seas}
+\section{Closed seas (\mdl{closea})}
 \label{MISC_closea}
+
+\colorbox{yellow}{Add here a short description of the way closed seas are managed}
 
 
@@ -81,5 +108,5 @@
 model configuration while having a small computer memory requirement. 
 It can also be used to easily test specific physics in a sub-domain (for example, 
-see \citep{Madec1996} for a test of the coupling used in the global ocean 
+see \citep{Madec_al_JPO96} for a test of the coupling used in the global ocean 
 version of OPA between sea-ice and ocean model over the Arctic or Antarctic 
 ocean, using a sub-domain). In the standard model, this option does not 
@@ -120,19 +147,21 @@
 \label{MISC_1D}
 
-The 1D model option simulates a stand alone water column within the 3D \NEMO 
-system. It can be applied to the ocean alone or to the ocean-ice system and can 
-include passive tracers or a biogeochemical model. It is set up by defining the 
-\key{cfg\_1d} CPP key. This 1D model is a very useful tool  \textit{(a)} to learn 
-about the physics and numerical treatment of vertical mixing processes ; \textit{(b)} 
-to investigate suitable parameterisations of unresolved turbulence (wind steering, 
-langmuir circulation, skin layers) ; \textit{(c)} to compare the behaviour of different 
-vertical mixing schemes  ; \textit{(d)} to perform sensitivity studies on the vertical 
-diffusion at a particular point of the ocean global domain ; \textit{(d)} to produce 
-extra diagnostics, without the large memory requirement of the full 3D model.
-
-The methodology is based on the use of the zoom functionality (see 
-\S\ref{MISC_zoom}), with some extra routines. There is no need to define a new 
-mesh, bathymetry, initial state or forcing, since the 1D model will use those of the 
-configuration it is a zoom of. 
+The 1D model option simulates a stand alone water column within the 3D \NEMO system. 
+It can be applied to the ocean alone or to the ocean-ice system and can include passive tracers 
+or a biogeochemical model. It is set up by defining the \key{cfg\_1d} CPP key. 
+The 1D model is a very useful tool  
+\textit{(a)} to learn about the physics and numerical treatment of vertical mixing processes ; 
+\textit{(b)} to investigate suitable parameterisations of unresolved turbulence (wind steering, 
+langmuir circulation, skin layers) ; 
+\textit{(c)} to compare the behaviour of different vertical mixing schemes  ; 
+\textit{(d)} to perform sensitivity studies on the vertical diffusion at a particular point of an ocean domain ; 
+\textit{(d)} to produce extra diagnostics, without the large memory requirement of the full 3D model.
+
+The methodology is based on the use of the zoom functionality over the smallest possible 
+domain : a 3 x 3 domain centred on the grid point of interest (see \S\ref{MISC_zoom}), 
+with some extra routines. There is no need to define a new mesh, bathymetry, 
+initial state or forcing, since the 1D model will use those of the configuration it is a zoom of. 
+The chosen grid point is set in par\_oce.F90 module by setting the jpizoom and jpjzoom 
+parameters to the indices of the location of the chosen grid point.
 
 % ================================================================
@@ -145,21 +174,21 @@
 %--------------------------------------------------------------------------------------------------------------
 
-% Steven update just bellow...   not sure it  is what I vant to say
-%Searching an equilibrium state with an ocean model requires repeated very long time 
-%integrations (each of a few thousand years for a global model). Due to the size of 
-Searching an equilibrium state with an ocean model requires very long time 
-integration (a few thousand years for a global model). Due to the size of 
-the time step required for numerical stability (less than a 
-few hours), this usually requires a large elapsed time. In order to overcome 
-this problem, \citet{Bryan1984} introduces a technique that is intended to accelerate 
+Searching an equilibrium state with an global ocean model requires a very long time 
+integration period (a few thousand years for a global model). Due to the size of 
+the time step required for numerical stability (less than a few hours), 
+this usually requires a large elapsed time. In order to overcome this problem, 
+\citet{Bryan1984} introduces a technique that is intended to accelerate 
 the spin up to equilibrium. It uses a larger time step in 
-the thermodynamic evolution equations than in the dynamic evolution 
+the tracer evolution equations than in the momentum evolution 
 equations. It does not affect the equilibrium solution but modifies the 
 trajectory to reach it.
 
 The acceleration of convergence option is used when \np{nn\_acc}=1. In that case, 
-$\Delta t=rdt$ is the time step of dynamics while $\widetilde{\Delta t}=rdttra$ is the 
-tracer time-step. Their values are set from the \np{rdt} and \np{rdttra}  namelist 
-parameters. The set of prognostic equations to solve becomes:
+$\Delta t=rn\_rdt$ is the time step of dynamics while $\widetilde{\Delta t}=rdttra$ is the 
+tracer time-step. the former is set from the \np{rn\_rdt} namelist parameter while the latter
+is computed using a hyperbolic tangent profile and the following namelist parameters : 
+\np{rn\_rdtmin}, \np{rn\_rdtmax} and \np{rn\_rdth}. Those three parameters correspond 
+to the surface value the deep ocean value and the depth at which the transition occurs, respectively. 
+The set of prognostic equations to solve becomes:
 \begin{equation} \label{Eq_acc}
 \begin{split}
@@ -183,7 +212,6 @@
 $\widetilde{ \Delta t}$ implies that the heat and salt contents are no longer 
 conserved due to the vertical coupling of the ocean level through both 
-advection and diffusion.
-% first mention of depth varying tilda-delta-t!   and namelist parameter associated to that are to be described
-
+advection and diffusion. Therefore \np{rn\_rdtmin} = \np{rn\_rdtmax} should be
+a more clever choice.
 
 % ================================================================
@@ -202,6 +230,7 @@
 $\bullet$ Vector and memory optimisation:
 
-\key{vectopt\_loop} enables internal loop collapse, a very efficient way to increase 
-the length of vector calculations and thus speed up the model on vector computers.
+\key{vectopt\_loop} enables the internal loops to collapse. This is very 
+a very efficient way to increase the length of vector calculations and thus 
+to speed up the model on vector computers.
  
 % Add here also one word on NPROMA technique that has been found useless, since compiler have made significant progress during the last decade.
@@ -209,12 +238,13 @@
 % Add also one word on NEC specific optimisation (Novercheck option for example)
 
-\key{vectopt\_memory} has been introduced in order to reduce the memory 
-requirement of the model. This is obviously done at the cost of increasing the 
-CPU time requirement, since it suppress intermediate computations which would 
-have been saved in in-core memory. This feature has not been intensively used. 
-In fact, currently it is only implemented for the TKE physics, in which,  when 
-\key{vectopt\_memory} is defined, the coefficients used for horizontal smoothing 
-of $A_v^T$ and $A_v^m$ are no longer computed once and for all. This reduces 
-the memory requirement by three 2D arrays.
+\key{vectopt\_memory} is an obsolescent option. It has been introduced in order 
+to reduce the memory requirement of the model at a time when in-core memory
+were rather limited. This is obviously done at the cost of increasing the CPU 
+time requirement, since it suppress intermediate computations which would have 
+been saved in in-core memory. Currently it is only used in the old implementation 
+of the TKE physics (\key{tke\_old}) where,  when \key{vectopt\_memory} 
+is defined, the coefficients used for horizontal smoothing of $A_v^T$ and $A_v^m$ 
+are no longer computed once and for all. This reduces the memory requirement by three 
+3D arrays. This option will disappear in the next \NEMO release.
 
  
@@ -231,18 +261,10 @@
 management. When defined, this key forces the activation of all options and 
 CPP keys. For example, all tracer and momentum advection schemes are called! 
-There is therefore no physical meaning associated with the model results. 
+Therefore the model results have no physical meaning. 
 However, this option forces both the compiler and the model to run through 
 all the \textsc{Fortran} lines of the model. This allows the user to check for obvious 
 compilation or execution errors with all CPP options, and errors in namelist options.
 
-3- \key{esopa} (to be rename key\_nemo) : which is another option for model 
-management. When defined, this key forces the activation of all options and CPP keys. 
-For example, all tracer and momentum advection schemes are called! There is 
-therefore no physical meaning associated with the model results. However, this option 
-forces both the compiler and the model to run through all the Fortran lines of the model. 
-This allows the user to check for obvious compilation or execution errors with all CPP 
-options, and errors in namelist options.
-
-4- last digit comparison (\np{nbit\_cmp}). In an MPP simulation, the computation of 
+4- last digit comparison (\np{nn\_bit\_cmp}). In an MPP simulation, the computation of 
 a sum over the whole domain is performed as the summation over all processors of 
 each of their sums over their interior domains. This double sum never gives exactly 
@@ -251,5 +273,5 @@
 mono-processor and multi-processor runs give exactly the same results. 
 
-$\bullet$  Benchmark (\np{nbench}). This option defines a benchmark run based on 
+$\bullet$  Benchmark (\np{nn\_bench}). This option defines a benchmark run based on 
 a GYRE configuration in which the resolution remains the same whatever the domain 
 size. This allows a very large model domain to be used, just by changing the domain 
@@ -261,5 +283,5 @@
 % Elliptic solvers (SOL)
 % ================================================================
-\section{Elliptic solvers (SOL directory)}
+\section{Elliptic solvers (SOL)}
 \label{MISC_sol}
 %--------------------------------------------namdom-------------------------------------------------------
@@ -267,11 +289,11 @@
 %--------------------------------------------------------------------------------------------------------------
 
-The computation of the surface pressure gradient with a rigid lid assumption 
-requires the computation of $\partial_t \psi$, the time evolution of the 
-barotropic streamfunction. $\partial_t \psi$ is the solution of an elliptic 
-equation \eqref{Eq_PE_psi} for which three solvers are available: a 
-Successive-Over-Relaxation scheme (SOR), a preconditioned conjugate gradient 
-scheme(PCG) \citep{Madec1988, Madec1990} and a Finite Elements Tearing and 
-Interconnecting scheme (FETI) \citep{Guyon_al_EP99, Guyon_al_CalPar99}. 
+When the filtered sea surface height option is used, the surface pressure gradient is 
+computed in \mdl{dynspg\_flt}. The force added in the momentum equation is solved implicitely.
+It is thus solution of an elliptic equation \eqref{Eq_PE_flt} for which two solvers are available: 
+a Successive-Over-Relaxation scheme (SOR) and a preconditioned conjugate gradient 
+scheme(PCG) \citep{Madec_al_OM88, Madec_PhD90}. The solver is selected trough the
+the value of \np{nn\_solv} (namelist parameter). 
+
 The PCG is a very efficient method for solving elliptic equations on vector computers. 
 It is a fast and rather easy method to use; which are attractive features for a large 
@@ -280,31 +302,26 @@
 a search for an optimal parameter as in the SOR method. However, the SOR has 
 been retained because it is a linear solver, which is a very useful property when 
-using the adjoint model of \NEMO. The FETI solver is a very efficient method on 
-massively parallel computers. However, it has never been used since OPA 8.0. 
-The current version in \NEMO should not even successfully go through the 
-compilation phase. 
-
-The surface pressure gradient is computed in \mdl{dynspg}. The solver used 
-(PCG or SOR) depending on the value of \np{nsolv} (namelist parameter). 
-At each time step the time derivative of the barotropic streamfunction is 
-the solution of \eqref{Eq_psi_total}. Introducing the following coefficients:
-
-\begin{equation} 
+using the adjoint model of \NEMO.
+
+At each time step, the time derivative of the sea surface height at time step $t+1$ 
+(or equivalently the divergence of the \textit{after} barotropic transport) that appears 
+in the filtering forced is the solution of the elliptic equation obtained from the horizontal 
+divergence of the vertical summation of \eqref{Eq_PE_flt}. 
+Introducing the following coefficients:
+\begin{equation}  \label{Eq_sol_matrix}
 \begin{aligned}
-&C_{i,j}^{NS} &&= \frac{e_{2v}(i,j)}{\left( H_v (i,j) e_{1v} (i,j)\right)}\\
-&C_{i,j}^{EW} &&= \frac{e_{1u}(i,j)}{\left( H_u (i,j) e_{2u} (i,j)\right)}\\
-&B_{i,j} &&= \delta_i \left( e_{2v}M_v \right) - \delta_j \left( e_{1u}M_u \right)\\
+&c_{i,j}^{NS}  &&= {2 \Delta t }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)}              \\
+&c_{i,j}^{EW} &&= {2 \Delta t }^2 \; \frac{H_u (i,j) \; e_{2u} (i,j)}{e_{1u}(i,j)}            \\
+&b_{i,j} &&= \delta_i \left[ e_{2u}M_u \right] - \delta_j \left[ e_{1v}M_v \right]\ ,   \\
 \end{aligned}
 \end{equation}
-
 the five-point finite difference equation \eqref{Eq_psi_total} can be rewritten as:
-
-\begin{multline} \label{Eq_solmat}
-C_{i+1,j}^{NS} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j} + \;
-C_{i,j+1}^{EW}{   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} + \;
-C_{i,j}    ^{NS} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} + \;
-C_{i,j}    ^{EW}{   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1}\\
--\left(C_{i+1,j}^{NS} + C_{i,j+1}^{EW} + C_{i,j}^{NS} + C_{i,j}^{EW} \right)   {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}  =  B_{i,j}
-\end{multline}
+\begin{equation}  \label{Eq_solmat}
+\begin{split}
+       c_{i+1,j}^{NS} D_{i+1,j}  + \;  c_{i,j+1}^{EW} D_{i,j+1}   
+  +   c_{i,j}    ^{NS} D_{i-1,j}   + \;  c_{i,j}    ^{EW} D_{i,j-1}                                          &    \\
+  -    \left(c_{i+1,j}^{NS} + c_{i,j+1}^{EW} + c_{i,j}^{NS} + c_{i,j}^{EW} \right)   D_{i,j}  &=  b_{i,j}
+\end{split}
+\end{equation}
 \eqref{Eq_solmat} is a linear symmetric system of equations. All the elements of 
 the corresponding matrix \textbf{A} vanish except those of five diagonals. With 
@@ -315,43 +332,42 @@
 \eqref{Eq_solmat}, is a vector.
 
+Note that in the linear free surface case, the depth that appears in \eqref{Eq_sol_matrix}
+does not vary with time, and thus the matrix can be computed once for all. In non-linear free surface 
+(\key{vvl} defined) the matrix have to be updated at each time step.
+
 % -------------------------------------------------------------------------------------------------------------
 %       Successive Over Relaxation
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Successive Over Relaxation \np{nsolv}=2}
+\subsection{Successive Over Relaxation (\np{nn\_solv}=2, \mdl{solsor})}
 \label{MISC_solsor}
 
-Let us introduce the four cardinal coefficients: $A_{i,j}^S = C_{i,j}^{NS}/D_{i,j}\,$, 
-$A_{i,j}^W = C_{i,j}^{EW}/D_{i,j}\,$, $A_{i,j}^E = C_{i,j+1}^{EW}/D_{i,j}\,$ and $A_{i,j}^N = C_{i+1,j}^{NS}/D_{i,j}\,$, and define 
-$\tilde B_{i,j} = B_{i,j}/D_{i,j}\,$, where $D_{i,j} = C_{i,j}^{NS}+ C_{i+1,j}^{NS} + C_{i,j}^{EW} + C_{i,j+1}^{EW} $ (i.e. the diagonal of 
-\textbf{A}). (VII.5.1) can be rewritten as:
-
-\begin{multline} \label{Eq_solmat_p}
-    A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j   } 
-+\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i    ,j+1} 		\\
-+\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1 ,j    } 
-+\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i    ,j-1} 	
-               -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} = \tilde B_{i,j}
-\end{multline}
-
-The SOR method used is an iterative method. Its algorithm can be summarised 
-as follows [see \citet{Haltiner1980} for a further discussion]:
+Let us introduce the four cardinal coefficients: 
+\begin{align*}
+a_{i,j}^S &= c_{i,j    }^{NS}/d_{i,j}     &\qquad  a_{i,j}^W &= c_{i,j}^{EW}/d_{i,j}       \\
+a_{i,j}^E &= c_{i,j+1}^{EW}/d_{i,j}    &\qquad   a_{i,j}^N &= c_{i+1,j}^{NS}/d_{i,j}   
+\end{align*}
+where $d_{i,j} = c_{i,j}^{NS}+ c_{i+1,j}^{NS} + c_{i,j}^{EW} + c_{i,j+1}^{EW}$ 
+(i.e. the diagonal of the matrix). \eqref{Eq_solmat} can be rewritten as:
+\begin{equation}  \label{Eq_solmat_p}
+\begin{split}
+a_{i,j}^{N}  D_{i+1,j} +\,a_{i,j}^{E}  D_{i,j+1} +\, a_{i,j}^{S}  D_{i-1,j} +\,a_{i,j}^{W} D_{i,j-1}  -  D_{i,j} = \tilde{b}_{i,j}
+\end{split}
+\end{equation}
+with $\tilde b_{i,j} = b_{i,j}/d_{i,j}$. \eqref{Eq_solmat_p} is the equation actually solved 
+with the SOR method. This method used is an iterative one. Its algorithm can be 
+summarised as follows (see \citet{Haltiner1980} for a further discussion):
 
 initialisation (evaluate a first guess from previous time step computations)
 \begin{equation}
-\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^0 = 2{\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^t} - {\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^{t-1}}
+D_{i,j}^0 = 2 \, D_{i,j}^t - D_{i,j}^{t-1}
 \end{equation}
 iteration $n$, from $n=0$ until convergence, do :
-\begin{multline} 
-R_{i,j}^n
-= A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j}^n 
-+\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} ^n		\\
-+\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} ^{n+1}
-+\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1} ^{n+1}	
-                   -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}^n - \tilde B_{i,j}
-\end{multline}
-\begin{equation} 
-     {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} ^{n+1}
-  = {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j} ^{n}
-  + \omega \;R_{i,j}^n
+\begin{equation} \label{Eq_sor_algo}
+\begin{split}
+R_{i,j}^n  = &a_{i,j}^{N} D_{i+1,j}^n       +\,a_{i,j}^{E}  D_{i,j+1} ^n	    	
+			+\, a_{i,j}^{S}  D_{i-1,j} ^{n+1}+\,a_{i,j}^{W} D_{i,j-1} ^{n+1}
+                 -  D_{i,j}^n - \tilde{b}_{i,j}                                           \\
+D_{i,j} ^{n+1}  = &D_{i,j} ^{n}   + \omega \;R_{i,j}^n     
+\end{split}
 \end{equation}
 where \textit{$\omega $ }satisfies $1\leq \omega \leq 2$. An optimal value exists for 
@@ -359,59 +375,52 @@
 adjusted empirically for each model domain (except for a uniform grid where an 
 analytical expression for \textit{$\omega$} can be found \citep{Richtmyer1967}). 
-The value of $\omega$ is set using \textit{sor}, a \textbf{namelist} parameter. 
+The value of $\omega$ is set using \np{rn\_sor}, a \textbf{namelist} parameter. 
 The convergence test is of the form:
 \begin{equation}
-\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}{\sum\limits_{i,j}{ \tilde B_{i,j}^n}{\tilde B_{i,j}^n}} \leq \epsilon
+\delta = \frac{\sum\limits_{i,j}{R_{i,j}^n}{R_{i,j}^n}}
+                    {\sum\limits_{i,j}{ \tilde{b}_{i,j}^n}{\tilde{b}_{i,j}^n}} \leq \epsilon
 \end{equation}
 where $\epsilon$ is the absolute precision that is required. It is recommended 
-that a value smaller or equal to $10^{-3}$ is used for $\epsilon$ since larger 
-values may lead to numerically induced basin scale barotropic oscillations. In fact, 
-for an eddy resolving configuration or in a filtered free surface case, a value three 
-orders of magnitude smaller than this should be used. The precision is specified by 
-setting \textit{eps} (\textbf{namelist parameter}). In addition, two other tests are 
-used to halt the iterative algorithm. They involve the number of iterations and the 
-modulus of the right hand side. If the former exceeds a specified value, \textit{nmax} 
-(\textbf{namelist parameter}), or the latter is greater than $10^{15}$, the whole 
-model computation is stopped and the last computed time step fields are saved 
-in the standard output file. In both cases, this usually indicates that there is something 
-wrong in the model configuration (an error in the mesh, the initial state, the input forcing, 
+that a value smaller or equal to $10^{-6}$ is used for $\epsilon$ since larger 
+values may lead to numerically induced basin scale barotropic oscillations. 
+The precision is specified by setting \np{rn\_eps} (\textbf{namelist} parameter). 
+In addition, two other tests are used to halt the iterative algorithm. They involve 
+the number of iterations and the modulus of the right hand side. If the former 
+exceeds a specified value, \np{nn\_max} (\textbf{namelist} parameter), 
+or the latter is greater than $10^{15}$, the whole model computation is stopped 
+and the last computed time step fields are saved in a abort.nc NetCDF file. 
+In both cases, this usually indicates that there is something wrong in the model 
+configuration (an error in the mesh, the initial state, the input forcing, 
 or the magnitude of the time step or of the mixing coefficients). A typical value of 
-$nmax$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few 
+$nn\_max$ is a few hundred when $\epsilon = 10^{-6}$, increasing to a few 
 thousand when $\epsilon = 10^{-12}$.
 The vectorization of the SOR algorithm is not straightforward. The scheme
 contains two linear recurrences on $i$ and $j$. This inhibits the vectorisation. 
-Therefore it has been rewritten as:
-
-\begin{multline} 
+\eqref{Eq_sor_algo} can be been rewritten as:
+\begin{equation} 
+\begin{split}
 R_{i,j}^n
-= A_{i,j}^{N}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i+1,j}^n 
-+\,A_{i,j}^{E}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j+1} ^n		\\
-+\,A_{i,j}^{S}  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i-1,j} ^{n}
-+\,A_{i,j}^{W} {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j-1} ^{n}	
-                   -  {   \left(  \frac{\partial \psi}{\partial t } \right)    }_{i,j}^n - \tilde B_{i,j}
-\end{multline}
-
-\begin{equation}
-R_{i,j}^n = R_{i,j}^n - \omega \;A_{i,j}^{S}\; R_{i,j-1}^n
-\end{equation}
-
-\begin{equation}
-R_{i,j}^n = R_{i,j}^n - \omega \;A_{i,j}^{W}\; R_{i-1,j}^n
-\end{equation}
-
-The SOR method is very flexible and can be used under a wide 
-range of conditions, including irregular boundaries, interior boundary 
-points, etc. Proofs of convergence, etc. may be found in the standard 
-numerical methods texts for partial differential equations.
+= &a_{i,j}^{N}  D_{i+1,j}^n +\,a_{i,j}^{E}  D_{i,j+1} ^n
+ +\,a_{i,j}^{S}  D_{i-1,j} ^{n}+\,_{i,j}^{W} D_{i,j-1} ^{n} -  D_{i,j}^n - \tilde{b}_{i,j}      \\
+R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{S}\; R_{i,j-1}^n                                             \\
+R_{i,j}^n = &R_{i,j}^n - \omega \;a_{i,j}^{W}\; R_{i-1,j}^n
+\end{split}
+\end{equation}
+This technique slightly increases the number of iteration required to reach the convergence,
+but this is largely compensated by the gain obtained by the suppression of the recurrences.
+
+Another technique have been chosen, the so-called red-black SOR. It consist in solving successively 
+\eqref{Eq_sor_algo} for odd and even grid points. It also slightly reduced the convergence rate
+but allows the vectorisation. In addition, and this is the reason why it has been chosen, it is able to handle the north fold boundary condition used in ORCA configuration ($i.e.$ tri-polar global ocean mesh).
+
+The SOR method is very flexible and can be used under a wide range of conditions, 
+including irregular boundaries, interior boundary points, etc. Proofs of convergence, etc. 
+may be found in the standard numerical methods texts for partial differential equations.
 
 % -------------------------------------------------------------------------------------------------------------
 %       Preconditioned Conjugate Gradient
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Preconditioned Conjugate Gradient}
+\subsection{Preconditioned Conjugate Gradient  (\np{nn\_solv}=1, \mdl{solpcg}) }
 \label{MISC_solpcg}
-
-\begin{flushright}
-(\textit{nbsfs=1}, \textbf{namelist parameter})
-\end{flushright}
 
 \textbf{A} is a definite positive symmetric matrix, thus solving the linear 
@@ -448,22 +457,17 @@
 
 initialisation :
-
 \begin{equation*} 
-\textbf{x}^0 = \left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^0 = 2{\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^t} - {\left(  \frac{\partial \psi}{\partial t } \right)_{i,j}^{t-1}}
-, \text{the initial guess }
+\begin{split}
+\textbf{x}^0 &= D_{i,j}^0   = 2 D_{i,j}^t - D_{i,j}^{t-1}       \quad, \text{the initial guess }     \\
+\textbf{r}^0 &= \textbf{d}^0 = \textbf{b} - \textbf{A x}^0       \\
+\gamma_0 &= \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
+\end{split}
 \end{equation*}
 
-\begin{equation*}
-\textbf{r}^0 = \textbf{d}^0 = \textbf{b} - \textbf{A x}^0
-\end{equation*}
-\begin{equation*}
-\gamma_0 = \langle{ \textbf{r}^0 , \textbf{r}^0} \rangle
-\end{equation*}
 iteration $n,$ from $n=0$ until convergence, do :
-
 \begin{equation} 
 \begin{split}
 \text{z}^n& = \textbf{A d}^n \\
-\alpha_n &= \gamma_n  \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
+\alpha_n &= \gamma_n /  \langle{ \textbf{z}^n , \textbf{d}^n} \rangle \\
 \textbf{x}^{n+1} &= \textbf{x}^n + \alpha_n \,\textbf{d}^n \\
 \textbf{r}^{n+1} &= \textbf{r}^n - \alpha_n \,\textbf{z}^n \\
@@ -476,22 +480,21 @@
 
 The convergence test is:
-
 \begin{equation}
 \delta = \gamma_{n}\; / \langle{ \textbf{b} , \textbf{b}} \rangle \leq \epsilon
 \end{equation}
 where $\epsilon $ is the absolute precision that is required. As for the SOR algorithm, 
-the whole model computation is stopped when the number of iterations, $nmax$, or 
+the whole model computation is stopped when the number of iterations, \np{nn\_max}, or 
 the modulus of the right hand side of the convergence equation exceeds a 
 specified value (see \S\ref{MISC_solsor} for a further discussion). The required 
 precision and the maximum number of iterations allowed are specified by setting 
-$eps$ and $nmax$ (\textbf{namelist parameters}).
+\np{rn\_eps} and \np{nn\_max} (\textbf{namelist} parameters).
 
 It can be demonstrated that the above algorithm is optimal, provides the exact 
 solution in a number of iterations equal to the size of the matrix, and that 
 the convergence rate is faster as the matrix is closer to the identity matrix,
-$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve a 
-better conditioned system which has the same solution. For that purpose, we 
-introduce a preconditioning matrix \textbf{Q} which is an approximation of 
-\textbf{A} but much easier to invert than \textbf{A}, and solve the system:
+$i.e.$ its eigenvalues are closer to 1. Therefore, it is more efficient to solve 
+a better conditioned system which has the same solution. For that purpose, 
+we introduce a preconditioning matrix \textbf{Q} which is an approximation 
+of \textbf{A} but much easier to invert than \textbf{A}, and solve the system:
 \begin{equation} \label{Eq_pmat}
 \textbf{Q}^{-1} \textbf{A x} = \textbf{Q}^{-1} \textbf{b}
@@ -502,5 +505,5 @@
 ${\langle{ \textbf{a} , \textbf{b}} \rangle}_Q = \langle{ \textbf{a} , \textbf{Q b}} \rangle$, and 
 if $\textbf{\~{b}} = \textbf{Q}^{-1}\;\textbf{b}$ and $\textbf{\~{A}} = \textbf{Q}^{-1}\;\textbf{A}$ 
-are substituted to \textbf{b} and \textbf{A} \citep{Madec1988}. 
+are substituted to \textbf{b} and \textbf{A} \citep{Madec_al_OM88}. 
 In \NEMO, \textbf{Q} is chosen as the diagonal of \textbf{ A}, i.e. the simplest form for 
 \textbf{Q} so that it can be easily inverted. In this case, the discrete formulation of 
@@ -508,64 +511,8 @@
 right hand side are computed independently from the solver used.
 
-% -------------------------------------------------------------------------------------------------------------
-%       FETI
-% -------------------------------------------------------------------------------------------------------------
-\subsection{FETI}
-\label{MISC_solfeti}
-
-FETI is a powerful solver that was developed by Marc Guyon  \citep{Guyon_al_EP99, 
-Guyon_al_CalPar99}. It has been converted from Fortan 77 to 90, but never 
-successfully tested after that. 
-
-Its main advantage is to save a lot of CPU time when compared to the SOR or PCG 
-algorithms. However, its main drawback is that the solution is dependent on the 
-domain decomposition chosen. Using a different number of processors, the solution 
-is the same at the precision required, but not the same at the computer precision. 
-This make it hard to debug.  
-
-% -------------------------------------------------------------------------------------------------------------
-%       Boundary Conditions --- Islands
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Boundary Conditions --- Islands (\key{islands})}
-\label{MISC_solisl}
-
-The boundary condition used for both recommended solvers is that the time 
-derivative of the barotropic streamfunction is zero along all the coastlines. 
-When islands are present in the model domain, additional computations must 
-be performed to determine the barotropic streamfunction with the correct boundary 
-conditions. This is detailed below.
-
-The model does not have specialised code for islands. They must instead be 
-identified to the solvers by the user via bathymetry information, i.e. the $mbathy$ 
-array should equal $-1$ over the first island, $-2$ over the second, ... , $-N$ over 
-the $N^{th}$ island. The model determines the position of island grid-points and 
-defines a closed contour around each island which is used to compute the circulation 
-around each one. The closed contour is formed from the ocean grid-points which 
-are the closest to the island.
-
-First, the island barotropic streamfunctions $\psi_n$ are computed using the SOR 
-or PCG scheme (they are solutions of \eqref{Eq_solmat} with the right-hand side 
-equal to zero and with $\psi_n = 1$ along the island $n$ and $\psi_n = 0$ along 
-the other coastlines) (Note that specifying $1$ as boundary condition on an island 
-for $\psi$ is equivalent to defining a specific right hand side for \eqref{Eq_solmat}). 
-The requested precision of this computation can be very high since it is only 
-performed once. The absolute precision, $epsisl$, and the maximum number of 
-iterations allowed, $nmisl$, are the \textbf{namelist parameters} used for this 
-computation. Their typical values are $epsisl = 10^{-10}$ and $nmisl = 4000$. 
-Then the island matrix A is computed from \eqref{Eq_PE_psi_isl_matrix} and 
-inverted. At each time step, $\psi_0$, the solution of \eqref{Eq_solmat} with $\psi = 0$ 
-along all coastlines, is computed using either SOR or PCG. (It should
-be noted that the first guess of this computation remains as usual except that 
-$\partial_t \psi_0$ is used, instead of $\partial_t \psi$. Indeed we are 
-computing $\partial_t \psi_0$ which is usually very different from $\partial_t \psi$.)
-Then, it is easy to find the time evolution of the barotropic streamfunction on each 
-island and to deduce $\partial_t \psi$, and to use it to compute the surface pressure 
-gradient. Note that the value of the barotropic streamfunction itself is also computed 
-as the time integration of $\partial_t \psi$ for further diagnostics.
-
 % ================================================================
 % Diagnostics
 % ================================================================
-\section{Diagnostics }
+\section{Diagnostics (DIA, IOM)}
 \label{MISC_diag}
 
@@ -577,5 +524,5 @@
 
 %to be updated with Seb documentation on the IO
-\amtcomment{ 
+
 The model outputs are of three types: the restart file, the output listing, 
 and the output file(s). The restart file is used internally by the code when 
@@ -592,22 +539,17 @@
 the $ocean.output$ file. The information is printed from within the code on the 
 logical unit $numout$. To locate these prints, use the UNIX command 
-"$grep -i numout^*$" in the source code directory.
-
-In the standard configuration, the user will find the model results in two 
-output files containing values for every time-step where output is demanded: 
-a \textbf{VO} file containing all the three dimensional fields in logical unit 
-$numwvo$, and a \textbf{SO} file containing all the two dimensional (horizontal) 
-fields in logical unit $numwso$. These outputs are defined in the $diawri.F$ 
-routine. The default and user-selectable diagnostics are described in {\S}III-12-c.
-
-The default output (for all output files apart from the restart file) is 32 bits binary 
-IEEE format, compatible with the Vairmer software (see the Climate Modelling 
-and global Change team WEB server at CERFACS: http://www.cerfacs.fr). 
-The model's reference directory also contains a visualisation tool based on 
-\textbf{NCAR Graphics} (http://ngwww.ucar.edu). If a user has access to the 
-NCAR software, he or she can copy the \textbf{LODMODEL/UTILS/OPADRA} 
-directory from the reference and, following the \textbf{README}, create 
-graphical outputs from the model's results.
-}
+"\textit{grep -i numout}" in the source code directory.
+
+In the standard configuration, the user will find the model results in 
+NetCDF files containing mean values (or instantaneous values if 
+\key{diainstant} is defined) for every time-step where output is demanded.
+These outputs are defined in the \mdl{diawri} module. 
+When defining \key{dimgout}, the output are written in DIMG format,
+an IEEE output format.
+
+Since version 3.2, an I/O server has been added which provides more
+flexibility in the choice of the fields to be output as well as how the 
+writing work is distributed over the processors in massively parallel
+computing. It is activated when \key{dimgout} is defined.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -635,5 +577,5 @@
 %       On-line Floats trajectories
 % -------------------------------------------------------------------------------------------------------------
-\subsection{On-line Floats trajectories}
+\subsection{On-line Floats trajectories (FLO)}
 \label{FLO}
 %--------------------------------------------namflo-------------------------------------------------------
@@ -669,3 +611,162 @@
 - the meridional heat and salt transports and their decomposition (\mdl{diamfl})
 
-- the surface pressure (\mdl{diaspr})
+In addition, a series of diagnostics has been added in the \mdl{diaar5}. 
+They corresponds to outputs that are required for AR5 simulations 
+(see Section \ref{MISC_steric} below for one of them). 
+Activating those outputs requires to define the \key{diaar5} CPP key.
+
+
+
+% ================================================================
+% Steric effect in sea surface height
+% ================================================================
+\section{Steric effect in sea surface height}
+\label{MISC_steric}
+
+
+Changes in steric sea level are caused when changes in the density of the water 
+column imply an expansion or contraction of the column. It is essentially produced 
+through surface heating/cooling and to a lesser extent through non-linear effects of 
+the equation of state (cabbeling, thermobaricity...).
+Non-Boussinesq models contain all ocean effects within the ocean acting 
+on the sea level. In particular, they include the steric effect. In contrast, 
+Boussinesq models, such as \NEMO, conserve volume, rather than mass, 
+and so do not properly represent expansion or contraction. The steric effect is 
+therefore not explicitely represented.
+This approximation does not represent a serious error with respect to the flow field 
+calculated by the model \citep{Greatbatch_JGR94}, but extra attention is required
+when investigating sea level, as steric changes are an important 
+contribution to local changes in sea level on seasonal and climatic time scales.
+This is especially true for investigation into sea level rise due to global warming. 
+
+Fortunately, the steric contribution to the sea level consists of a spatially uniform 
+component that can be diagnosed by considering the mass budget of the world 
+ocean \citep{Greatbatch_JGR94}. 
+In order to better understand how global mean sea level evolves and thus how
+the steric sea level can be diagnosed, we compare, in the following, the 
+non-Boussinesq and Boussinesq cases.
+
+Let denote 
+$\mathcal{M}$ the total mass of liquid seawater ($\mathcal{M}=\int_D \rho dv$), 
+$\mathcal{V}$ the total volume of seawater ($\mathcal{V}=\int_D dv$), 
+$\mathcal{A}$ the total surface of the ocean ($\mathcal{A}=\int_S ds$), 
+$\bar{\rho}$ the global mean seawater (\textit{in situ}) density ($\bar{\rho}= 1/\mathcal{V} \int_D \rho \,dv$), and
+$\bar{\eta}$ the global mean sea level ($\bar{\eta}=1/\mathcal{A}\int_S \eta \,ds$).
+
+A non-Boussinesq fluid conserves mass. It satisfies the following relations:
+\begin{equation} \label{Eq_MV_nBq} 
+\begin{split} 
+\mathcal{M} &=  \mathcal{V}  \;\bar{\rho}      	\\
+\mathcal{V} &=  \mathcal{A}  \;\bar{\eta}	
+\end{split}
+\end{equation}
+Temporal changes in total mass is obtained from the density conservation equation :
+\begin{equation}  \label{Eq_Co_nBq}
+\frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface}
+\end{equation}
+where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass 
+exchanges with the other media of the Earth system (atmosphere, sea-ice, land). 
+Its global averaged leads to the total mass change 
+\begin{equation}  \label{Eq_Mass_nBq}
+\partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}}
+\end{equation}
+where $\overline{\textit{emp}}=\int_S \textit{emp}\,ds$ is the net mass flux 
+through the ocean surface.
+Bringing \eqref{Eq_Mass_nBq} and the time derivative of \eqref{Eq_MV_nBq} 
+together leads to the evolution equation of the mean sea level
+\begin{equation} \label{Eq_ssh_nBq}
+  \partial_t \bar{\eta} =  \frac{\overline{\textit{emp}}}{ \bar{\rho}} 
+  					- \frac{\mathcal{V}}{\mathcal{A}}  \;\frac{\partial_t \bar{\rho} }{\bar{\rho}}
+\end{equation}
+The first term in equation \eqref{Eq_ssh_nBq} alters sea level by adding or 
+subtracting mass from the ocean. 
+The second term arises from temporal changes in the global mean 
+density; $i.e.$ from steric effects. 
+
+In a Boussinesq fluid, $\rho$ is replaced by $\rho_o$ in all the equation except when $\rho$ 
+appears multiplied by the gravity ($i.e.$ in the hydrostatic balance of the primitive Equations). 
+In particular, the mass conservation equation, \eqref{Eq_Co_nBq}, degenerates into 
+the incompressibility equation:
+\begin{equation}  \label{Eq_Co_Bq}
+\frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) =  \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface}
+\end{equation}
+and the global average of this equation now gives the temporal change of the total volume,
+\begin{equation}  \label{Eq_V_Bq}
+  \partial_t \mathcal{V} =   \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o} 
+\end{equation}
+Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the 
+Boussinesq mass, $\mathcal{M}_o = \rho_o \mathcal{V}$. The total volume (or equivalently  
+the global mean sea level) is altered only by net volume fluxes across the ocean surface,  
+not by changes in mean mass of the ocean: the steric effect is missing in a Boussinesq fluid.
+ 
+Nevertheless, following \citep{Greatbatch_JGR94}, the steric effect on the volume can be 
+diagnosed by considering the mass budget of the ocean. 
+The apparent changes in $\mathcal{M}$, mass of the ocean, which are not induced by surface 
+mass flux must be compensated by a spatially uniform change in the mean sea level due to 
+expansion/contraction of the ocean \citep{Greatbatch_JGR94}. In others words, the Boussinesq 
+mass, $\mathcal{M}_o$, can be related to $\mathcal{M}$, the  total mass of the ocean seen 
+by the Boussinesq model, via the steric contribution to the sea level, $\eta_s$, a spatially 
+uniform variable, as follows:
+\begin{equation}  \label{Eq_M_Bq}
+   \mathcal{M}_o  =  \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A} 
+\end{equation}
+Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through 
+the ocean surface is converted into a mean change in sea level. Introducing the total density 
+anomaly, $\mathcal{D}= \int_D d_a \,dv$, where $d_a= (\rho -\rho_o ) / \rho_o$  
+is the density anomaly used in \NEMO (cf. \S\ref{TRA_eos}) in \eqref{Eq_M_Bq}
+leads to a very simple form for the steric height:
+\begin{equation}  \label{Eq_steric_Bq}
+   \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D} 
+\end{equation}
+
+The above formulation of the steric height of a Boussinesq ocean requires four remarks.
+First, one can be tempted to define $\rho_o$ as the initial value of $\mathcal{M}/\mathcal{V}$,
+$i.e.$ set $\mathcal{D}_{t=0}=0$, so that the initial steric height is zero. We do not
+recommend that. Indeed, in this case $\rho_o$ depends on the initial state of the ocean. 
+Since $\rho_o$ has a direct effect on the dynamics of the ocean (it appears in the pressure 
+gradient term of the momentum equation) it is definitively not a good idea when 
+inter-comparing experiments. 
+We better recommend to fixe once for all $\rho_o$ to $1035\;Kg\,m^{-3}$. This value is a 
+sensible choice for the reference density used in a Boussinesq ocean climate model since, 
+with the exception of only a small percentage of the ocean, density in the World Ocean 
+varies by no more than 2$\%$ from this value (\cite{Gill1982}, page 47).
+
+Second, we have assumed here that the total ocean surface, $\mathcal{A}$, does not
+change when the sea level is changing as it is the case in all global ocean GCMs 
+(wetting and drying of grid point is not allowed). 
+  
+Third, the discretisation of \eqref{Eq_steric_Bq} depends on the type of free surface
+which is considered. In the non linear free surface case, $i.e.$ \key{vvl} defined, it is
+given by
+\begin{equation}  \label{Eq_discrete_steric_Bq}
+   \eta_s =  - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }
+   					{ \sum_{i,\,j,\,k}         e_{1t} e_{2t} e_{3t} } 
+\end{equation}
+whereas in the linear free surface, the volume above the \textit{z=0} surface must be explicitly taken 
+into account to better approximate the total ocean mass and thus the steric sea level:
+\begin{equation}  \label{Eq_discrete_steric_Bq}
+   \eta_s =  - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
+   						{\sum_{i,\,j,\,k} e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}           e_{1t}e_{2t} \eta } 
+\end{equation}
+
+The fourth and last remark concerns the effective sea level and the presence of sea-ice.
+In the real ocean, sea ice (and snow above it)  depresses the liquid seawater through 
+its mass loading. This depression is a result of the mass of sea ice/snow system acting 
+on the liquid ocean. There is, however, no dynamical effect associated with these depressions 
+in the liquid ocean sea level, so that there are no associated ocean currents. Hence, the 
+dynamically relevant sea level is the effective sea level, $i.e.$ the sea level as if sea ice 
+(and snow) were converted to liquid seawater \citep{Campin_al_OM08}. However,
+in the current version of \NEMO the sea-ice is levitating above the ocean without 
+mass exchanges between ice and ocean. Therefore the model effective sea level
+is always given by $\eta + \eta_s$, whether or not there is sea ice present.
+
+In AR5 outputs, the thermosteric sea level is demanded. It is steric sea level due to 
+changes in ocean density arising just from changes in temperature. It is given by:
+\begin{equation}  \label{Eq_thermosteric_Bq}
+   \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv
+\end{equation}
+where $S_o$ and $p_o$ are the initial salinity and pressure, respectively.
+
+Both steric and thermosteric sea level are computed in \mdl{diaar5} which needs
+the \key{diaar5} defined to be called.
+
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 1831)
@@ -7,4 +7,7 @@
 \minitoc
 
+
+\newpage
+$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -164,4 +167,7 @@
 
 
+\newpage
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % The Horizontal Pressure Gradient
@@ -196,6 +202,8 @@
  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.
+ Rossby or planetary waves). The rigid-lid hypothesis is an obsolescent feature in modern 
+ OGCMs. It has been available until the release 3.1 of  \NEMO, and it has been removed
+ in release 3.2 and followings. Only the free surface formulation is now described in the
+ this document (see the next sub-section).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -221,5 +229,5 @@
 short with respect to the other processes described by the primitive equations.
 
-Three choices can be made regarding the implementation of the free surface in the model, 
+Two choices can be made regarding the implementation of the free surface in the model, 
 depending on the physical processes of interest. 
 
@@ -238,15 +246,10 @@
 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 then considered as a fixed surface, so that all 
-external waves are removed from the system. 
+to take into account freshwater fluxes applied at the ocean surface \citep{Roullet_Madec_JGR00}.
 
 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} 
+of the temporal derivatives, using the time splitting method \citep{Killworth_al_JPO91, Zhang_Endoh_JGR92} 
 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 
+developed by \citet{Roullet_Madec_JGR00}: 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}
@@ -266,5 +269,5 @@
 $i.e.$ 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 
+can be damped by choosing $T_c > \Delta t$. \citet{Roullet_Madec_JGR00} 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 
@@ -276,5 +279,5 @@
 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 
+by \citet{Roullet_Madec_JGR00} 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 
@@ -287,5 +290,5 @@
 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 
+linearized, \citet{Roullet_Madec_JGR00} 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}
@@ -295,5 +298,5 @@
 
 The right hand side of (\ref{Eq_PE_salt_content_linear}) is small in equilibrium solutions 
-\citep{Roullet2000}. It can be significant when the freshwater forcing is not balanced and 
+\citep{Roullet_Madec_JGR00}. It can be significant when the freshwater forcing is not balanced and 
 the globally averaged free surface is drifting. An increase in sea surface height \textit{$\eta $} 
 results in a decrease of the salinity in the fixed volume $D$. Even in that case though, 
@@ -309,107 +312,7 @@
 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 a spurious source of salt at the ocean 
-surface \citep{Roullet2000}.
-
-% -------------------------------------------------------------------------------------------------------------
-% Rigid-Lid Formulation
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Rigid-Lid formulation}
-\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 \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)
-\end{equation}
-
-As $p_s$ does not depend on depth, its horizontal gradient is obtained by forming the 
-vertical average of \eqref{Eq_PE_dyn} and using \eqref{Eq_PE_u_psi}:
-
-\begin{equation} \label{Eq_PE_u_barotrope}
-\frac{1}{\rho _o }\nabla _h p_s 
-=\overline{\vect{M}} -\frac{\partial \overline{\vect{U}} _h }{\partial t}
-=\overline{\vect{M}} 
--\frac{1}{H} \left[   \vect{k} \times \nabla \left( \frac{\partial \psi}{\partial t} \right)   \right]
-\end{equation}
-
-Here ${\rm {\bf M}} = \left( M_u,M_v \right)$ represents the collected contributions of the 
-Coriolis, hydrostatic pressure gradient, nonlinear and viscous terms in \eqref{Eq_PE_dyn}. 
-The time derivative of $\psi $ is the solution of an elliptic equation which is obtained from 
-the vertical component of the curl of (\ref{Eq_PE_u_barotrope}):
-\begin{equation} \label{Eq_PE_psi}
-\left[   {\nabla \times \left[ {\frac{1}{H} \vect{\bf k} 
-  \times \nabla \left(   {\frac{\partial \psi }{\partial t}} \right)}   \right]} \; \right]_z 
-=\left[   {\nabla \times \overline{\vect{M}} }   \right]_z 
-\end{equation}
-
-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( 
-{\frac{\partial \psi }{\partial t}} \right)} \right] \cdot {\rm {\bf d}}\ell } 
-= \oint_n {\overline {\rm {\bf M}} \cdot {\rm {\bf d}}\ell } 
-\qquad  1 \leq n \leq Q
-\end{equation}
-
-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 \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( {\frac{\partial \mu _n }{\partial t}} 
-\right)_{1\leqslant n\leqslant Q}        \\
- =\left( {\oint_n {\left[ {\overline {\rm 
-{\bf M}} -\frac{1}{H}\left[ {{\rm {\bf k}}\times \nabla \left( 
-{\frac{\partial \psi _o }{\partial t}} \right)} \right]} \right]\cdot {\rm 
-{\bf d}}\ell } } \right)_{1\leqslant n\leqslant Q} 
-\end{multline}
-which can be rewritten as:
-\begin{equation} \label{Eq_PE_psi_isl_matrix}
-{\rm {\bf A}}\;\left( {\frac{\partial \mu _n }{\partial t}} 
-\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:
-\begin{equation} \label{Eq_PE_psi_isl_dt}
-\frac{\partial \psi }{\partial t}=\frac{\partial \psi _o }{\partial 
-t}+\sum\limits_{n=1}^{n=Q} {{\rm {\bf A}}^{-1}{\rm {\bf B}}\;\psi _n } 
-\end{equation}
-
+the rigid lid approximation in which case freshwater forcing is represented by a virtual 
+salt flux, leading to a spurious source of salt at the ocean surface 
+\citep{Huang_JPO93, Roullet_Madec_JGR00}.
 
 
@@ -435,5 +338,5 @@
 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 
+\citep{Madec_Imbard_CD96, Murray_JCP96}. 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. 
@@ -444,5 +347,5 @@
 of the conservation laws of fluid dynamics.
 
-Let (\textbf{i},\textbf{j},\textbf{k}) be a set of orthogonal curvilinear coordinates on the sphere 
+Let (\textit{i},\textit{j},\textit{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 
@@ -682,6 +585,6 @@
 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}
-										-u \frac{\partial e_1}{\partial j} 	\right)
+f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i}
+					         -u \frac{\partial e_1}{\partial j}  \right)
 \end{equation}
 
@@ -690,29 +593,33 @@
 the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
 
+
+$\ $\newline    % force a new ligne
+
 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 :
+$\bullet$ \textbf{Vector invariant form of the momentum equations} :
+
 \begin{subequations} \label{Eq_PE_dyn_vect}
-\begin{multline} \label{Eq_PE_dyn_vect_u}
-\frac{\partial u}{\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} 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}{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}
+\begin{equation} \label{Eq_PE_dyn_vect_u} \begin{split}
+\frac{\partial u}{\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    }  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}}      \\
+\\
+\frac{\partial v}{\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     }   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{split} \end{equation}
 \end{subequations}
 
-flux form:
+
+\vspace{+10pt}
+$\bullet$ \textbf{flux form of the momentum equations} :
 \begin{subequations} \label{Eq_PE_dyn_flux}
 \begin{multline} \label{Eq_PE_dyn_flux_u}
@@ -741,38 +648,14 @@
 \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
-\begin{equation}\label{Eq_PE_dyn_sco}
-\frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c}
- {\frac{g}{\;e_1 }\frac{\partial \eta }{\partial i}} \hfill \\
- {\frac{g}{\;e_2 }\frac{\partial \eta }{\partial j}} \hfill \\
-\end{array} }} \right)
-\qquad \text{where $\eta$ is solution of \eqref{Eq_PE_ssh} }
-\end{equation}
-
-$*$ rigid-lid approximation
-\begin{equation}\label{Eq_PE_dyn_zco}
-\frac{1}{\rho _o }\nabla _h p_s =\left( {{\begin{array}{*{20}c}
- {\overline M _u +\frac{1}{H\;e_2 }\frac{\partial }{\partial j}\left( 
-{\frac{\partial \psi }{\partial t}} \right)}		 \\
- {\overline M _v -\frac{1}{H\;e_1 }\frac{\partial }{\partial i}\left( 
-{\frac{\partial \psi }{\partial t}} \right)} 		 \\
-\end{array} }} \right)
-\end{equation}
-where ${\vect{M}}= \left( M_u,M_v \right)$ represents the collected contributions of nonlinear, 
-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}
-								 \left( {\frac{\partial \psi }{\partial t}} \right)}   \right]
-+\frac{\partial }{\partial j}\left[ {\frac{e_1 }{H\,e_2}\frac{\partial }{\partial j}
-								 \left( {\frac{\partial \psi }{\partial t}} \right)} \right]
-= \\
-  \frac{\partial }{\partial i}\left( {e_2 \overline M _v } \right)
-- \frac{\partial }{\partial j}\left( {e_1 \overline M _u } \right)
-\end{multline}
+where $\zeta$, the relative vorticity, is given by \eqref{Eq_PE_curl_Uh} and $p_s $, 
+the surface pressure, is given by:
+\begin{equation} \label{Eq_PE_spg}
+p_s = \left\{ \begin{split} 
+\rho \,g \,\eta &                                 \qquad  \qquad  \;   \qquad \text{ standard free surface} \\ 
+\rho \,g \,\eta &+ \rho_o \,\mu \,\frac{\partial \eta }{\partial t}      \qquad \text{ filtered     free surface}    
+\end{split} 
+\right.
+\end{equation}
+with $\eta$ is solution of \eqref{Eq_PE_ssh}
 
 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
@@ -783,5 +666,4 @@
 \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 
 \end{equation}
-
 where the divergence of the horizontal velocity, $\chi$ is given by \eqref{Eq_PE_div_Uh}.
 
@@ -809,9 +691,21 @@
 in Chapter~\ref{SBC}.
 
+
 \newpage 
 % ================================================================
+% Curvilinear s-coordinate System
+% ================================================================
+\section{Curvilinear \textit{s}-coordinate System}
+\label{PE_sco}
+
+
+
+
+
+% -------------------------------------------------------------------------------------------------------------
 % Curvilinear z*-coordinate System
-% ================================================================
-\section{Curvilinear \textit{z*}-coordinate System}
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear \textit{z*}-coordinate System}
+\label{PE_zco_star}
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -858,8 +752,8 @@
 
 \newpage 
-% ================================================================
-% Curvilinear s-coordinate System
-% ================================================================
-\section{Curvilinear \textit{s}-coordinate System}
+% -------------------------------------------------------------------------------------------------------------
+% Terrain following  coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Terrain following \textit{s-}coordinate}
 \label{PE_sco}
 
@@ -867,5 +761,5 @@
 % Introduction
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Introduction}
+\subsubsection{Introduction}
 
 Several important aspects of the ocean circulation are influenced by bottom topography. 
@@ -914,5 +808,5 @@
 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 
+overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec_al_JPO96}. 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 
@@ -932,10 +826,10 @@
 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 
+($i.e.$ the main thermocline) \citep{Madec_al_JPO96}. 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 
+The $s-$coordinates introduced here \citep{Lott_al_OM90,Madec_al_JPO96} 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 
@@ -945,5 +839,5 @@
 % The s-coordinate Formulation
 % -------------------------------------------------------------------------------------------------------------
-\subsection{The \textit{s-}coordinate Formulation}
+\subsubsection{The \textit{s-}coordinate Formulation}
 
 Starting from the set of equations established in \S\ref{PE_zco} for the special case $k=z$ 
@@ -1023,4 +917,6 @@
 Add a few works on z and zps and s and underlies the differences between all of them
 \colorbox{yellow}{ $< = =$ end update}  }
+
+
 
 \newpage 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_SBC.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 1831)
@@ -17,5 +17,5 @@
 \item the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
 \item the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
-\item the surface freshwater budget $\left( {\text{EMP},\;\text{EMP}_S } \right)$
+\item the surface freshwater budget $\left( {\textit{emp},\;\textit{emp}_S } \right)$
 \end{itemize}
 
@@ -28,11 +28,11 @@
 the  \np{nf\_sbc} namelist parameter. 
 When the fields are supplied from data files (flux and bulk formulations), the input fields 
-need not be supplied on the model grid.  Instead a file of coordinates and weights can be supplied which
-maps the data from the supplied grid to the model points (so called "Interpolation on the Fly").
-In addition, the resulting fields can be further modified using 
-several namelist options. These options control  the rotation of vector components
-supplied relative to an east-north coordinate system onto the local grid directions in the model;
-the addition of a surface restoring 
-term to observed SST and/or SSS (\np{ln\_ssr}=true); the modification of fluxes 
+need not be supplied on the model grid.  Instead a file of coordinates and weights can 
+be supplied which maps the data from the supplied grid to the model points 
+(so called "Interpolation on the Fly").
+In addition, the resulting fields can be further modified using several namelist options. 
+These options control  the rotation of vector components supplied relative to an east-north 
+coordinate system onto the local grid directions in the model; the addition of a surface 
+restoring term to observed SST and/or SSS (\np{ln\_ssr}=true); the modification of fluxes 
 below ice-covered areas (using observed ice-cover or a sea-ice model) 
 (\np{nn\_ice}=0,1, 2 or 3); the addition of river runoffs as surface freshwater 
@@ -42,9 +42,8 @@
 cycle (\np{ln\_dm2dc}=true).
 
-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. Next the scheme for interpolation on the fly is described.
-Finally, the different options that further modify
-the fluxes applied to the ocean are discussed.
+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. 
+Next the scheme for interpolation on the fly is described.
+Finally, the different options that further modify the fluxes applied to the ocean are discussed.
 
 
@@ -75,5 +74,5 @@
 \begin{equation} \label{Eq_sbc_trasbc_q}
 \frac{\partial T}{\partial t}\equiv \cdots \;+\;\left. {\frac{Q_{ns} }{\rho 
-_o \;C_p \;e_{3T} }} \right|_{k=1} \quad
+_o \;C_p \;e_{3t} }} \right|_{k=1} \quad
 \end{equation}
 $Q_{sr}$ is the penetrative part of the heat flux. It is applied as a 3D 
@@ -81,6 +80,5 @@
 
 \begin{equation} \label{Eq_sbc_traqsr}
-\frac{\partial T}{\partial t}\equiv \cdots \;+\frac{Q_{sr} }{\rho _o C_p 
-\,e_{3T} }\delta _k \left[ {I_w } \right]
+\frac{\partial T}{\partial t}\equiv \cdots \;+\frac{Q_{sr} }{\rho_o C_p \,e_{3t} }\delta _k \left[ {I_w } \right]
 \end{equation}
 where $I_w$ is a non-dimensional function that describes the way the light 
@@ -88,33 +86,33 @@
 exponentials (see \S\ref{TRA_qsr}).
 
-The surface freshwater budget is provided by fields: EMP and EMP$_S$ which 
+The surface freshwater budget is provided by fields: \textit{emp} and $\textit{emp}_S$ which 
 may or may not be identical. Indeed, a surface freshwater flux has two effects: 
 it changes the volume of the ocean and it changes the surface concentration of 
 salt (and other tracers). Therefore it appears in the sea surface height as a volume 
-flux, EMP (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations 
+flux, \textit{emp} (\textit{dynspg\_xxx} modules), and in the salinity time evolution equations 
 as a concentration/dilution effect, 
-EMP$_{S}$ (\mdl{trasbc} module). 
+$\textit{emp}_{S}$ (\mdl{trasbc} module). 
 \begin{equation} \label{Eq_trasbc_emp}
 \begin{aligned}
-&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\text{EMP}\quad  \\ 
+&\frac{\partial \eta }{\partial t}\equiv \cdots \;+\;\textit{emp}\quad  \\ 
 \\
- &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\text{EMP}_S \;S}{e_{3T} }} \right|_{k=1} \\ 
+ &\frac{\partial S}{\partial t}\equiv \cdots \;+\left. {\frac{\textit{emp}_S \;S}{e_{3t} }} \right|_{k=1} \\ 
  \end{aligned}
 \end{equation} 
 
-In the real ocean, EMP$=$EMP$_S$ and the ocean salt content is conserved, 
+In the real ocean, $\textit{emp}=\textit{emp}_S$ and the ocean salt content is conserved, 
 but it exist several numerical reasons why this equality should be broken. 
 For example:
 
 When the rigid-lid assumption is made, the ocean volume becomes constant and 
-thus, EMP$=$0, not EMP$_{S }$.
+thus, $\textit{emp}=0$, not $\textit{emp}_S$.
 
 When the ocean is coupled to a sea-ice model, the water exchanged between ice and 
 ocean is slightly 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 and the salt flux between ice and ocean, while EMP is 
-only the volume flux. In addition, in the current version of \NEMO, the 
-sea-ice is assumed to be above the ocean. Freezing/melting does not change 
-the ocean volume (no impact on EMP) but it modifies the SSS.
+$\textit{emp}_{S}$ take into account both concentration/dilution effect associated with 
+freezing/melting and the salt flux between ice and ocean, while \textit{emp} is 
+only the volume flux. In addition, in the current version of \NEMO, the sea-ice is 
+assumed to be above the ocean (the so-called levitating sea-ice). Freezing/melting does 
+not change the ocean volume (no impact on \textit{emp}) but it modifies the SSS.
 %gm  \colorbox{yellow}{(see {\S} on LIM sea-ice model)}.
 
@@ -127,5 +125,5 @@
 associated with precipitation! Precipitation can change the ocean volume and thus the
 ocean heat content. It is therefore associated with a heat flux (not yet  
-diagnosed in the model) \citep{Roullet2000}).
+diagnosed in the model) \citep{Roullet_Madec_JGR00}).
 
 %\colorbox{yellow}{Miss: }
@@ -193,5 +191,5 @@
 be uniform in space. They take constant values given in the namelist 
 namsbc{\_}ana by the variables \np{rn\_utau0}, \np{rn\_vtau0}, \np{rn\_qns0}, 
-\np{rn\_qsr0}, and \np{rn\_emp0} (EMP$=$EMP$_S$). The runoff is set to zero. 
+\np{rn\_qsr0}, and \np{rn\_emp0} ($\textit{emp}=\textit{emp}_S$). The runoff is set to zero. 
 In addition, the wind is allowed to reach its nominal value within a given number 
 of time steps (\np{nn\_tau000}).
@@ -267,5 +265,5 @@
 %-------------------------------------------------------------------------------------------------------------
 
-The CORE bulk formulae have been developed by \citet{LargeYeager2004}. 
+The CORE bulk formulae have been developed by \citet{Large_Yeager_Rep04}. 
 They have been designed to handle the CORE forcing, a mixture of NCEP 
 reanalysis and satellite data. They use an inertial dissipative method to compute 
@@ -408,13 +406,11 @@
 The symbolic algorithm used to calculate values on the model grid is now:
 
-\begin{multline*}
-f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
-\\
-                        + \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }
-\\
-                        + \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }
-\\
-                        + \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
-\end{multline*}
+\begin{equation*} \begin{split}
+f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}     
+              +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\
+              +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }   
+              +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
+\end{split}
+\end{equation*}
 The gradients here are taken with respect to the horizontal indices and not distances since the spatial dependency has been absorbed into the weights.
 
@@ -515,9 +511,9 @@
 
 \begin{equation} \label{Eq_sbc_dmp_emp}
-EMP = EMP_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
+\textit{emp} = \textit{emp}_o + \gamma_s^{-1} e_{3t}  \frac{  \left(\left.S\right|_{k=1}-SSS_{Obs}\right)}
 												         {\left.S\right|_{k=1}}
 \end{equation}
 
-where EMP$_{o }$ is a net surface fresh water flux (observed, climatological or an
+where $\textit{emp}_{o }$ is a net surface fresh water flux (observed, climatological or an
 atmospheric model product), \textit{SSS}$_{Obs}$ is a sea surface salinity (usually a time 
 interpolation of the monthly mean Polar Hydrographic Climatology \citep{Steele2001}), 
@@ -602,5 +598,5 @@
 \item[\np{nn\_fwb}=0]  no control at all. The mean sea level is free to drift, and will 
 certainly do so.
-\item[\np{nn\_fwb}=1]  global mean EMP set to zero at each model time step. 
+\item[\np{nn\_fwb}=1]  global mean \textit{emp} set to zero at each model time step. 
 %Note that with a sea-ice model, this technique only control the mean sea level with linear free surface (\key{vvl} not defined) and no mass flux between ocean and ice (as it is implemented in the current ice-ocean coupling). 
 \item[\np{nn\_fwb}=2]  freshwater budget is adjusted from the previous year annual 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 1831)
@@ -71,5 +71,5 @@
 \label{TRA_adv}
 %------------------------------------------nam_traadv-----------------------------------------------------
-\namdisplay{nam_traadv}
+\namdisplay{namtra_adv}
 %-------------------------------------------------------------------------------------------------------------
 
@@ -77,19 +77,18 @@
 fluxes. Its discrete expression is given by :
 \begin{equation} \label{Eq_tra_adv}
-ADV_\tau =-\frac{1}{b_T} \left( 
+ADV_\tau =-\frac{1}{b_t} \left( 
 \;\delta _i \left[ e_{2u}\,e_{3u} \;  u\; \tau _u  \right]
 +\delta _j \left[ e_{1v}\,e_{3v}  \;  v\; \tau _v  \right] \; \right)
--\frac{1}{e_{3T}} \;\delta _k \left[ w\; \tau _w \right]
-\end{equation}
-where $\tau$ is either T or S, and $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 
+-\frac{1}{e_{3t}} \;\delta _k \left[ w\; \tau _w \right]
+\end{equation}
+where $\tau$ is either T or S, and $b_t= e_{1t}\,e_{2t}\,e_{3t}$ is the volume of $T$-cells. 
 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 
-\left[ {e_{2u} {\kern 1pt}{\kern 1pt}\;u\;\tau _u } \right]+\delta _j \left[ 
-{e_{1v} {\kern 1pt}v\;\tau _v } \right]\;} \right)-\frac{1}{\mathop 
-e\nolimits_{3T} }\delta _k \left[ {w\;\tau _w } \right]
+ADV_\tau = - \frac{1}{e_{1t}\,e_{2t}} \left( \; \delta_i \left[ e_{2u} \;u \;\tau_u \right] 
+                                                                 + \delta_j \left[ e_{1v} \;v \;\tau_v  \right] \; \right)
+                   -  \frac{1}{e_{3t}}                      \delta_k \left[             w \;\tau_w \right]
 \end{equation}
 since the vertical scale factors are functions of $k$ only, and thus 
-$e_{3u} =e_{3v} =e_{3T} $. The flux form in \eqref{Eq_tra_adv} 
+$e_{3u} =e_{3v} =e_{3t} $. The flux form in \eqref{Eq_tra_adv} 
 requires implicitly the use of the continuity equation. Indeed, it is obtained
 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
@@ -125,12 +124,10 @@
 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} $. 
+flux which is set for all advection schemes as 
+$\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $, $i.e.$ 
+the product of surface velocity (at $z=0$) by the first level tracer value.
 \item [non-linear free surface:] (\key{vvl} is defined) 
 convergence/divergence in the first ocean level moves the free surface 
@@ -144,5 +141,5 @@
 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}).
+and also \citet{Roullet_Madec_JGR00,Griffies_al_MWR01,Campin2004}).
 
 The velocity field that appears in (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_zco}) 
@@ -266,9 +263,9 @@
 There 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 
+\citet{Zalesak_JCP79}. $c_u$ 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 
+\citet{Levy_al_GRL01}; note that in this paper it is referred to as "FCT" (Flux corrected 
 transport) rather than TVD. The TVD scheme is computed in the \mdl{traadv\_tvd} module.
 
@@ -287,5 +284,5 @@
 
 The Monotone Upstream Scheme for Conservative Laws (MUSCL) has been 
-implemented by \citet{Levy2001}. In its formulation, the tracer at velocity points 
+implemented by \citet{Levy_al_GRL01}. 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 :
@@ -351,8 +348,8 @@
 second term (which is the diffusive part of the scheme), is 
 evaluated using the \textit{before} tracer (forward in time). 
-This choice is discussed by \citet{Webb1998} in the context of the 
+This choice is discussed by \citet{Webb_al_JAOT98} 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}. 
+leads to the QUICK advection scheme \citep{Webb_al_JAOT98}. 
 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 
@@ -446,5 +443,5 @@
 \label{TRA_ldf}
 %-----------------------------------------nam_traldf------------------------------------------------------
-\namdisplay{nam_traldf}
+\namdisplay{namtra_ldf}
 %-------------------------------------------------------------------------------------------------------------
  
@@ -471,9 +468,9 @@
 surfaces is given by: 
 \begin{equation} \label{Eq_tra_ldf_lap}
-D_T^{lT} =\frac{1}{b_T} \left( \;
+D_T^{lT} =\frac{1}{b_tT} \left( \;
    \delta _{i}\left[ A_u^{lT} \; \frac{e_{2u}\,e_{3u}}{e_{1u}} \;\delta _{i+1/2} [T] \right] 
 + \delta _{j}\left[ A_v^{lT} \;  \frac{e_{1v}\,e_{3v}}{e_{2v}} \;\delta _{j+1/2} [T] \right]  \;\right)
 \end{equation}
-where  $b_T= e_{1T}\,e_{2T}\,e_{3T}$  is the volume of $T$-cells. 
+where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells. 
 It can be found in the \mdl{traadv\_lap} module.
 
@@ -487,5 +484,5 @@
 
 Note that 
-(a) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}=e_{3v}=e_{3T}$, 
+(a) 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}) ; 
 (b) In partial step $z$-coordinate (\np{ln\_zps}=.true.), tracers in horizontally 
@@ -507,6 +504,6 @@
 \begin{equation} \label{Eq_tra_ldf_iso}
 \begin{split}
- D_T^{lT} = \frac{1}{b_T}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left( 
-	  \frac{e_{2u}\;e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T]
+ D_T^{lT} = \frac{1}{b_t}   & \left\{   \,\;\delta_i \left[   A_u^{lT}  \left( 
+	  \frac{e_{2u}\,e_{3u}}{e_{1u}} \,\delta_{i+1/2}[T]
 	- e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k}
                                                      \right)   \right]   \right.    \\ 
@@ -525,5 +522,5 @@
  \end{split}
  \end{equation}
-where $b_T= e_{1T}\,e_{2T}\,e_{3T}$  is the volume of $T$-cells, 
+where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells, 
 $r_1$ and $r_2$ are the slopes between the surface of computation 
 ($z$- or $s$-surfaces) and the surface along which the diffusion operator 
@@ -546,6 +543,6 @@
 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 
-developed by \cite{Griffies1998} which preserves both tracer and its variance 
+background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme 
+developed by \cite{Griffies_al_JPO98} which preserves both tracer and its variance 
 is currently been tested in \NEMO. It should be available in a forthcoming
 release.
@@ -563,6 +560,6 @@
 
 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 
+applying (\ref{Eq_tra_ldf_lap}) twice. The operator requires an additional assumption 
+on boundary conditions: both 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 
@@ -618,7 +615,7 @@
 \begin{equation} \label{Eq_tra_zdf}
 \begin{split}
-D^{vT}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right] 
+D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right] 
 \\
-D^{vS}_T &= \frac{1}{e_{3T}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right] 
+D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right] 
 \end{split}
 \end{equation}
@@ -645,6 +642,6 @@
 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 
+splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. 
+Namelist variables \np{lnn\_zdfexp} and \np{nn\_zdfexp} apply to both 
 tracers and dynamics. 
 
@@ -667,5 +664,5 @@
 enhance readability of the code. The two formulations are completely 
 equivalent; the forcing terms in trasbc are the surface fluxes divided by 
-the thickness of the top model layer. Following \citet{Roullet2000} the 
+the thickness of the top model layer. Following \citet{Roullet_Madec_JGR00} 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 
@@ -699,15 +696,15 @@
 \begin{equation} \label{Eq_tra_forcing}
 \begin{aligned}
- &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }-\frac{\text{EMP}\;\left. T 
-\right|_{k=1} }{e_{3T} }  & \\ 
+ &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3t} }-\frac{\textit{emp}\;\left. T 
+\right|_{k=1} }{e_{3t} }  & \\ 
 \\
-& F^S =\frac{\text{EMP}_S\;\left. S \right|_{k=1} }{e_{3T} }   &
+& F^S =\frac{\textit{emp}_S\;\left. S \right|_{k=1} }{e_{3t} }   &
  \end{aligned}
 \end{equation} 
-where EMP is the freshwater budget (evaporation minus precipitation 
+where \textit{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$_S\;.\left. S \right|_{k=1}$ 
+penetrates into the water column), the product $\textit{emp}_S\;.\left. S \right|_{k=1}$ 
 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 
@@ -722,21 +719,21 @@
 for temperature. The resulting forcing term for temperature is: 
 \begin{equation} \label{Eq_tra_forcing_q}
-F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }
+F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3t} }
 \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 
+definition of $\textit{emp}_S$ 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}. 
+is not exactly conserved (\citet{Roullet_Madec_JGR00}. 
 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 of EMP and S, is not compensated by the advection of fluid 
+the product of \textit{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 
+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 
@@ -753,5 +750,5 @@
 \label{TRA_qsr}
 %--------------------------------------------namqsr--------------------------------------------------------
-\namdisplay{namqsr}
+\namdisplay{namtra_qsr}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -773,5 +770,5 @@
 is discretized as follows:
 \begin{equation} \label{Eq_tra_qsr}
-\frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3T}} \delta_k \left[ I_w \right]
+\frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k} \equiv \frac{1}{\rho_o\, C_p\, e_{3t}} \delta_k \left[ I_w \right]
 \end{equation}
 
@@ -824,11 +821,11 @@
 \label{TRA_bbc}
 %--------------------------------------------nambbc--------------------------------------------------------
-\namdisplay{nambbc}
+\namdisplay{namtra_bbc}
 %--------------------------------------------------------------------------------------------------------------
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t] \label{Fig_geothermal}  \begin{center}
+\begin{figure}[!b] \label{Fig_geothermal}  \begin{center}
 \includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_geoth.pdf}
-\caption{Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OSD08}.
-It is inferred from the age of the sea floor and the formulae of \citet{Stein1992}.}
+\caption{Geothermal Heat flux (in $mW.m^{-2}$) used by \cite{Emile-Geay_Madec_OS09}.
+It is inferred from the age of the sea floor and the formulae of \citet{Stein_Stein_Nat92}.}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -840,9 +837,9 @@
 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 
+global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), 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) by a few Sverdrups  \citep{Emile-Geay_Madec_OSD08}. 
+Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}. 
 
 The presence or not of geothermal heating is controlled by the namelist 
@@ -860,5 +857,5 @@
 \label{TRA_bbl}
 %--------------------------------------------nambbl---------------------------------------------------------
-\namdisplay{nambbl}
+\namdisplay{namtra_bbl}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -876,5 +873,5 @@
 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. 
+a sill \citep{Willebrand_al_PO01}, and the thickness of the plume is not resolved. 
 
 The idea of the bottom boundary layer (BBL) parameterisation first introduced by 
@@ -960,6 +957,6 @@
 		{Tracer damping (\mdl{tradmp})}
 \label{TRA_dmp}
-%--------------------------------------------namtdp-----------------------------------------------------
-\namdisplay{namtdp}
+%--------------------------------------------namtra_dmp-------------------------------------------------
+\namdisplay{namtra_dmp}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -997,13 +994,19 @@
 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 
+\citep{Madec_al_JPO96} and in the equatorial region 
+\citep{Reverdin1991, Fujio1991, Marti_PhD92} 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}.
+structure in equilibrium with its physics. 
+The choice of the shape of the Newtonian damping is controlled by two 
+namelist parameters \np{nn\_zdmp}. The former allows to specified the 
+width of the equatorial band in which no damping is applied as well as a decrease 
+in vicinity of the coast and a damping everywhere in the Red and Med Seas, 
+whereas the latter set a damping acting in the mixed layer or not. 
+The time scale associated with the damping depends on the depth as
+a hyperbolic tangent, with \np{rn\_surf} as surface value, \np{rn\_bot} as 
+bottom value and a transition depth of \np{rn\_dep}.  
 
 The robust diagnostic method is very efficient in preventing temperature 
@@ -1013,5 +1016,5 @@
 by stabilising the water column too much.
 
-An example of the computation of $S_o$ for robust diagnostic experiments 
+An example of the computation of $\gamma$ for a robust diagnostic experiment 
 with the ORCA2 model is provided in the \mdl{tradmp} module 
 (subroutines \rou{dtacof} and \rou{cofdis} which compute the coefficient 
@@ -1036,5 +1039,5 @@
 T^{t+\Delta t} &= T^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_T^t 	\\
 \\
-T_f^t  \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2T^t+T^{t+\Delta t}} \right]
+T_f^t  \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2t^t+T^{t+\Delta t}} \right]
 \end{split}
 \end{equation} 
@@ -1069,49 +1072,54 @@
 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):
+particularly in the deep ocean. The ocean seawater volumic mass, $\rho$, 
+abusively called 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 
+\textit{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 anomaly, $d_a= \rho / \rho_o - 1$, 
+is computed, with $\rho_o$ a reference volumic mass. Called \textit{rau0} 
+in the code, $\rho_o$ is defined in \mdl{phycst}, and a value of $1,035~Kg/m^3$. 
+This is a sensible choice for the reference density used in a Boussinesq ocean 
+climate model, as, with the exception of only a small percentage of the ocean, 
+density in the World Ocean varies by no more than 2$\%$ from $1,035~kg/m^3$ 
+\citep{Gill1982}.
+
+The default option (namelist parameter \np{nn\_eos}=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.
+With such an equation of state there is no longer a distinction between 
+\textit{in situ} and \textit{potential} density and both cabbeling and thermobaric
+effects are removed.
+Two linear formulations are available: a function of $T$ only (\np{nn\_eos}=1) 
+and a function of both $T$ and $S$ (\np{nn\_eos}=2):
 \begin{equation} \label{Eq_tra_eos_linear}
-\begin{aligned}
- d(T)    &= {\rho (T)} / {\rho _0 } &&= 1.028 - \alpha \;T     \\ 
- d(T,S) &= {\rho (T,S)}                &&= \ \ \ \beta \;S - \alpha \;T 
-\end{aligned}
+\begin{split}
+  d_a(T)       &=  \rho (T)      /  \rho_o   - 1     =  \  0.0285         -  \alpha   \;T     \\ 
+  d_a(T,S)    &=  \rho (T,S)   /  \rho_o   - 1     =  \  \beta \; S       -  \alpha   \;T    
+\end{split}
 \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 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 
+($\alpha$ and $\beta$ can be modified through the \np{rn\_alpha} and 
+\np{rn\_beta} namelist parameters). Note that when $d_a$ is a function 
+of $T$ only (\np{nn\_eos}=1), the salinity is a passive tracer and can be 
 used as such.
 
@@ -1119,5 +1127,5 @@
 %        Brunt-Vais\"{a}l\"{a} Frequency
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{neos} = 0, 1 or 2)}
+\subsection{Brunt-Vais\"{a}l\"{a} Frequency (\np{nn\_eos} = 0, 1 or 2)}
 \label{TRA_bn2}
 
@@ -1128,7 +1136,7 @@
  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} 
+ depends on the type of equation of state used (\np{nn\_eos} namelist parameter).
+
+For \np{nn\_eos}=0 (\citet{JackMcD1995} equation of state), the \citet{McDougall1987} 
 polynomial expression is used (with the pressure in decibar approximated by 
 the depth in meters): 
@@ -1137,9 +1145,7 @@
 	   \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. 
+where $\alpha$ and $\beta$ are the thermal and haline expansion coefficients. 
+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 
@@ -1147,5 +1153,5 @@
 then averaged to $w$-points.
 
-When a linear equation of state is used (\np{neos}=1 or 2, 
+When a linear equation of state is used (\np{nn\_eos}=1 or 2, 
 \eqref{Eq_tra_bn2} reduces to:
 \begin{equation} \label{Eq_tra_bn2_linear}
@@ -1158,5 +1164,5 @@
 %        Specific Heat
 % -------------------------------------------------------------------------------------------------------------
-\subsection   [Specific Heat (\textit{phycst})]
+\subsection   	[Specific Heat (\textit{phycst})]
 			{Specific Heat (\mdl{phycst})}
 \label{TRA_adv_ldf}
@@ -1171,14 +1177,10 @@
 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 (\textit{ocfzpt})]
-			{Freezing Point of Seawater (\mdl{ocfzpt})}
+\subsection   [Freezing Point of Seawater]
+			{Freezing Point of Seawater}
 \label{TRA_fzp}
 
@@ -1194,14 +1196,7 @@
 \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}
-%%%
+terms in \eqref{Eq_tra_eos_fzp} (last term) have been dropped. The freezing
+point is computed through \textit{tfreez}, a \textsc{Fortran} function that can be found 
+in \mdl{eosbn2}.  
 
 % ================================================================
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 1831)
@@ -7,6 +7,8 @@
 
 %gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
-\gmcomment{Steven remark (not taken into account : problem here with turbulent vs turbulence.  I've changed "turbulent closure" to "turbulence closure", "turbulent mixing length" to "turbulence mixing length", but I've left "turbulent kinetic energy" alone - though I think it is an historical abberation!
-Gurvan :  I kept "turbulent closure etc "...}
+
+
+\newpage
+$\ $\newline    % force a new ligne
 
 
@@ -74,5 +76,5 @@
 
 %--------------------------------------------namric---------------------------------------------------------
-\namdisplay{namric}
+\namdisplay{namzdf_ric}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -84,5 +86,5 @@
 growth of Kelvin-Helmholtz like instabilities leads to a dependency between the 
 vertical eddy coefficients and the local Richardson number ($i.e.$ the 
-ratio of stratification to vertical shear). Following \citet{PacPhil1981}, the following 
+ratio of stratification to vertical shear). Following \citet{Pacanowski_Philander_JPO81}, the following 
 formulation has been implemented:
 \begin{equation} \label{Eq_zdfric}
@@ -107,22 +109,22 @@
 \label{ZDF_tke}
 
-%--------------------------------------------namtke---------------------------------------------------------
-\namdisplay{namtke}
+%--------------------------------------------namzdf_tke--------------------------------------------------
+\namdisplay{namzdf_tke}
 %--------------------------------------------------------------------------------------------------------------
 
 The vertical eddy viscosity and diffusivity coefficients are computed from a TKE 
-turbulent closure model based on a prognostic equation for $\bar {e}$, the turbulent 
+turbulent closure model based on a prognostic equation for $\bar{e}$, the turbulent 
 kinetic energy, and a closure assumption for the turbulent length scales. This 
 turbulent closure model has been developed by \citet{Bougeault1989} in the 
 atmospheric case, adapted by \citet{Gaspar1990} for the oceanic case, and 
-embedded in OPA by \citet{Blanke1993} for equatorial Atlantic simulations. Since 
-then, significant modifications have been introduced by \citet{Madec1998} in both 
-the implementation and the formulation of the mixing length scale. The time 
-evolution of $\bar{e}$ is the result of the production of $\bar{e}$ through vertical 
-shear, its destruction through stratification, its vertical diffusion, and its dissipation 
-of \citet{Kolmogorov1942} type:
+embedded in OPA, the ancestor of NEMO, by \citet{Blanke1993} for equatorial Atlantic 
+simulations. Since then, significant modifications have been introduced by 
+\citet{Madec1998} in both the implementation and the formulation of the mixing 
+length scale. The time evolution of $\bar{e}$ is the result of the production of 
+$\bar{e}$ through vertical shear, its destruction through stratification, its vertical 
+diffusion, and its dissipation of \citet{Kolmogorov1942} type:
 \begin{equation} \label{Eq_zdftke_e}
 \frac{\partial \bar{e}}{\partial t} = 
-\frac{A^{vm}}{e_3 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
+\frac{A^{vm}}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
 							+\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
 -A^{vT}\,N^2
@@ -133,5 +135,5 @@
 \begin{equation} \label{Eq_zdftke_kz}
    \begin{split}
-         A^{vm} &= C_k\  l_k\  \sqrt {\bar{e}}  	\\
+         A^{vm} &= C_k\  l_k\  \sqrt {\bar{e}\; }  	\\
          A^{vT} &= A^{vm} / P_{rt}
    \end{split}
@@ -139,9 +141,9 @@
 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}), 
 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 
-$P_{rt} $ is the Prandtl number. The constants $C_k = \sqrt {2} /2$ and 
-$C_\epsilon = 0.1$ are designed to deal with vertical mixing at any depth 
-\citep{Gaspar1990}. They are set through namelist parameters \np{ediff} 
-and \np{ediss}. $P_{rt} $ can be set to unity or, following \citet{Blanke1993}, 
-be a function of the local Richardson number, $R_i $:
+$P_{rt}$ is the Prandtl number. The constants $C_k =  0.1$ and 
+$C_\epsilon = \sqrt {2} /2$  $\approx 0.7$ are designed to deal with vertical mixing 
+at any depth \citep{Gaspar1990}. They are set through namelist parameters 
+\np{nn\_ediff} and \np{nn\_ediss}. $P_{rt}$ can be set to unity or, following 
+\citet{Blanke1993}, be a function of the local Richardson number, $R_i$:
 \begin{align*} \label{Eq_prt}
 P_{rt} = \begin{cases}
@@ -151,23 +153,22 @@
             \end{cases}
 \end{align*}
-Note that a horizontal Shapiro filter can optionally be applied to $R_i$. 
-However it is an obsolescent option that is not recommended.  
-The choice of $P_{rt} $ is controlled by the \np{npdl} namelist parameter.
+The choice of $P_{rt}$ is controlled by the \np{nn\_pdl} namelist parameter.
 
 For computational efficiency, the original formulation of the turbulent length 
 scales proposed by \citet{Gaspar1990} has been simplified. Four formulations 
-are proposed, the choice of which is controlled by the \np{nmxl} namelist 
+are proposed, the choice of which is controlled by the \np{nn\_mxl} namelist 
 parameter. The first two are based on the following first order approximation 
 \citep{Blanke1993}:
 \begin{equation} \label{Eq_tke_mxl0_1}
-l_k = l_\epsilon = \sqrt {2 \bar e} / N
-\end{equation}
-which is valid in a stable stratified region with constant values of the brunt-
+l_k = l_\epsilon = \sqrt {2 \bar{e}\; } / N
+\end{equation}
+which is valid in a stable stratified region with constant values of the Brunt-
 Vais\"{a}l\"{a} frequency. The resulting length scale is bounded by the distance 
-to the surface or to the bottom (\np{nmxl}=0) or by the local vertical scale factor (\np{nmxl}=1). \citet{Blanke1993} notice that this simplification has two major 
+to the surface or to the bottom (\np{nn\_mxl}=0) or by the local vertical scale factor 
+(\np{nn\_mxl}=1). \citet{Blanke1993} notice that this simplification has two major 
 drawbacks: it makes no sense for locally unstable stratification and the 
 computation no longer uses all the information contained in the vertical density 
 profile. To overcome these drawbacks, \citet{Madec1998} introduces the 
-\np{nmxl}=2 or 3 cases, which add an extra assumption concerning the vertical 
+\np{nn\_mxl}=2 or 3 cases, which add an extra assumption concerning the vertical 
 gradient of the computed length scale. So, the length scales are first evaluated 
 as in \eqref{Eq_tke_mxl0_1} and then bounded such that:
@@ -185,6 +186,5 @@
 constraint, we introduce two additional length scales: $l_{up}$ and $l_{dwn}$, 
 the upward and downward length scales, and evaluate the dissipation and 
-mixing turbulent length scales as (and note that here we use numerical 
-indexing):
+mixing length scales as (and note that here we use numerical indexing):
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \label{Fig_mixing_length}  \begin{center}
@@ -196,16 +196,16 @@
 \begin{equation} \label{Eq_tke_mxl2}
 \begin{aligned}
- l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3T}^{(k)}\ \ \ \;  \right)
+ l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
     \quad &\text{ from $k=1$ to $jpk$ }\ \\
- l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3T}^{(k-1)}  \right)   
+ l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)   
     \quad &\text{ from $k=jpk$ to $1$ }\ \\
 \end{aligned}
 \end{equation}
 where $l^{(k)}$ is computed using \eqref{Eq_tke_mxl0_1}, 
-$i.e.$ $l^{(k)} = \sqrt {2 \bar e^{(k)} / N^{(k)} }$.
+$i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
 
 In the \np{nmxl}=2 case, the dissipation and mixing length scales take the same 
 value: $ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the 
-\np{nmxl}=2 case, the dissipation and mixing length scales are give 
+\np{nmxl}=2 case, the dissipation and mixing turbulent length scales are give 
 as in \citet{Gaspar1990}:
 \begin{equation} \label{Eq_tke_mxl_gaspar}
@@ -217,18 +217,135 @@
 
 At the sea surface the value of $\bar{e}$ is prescribed from the wind 
-stress field: $\bar{e}=ebb\;\left| \tau \right|$ ($ebb=60$ by default) 
-with a minimal threshold of $emin0=10^{-4}~m^2.s^{-2}$ (namelist
+stress field: $\bar{e}=rn\_ebb\;\left| \tau \right|$ (\np{rn\_ebb}=60 by default) 
+with a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist
 parameters). Its value at the bottom of the ocean is assumed to be 
 equal to the value of the level just above. The time integration of the 
 $\bar{e}$ equation may formally lead to negative values because the 
 numerical scheme does not ensure its positivity. To overcome this 
-problem, a cut-off in the minimum value of $\bar{e}$ is used. Following 
-\citet{Gaspar1990}, the cut-off value is set to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. 
-This allows the subsequent formulations to match that of\citet{Gargett1984} 
-for the diffusion in the thermocline and deep ocean :  $(A^{vT} = 10^{-3} / N)$. 
+problem, a cut-off in the minimum value of $\bar{e}$ is used (\np{rn\_emin} 
+namelist parameter). Following \citet{Gaspar1990}, the cut-off value is set 
+to $\sqrt{2}/2~10^{-6}~m^2.s^{-2}$. This allows the subsequent formulations 
+to match that of \citet{Gargett1984} for the diffusion in the thermocline and 
+deep ocean :  $(A^{vT} = 10^{-3} / N)$. 
 In addition, a cut-off is applied on $A^{vm}$ and $A^{vT}$ to avoid numerical 
 instabilities associated with too weak vertical diffusion. They must be 
 specified at least larger than the molecular values, and are set through 
-\textit{avm0} and \textit{avt0} (\textbf{namelist} parameters).
+\np{avm0} and \np{avt0} (namelist parameters).
+
+% -------------------------------------------------------------------------------------------------------------
+%        TKE Turbulent Closure Scheme : new organization to energetic considerations
+% -------------------------------------------------------------------------------------------------------------
+\subsection{TKE Turbulent Closure Scheme - time integration (\key{zdftke} and (\key{zdftke2})}
+\label{ZDF_tke2}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_TKE_time_scheme}  \begin{center}
+\includegraphics[width=1.00\textwidth]{./TexFiles/Figures/Fig_ZDF_TKE_time_scheme.pdf}
+\caption {Illustration of the TKE time integration and its links to the momentum and tracer time integration. }
+\end{center}  
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The production of turbulence by vertical shear (the first term of the right hand side 
+of \eqref{Eq_zdftke_e}) should balance the loss of kinetic energy associated with
+the vertical momentum diffusion (first line in \eqref{Eq_PE_zdf}). 
+The total loss of kinetic energy (in 1D for the demonstration) 
+due to this term is obtained by multiply this quantity by $u^n$ and verticaly integrating:   
+
+\begin{equation} \label{Eq_energ1}
+\int_{k_b}^{k_s} {u^t \frac{1}{e_3}
+                                 \frac{\partial   } 
+                                        {\partial k} \left( \frac{A^{vm}}{e_3} 
+                                                                 \frac{\partial{u^{t+1}}}
+                                                                        {\partial k           }     \right) \; e_3 \; dk }
+= \left[  \frac{u^t}{e_3}   A^{vm} \frac{\partial{u^{t+1}}}{\partial k} \right]_{k_b}^{k_s}
+ - \int_{k_b}^{k_s}{\frac{A^{vm}}{{e_3}}\frac{\partial{u^t}}{\partial k}\frac{\partial{u^{t+1}}}{\partial k}} \ dk
+\end{equation}
+
+The first term of the right hand side of \eqref{Eq_energ1} represents the kinetic 
+energy transfer at the surface (atmospheric forcing) and at the bottom (friction effect). 
+The second term is always negative and have to balance the term of \eqref{Eq_zdftke_e} 
+previously identified.
+
+The sink term (possibly a source term in statically unstable situations) of turbulence 
+by buoyancy (second term of the right hand side of \eqref{Eq_zdftke_e}) must balance
+the source of potential energy associated with the vertical diffusion 
+in the density equation (second line in \eqref{Eq_PE_zdf}). The source of potential 
+energy (in 1D for the demonstration) due to this term is obtained by multiply this quantity 
+by $gz{\rho_r}^{-1}$ and verticaly integrating:
+
+\begin{equation} \label{Eq_energ2}
+\begin{aligned}
+\int_{k_b}^{k_s}{\frac{g\;z}{e_3} \frac{\partial }{\partial k}
+                        \left( \frac{A^{vT}}{e_3} 
+                        \frac{\partial{\rho^{t+1}}}{\partial k}   \right)} \; e_3 \; dk
+=\left[  g\;z \frac{A^{vT}}{e_3}
+                 \frac{\partial{\rho^{t+1}}}{\partial k} \right]_{k_b}^{k_s}  
+- \int_{k_b}^{k_s}{  \frac{A^{vT}}{e_3} g \frac{\partial{\rho^{t+1}}}{\partial k}} \; dk\\
+\\
+= - \left[  z\,A^{vT} {N_{t+1}}^2 \right]_{k_b}^{k_s}
++ \int_{k_b}^{k_s}{  \rho^{t+1} \; A^{vT}{N_{t+1}}^2 \; e_3 \; dk  }\\
+\end{aligned}
+\end{equation}
+where $N^2_{t+1}$ is  the Brunt-Vaissala frequency at $t+1$ 
+and noting that $\frac{\partial z}{\partial k} = e_3$.
+
+The first term is always zero because the Brunt Vaissala frequency is set to zero at the 
+surface and the bottom. The second term is of opposite sign than the buoyancy term
+identified previously.
+
+Under these energetic considerations, \eqref{Eq_zdftke_e} have to be written like this 
+to be consistant: 
+
+\begin{equation} \label{Eq_zdftke_ene}
+\frac{\partial \bar{e}}{\partial t} =
+\frac{A^{vm}}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u^a}{\partial k}} \right)\left( {\frac{\partial u^n}{\partial k}} \right)
+                                                        +\left( {\frac{\partial v^a}{\partial k}} \right)\left( {\frac{\partial v^n}{\partial k}} \right)
+} \right]-A^{vT}\,{N_n}^2+...
+\end{equation}
+
+Note that during a time step, the equation \eqref{Eq_zdftke_e} is resolved before those 
+of momentum and density. So, the indice "a" (after) become "n" (now) and the indice "n" 
+(now) become "b" (before).
+
+Moreover, the vertical shear have to be multiply by the appropriate viscosity for numerical 
+stability. Thus, the vertical shear at U-point have to be multiply by the viscosity avmu and 
+the vertical shear at V-point have to be multiply by the viscosity avmv. Next, these two 
+quantities are averaged to obtain a production term by vertical shear at W-point : 
+
+\begin{equation} \label{Eq_zdftke_ene2}
+\frac{\partial \bar{e}}{\partial t} =
+\frac{1}{{e_3}^2 }\;\left[ {  
+    A^{vmu}\left({\frac{\partial u^a}{\partial k}} \right)\left( {\frac{\partial u^n}{\partial k}} \right)
+   +A^{vmv}\left({\frac{\partial v^a}{\partial k}} \right)\left( {\frac{\partial v^n}{\partial k}} \right)
+                           } \right]-A^{vT}\,{N_n}^2+...
+\end{equation} 
+
+The TKE equation is resolved before the mixing length, the viscosity and diffusivity. Two tabs 
+are then declared : dissl (dissipation length) (Remark : it's not only the dissipation lenght, 
+it's the root of the TKE divided by the dissipation lenght) and avmt (viscosity at the points T) 
+used for the vertical diffusion of the TKE. 
+
+This new organization needs also a reorganization of the routine step.F90 (controled by 
+the key \key{ene\_cons}). The bigger change is the estimation of the Brunt-Vaissala 
+frequency at "n" instead of "b". Moreover for energetic considerations, the call of tranxt.F90 
+is done after the density update. On the contrary, the density is updated with scalars fields 
+filtered by the Asselin filter. 
+
+This new organisation of the routine zdftke force to save five three dimensionnal tabs in 
+the restart : avmu, avmv, avt, avmt and dissl are needed to calculate $e_n$. At the end 
+of the run (time step = nitend), the alternative is to save only $e_n$ estimated at the 
+following time step (nitend+1). The next run using this restart file, the mixing length 
+and turbulents coefficients are directly calculated using $e_n$. It is the same thing 
+for the intermediate restart.
+
+
+%%GM for figure of the time scheme:
+\begin{equation} 
+  \rho^{t+1} \; A^{vT}{N_{t+1}}^2 \; dk  \\
+\end{equation}
+
+%%end GM 
+
 
 % -------------------------------------------------------------------------------------------------------------
@@ -239,9 +356,8 @@
 
 %--------------------------------------------namkpp--------------------------------------------------------
-\namdisplay{namkpp}
-%--------------------------------------------------------------------------------------------------------------
-
-The K-Profile Parametrization (KKP) developed by \cite{Large_al_RG94} has been 
-implemented in \NEMO by J. Chanut (PhD reference to be added here!).
+\namdisplay{namzdf_kpp}
+%--------------------------------------------------------------------------------------------------------------
+
+The KKP scheme has been implemented by J. Chanut ...
 
 \colorbox{yellow}{Add a description of KPP here.}
@@ -274,6 +390,6 @@
 \label{ZDF_npc}
 
-%--------------------------------------------namnpc--------------------------------------------------------
-\namdisplay{namnpc}
+%--------------------------------------------namzdf--------------------------------------------------------
+\namdisplay{namzdf}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -298,5 +414,5 @@
 the statically unstable portion of the water column, but only until the density 
 structure becomes neutrally stable ($i.e.$ until the mixed portion of the water 
-column has \textit{exactly} the density of the water just below) \citep{Madec1991a}. 
+column has \textit{exactly} the density of the water just below) \citep{Madec_al_JPO91}. 
 The associated algorithm is an iterative process used in the following way 
 (Fig. \ref{Fig_npc}): starting from the top of the ocean, the first instability is 
@@ -320,5 +436,5 @@
 convective algorithm has been proved successful in studies of the deep 
 water formation in the north-western Mediterranean Sea 
-\citep{Madec1991a, Madec1991b, Madec1991c}.
+\citep{Madec_al_JPO91, Madec_al_DAO91, Madec_Crepon_Bk91}.
 
 Note that in the current implementation of this algorithm presents several 
@@ -341,5 +457,5 @@
 after a static adjustment. In that case, we recommend the addition of momentum 
 mixing in a manner that mimics the mixing in temperature and salinity 
-\citep{Speich1992, Speich1996}.
+\citep{Speich_PhD92, Speich_al_JPO96}.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -357,5 +473,5 @@
 In this case, the vertical eddy mixing coefficients are assigned very large values 
 (a typical value is $10\;m^2s^{-1})$ in regions where the stratification is unstable 
-($i.e.$ when the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar1997, Lazar1999}. 
+($i.e.$ when the Brunt-Vais\"{a}l\"{a} frequency is negative) \citep{Lazar_PhD97, Lazar_al_JPO99}. 
 This is done either on tracers only (\np{n\_evdm}=0) or on both momentum and 
 tracers (\np{n\_evdm}=1).
@@ -405,5 +521,5 @@
 
 %-------------------------------------------namddm--------------------------------------------------------
-\namdisplay{namddm}
+\namdisplay{namzdf_ddm}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -422,6 +538,5 @@
 \end{align*}
 where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection, 
-and $o$ by processes other than double diffusion. The rates of double-diffusive mixing 
-depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 
+and $o$ by processes other than double diffusion. The rates of double-diffusive mixing depend on the buoyancy ratio $R_\rho = \alpha \partial_z T / \beta \partial_z S$, 
 where $\alpha$ and $\beta$ are coefficients of thermal expansion and saline 
 contraction (see \S\ref{TRA_eos}). To represent mixing of $S$ and $T$ by salt 
@@ -454,6 +569,5 @@
 we adopt $R_c = 1.6$, $n = 6$, and $A^{\ast v} = 10^{-4}~m^2.s^{-1}$.
 
-To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested 
-by Federov (1988) is used: 
+To represent mixing of S and T by diffusive layering,  the diapycnal diffusivities suggested by Federov (1988) is used: 
 \begin{align} 	\label{Eq_zdfddm_d}
 A_d^{vT} &= 	\begin{cases}
@@ -486,5 +600,5 @@
 
 %--------------------------------------------nambfr--------------------------------------------------------
-\namdisplay{nambfr}
+\namdisplay{namzdf_bfr}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -536,5 +650,5 @@
 is generally estimated by setting a typical decay time $\tau$ in the deep ocean, 
 and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted 
-values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly1984}. 
+values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}. 
 A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used 
 in quasi-geostrophic models. One may consider the linear friction as an 
@@ -585,5 +699,5 @@
 breaking and other short time scale currents. A typical value of the drag 
 coefficient is $C_D = 10^{-3} $. As an example, the CME experiment 
-\citep{Treguier1992} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$, 
+\citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$, 
 while the FRAM experiment \citep{Killworth1992} uses $e_b =0$ 
 and $e_b =2.5\;\;10^{-3}$m$^2$.s$^{-2}$. The FRAM choices have been 
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Introduction.tex
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Introduction.tex	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Introduction.tex	(revision 1831)
@@ -72,5 +72,5 @@
 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{Large_al_RG94}, or \citet{PacPhil1981} mixing 
+the coefficients with \citet{Blanke1993}, \citet{Large_al_RG94}, or \citet{Pacanowski_Philander_JPO81} mixing 
 schemes.
 
@@ -123,5 +123,5 @@
 
 OPA, like all research tools, is in perpetual evolution. The present document describes 
-the OPA version include in the release 3.0 of NEMO. This release differs significantly
+the OPA version include in the release 3.2 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 
@@ -133,7 +133,7 @@
 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  
+(4) more choices for the treatment of the free surface: full explicit, split-explicit and filtered.
+(5) suppression of the rigid-lid option;\\
+(6) 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; \\
@@ -141,10 +141,10 @@
 (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. \\
+(10) rewriting of the I/O management with the use of an I/O server; \\
+(11) generalized ocean-ice-atmosphere-CO2 coupling interface, interfaced with OASIS 3 coupler. \\
 (12) surface module (SBC) that simplify the way the ocean is forced and include two
-bulk formulea (CLIO and CORE)\\
+bulk formulea (CLIO and CORE) and which includes an on-the-fly interpolation of input forcing fields\\
 (13) introduction of LIM 3, the new Louvain-la-Neuve sea-ice model (C-grid rheology and
-new thermodynamics including bulk ice salinity) \citep{Vancoppenolle_al_OM08}
+new thermodynamics including bulk ice salinity) \citep{Vancoppenolle_al_OM09a, Vancoppenolle_al_OM09b}
 
 In addition, several minor modifications in the coding have been introduced with the constant concern of improving performance on both scalar and vector computers. 
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynadv
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynadv	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_dynadv    !   formulation of the momentum advection
-!-----------------------------------------------------------------------
-   ln_dynadv_vec = .true.  !  vector form (T) or flux form (F)  
-   ln_dynadv_cen2= .false. !  flux form - 2nd order centered scheme
-   ln_dynadv_ubs = .false. !  flux form - 3rd order UBS      scheme 
-/  
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynhpg
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynhpg	(revision 1830)
+++ 	(revision )
@@ -1,12 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_dynhpg    !   Hydrostatic pressure gradient option
-!-----------------------------------------------------------------------
-   ln_hpg_zco  = .false.   !  z-coordinate - full steps                   
-   ln_hpg_zps  = .true.    !  z-coordinate - partial steps (interpolation)
-   ln_hpg_sco  = .false.   !  s-coordinate (standard jacobian formulation)
-   ln_hpg_hel  = .false.   !  s-coordinate (helsinki modification)
-   ln_hpg_wdj  = .false.   !  s-coordinate (weighted density jacobian)
-   ln_hpg_djc  = .false.   !  s-coordinate (Density Jacobian with Cubic polynomial)
-   ln_hpg_rot  = .false.   !  s-coordinate (ROTated axes scheme)
-   gamm        = 0.e0      !  weighting coefficient (wdj scheme)
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynldf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynldf	(revision 1830)
+++ 	(revision )
@@ -1,14 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_dynldf    !   lateral diffusion on momentum
-!-----------------------------------------------------------------------
-!                               !  Type of the operator : 
-   ln_dynldf_lap    =  .true.   !     laplacian operator         
-   ln_dynldf_bilap  =  .false.  !     bilaplacian operator    
-!                               !  Direction of action  : 
-   ln_dynldf_level  =  .false.  !     iso-level               
-   ln_dynldf_hor    =  .true.   !     horizontal (geopotential)        (require "key_ldfslp" in s-coord.)
-   ln_dynldf_iso    =  .false.  !     iso-neutral                      (require "key_ldfslp")
-                                !  Coefficient
-   ahm0    = 40000.             !     horizontal eddy viscosity   [m2/s]
-   ahmb0   =     0.             !     background eddy viscosity for ldf_iso [m2/s]
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynspg
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynspg	(revision 1830)
+++ 	(revision )
@@ -1,9 +1,0 @@
-!-----------------------------------------------------------------------
-!nam_dynspg    !   surface pressure gradient   (CPP key only)
-!-----------------------------------------------------------------------
-!                          !  Non-linear free surface                   ("key_vvl")
-!                          !  explicit free surface                     ("key_dynspg_exp")
-!                          !  filtered free surface                     ("key_dynspg_flt")
-!                          !  split-explicit free surface               ("key_dynspg_ts")
-!                          !  rigid-lid                                 ("key_dynspg_rl")
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynvor
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_dynvor	(revision 1830)
+++ 	(revision )
@@ -1,8 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_dynvor    !   option of physics/algorithm (not control by CPP keys)
-!-----------------------------------------------------------------------
-   ln_dynvor_ene = .false. !  enstrophy conserving scheme  
-   ln_dynvor_ens = .true.  !  energy conserving scheme    
-   ln_dynvor_mix = .false. !  mixed scheme               
-   ln_dynvor_een = .false. !  energy & enstrophy scheme  
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_lbc
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_lbc	(revision 1830)
+++ 	(revision )
@@ -1,8 +1,0 @@
-!-----------------------------------------------------------------------
-&namlbc        !   lateral momentum boundary condition
-!-----------------------------------------------------------------------
-   shlat       =    2.     !      shlat = 0 : free slip
-                           !  0 < shlat < 2 : partial slip
-                           !      shlat = 2 : no slip
-                           !  2 < shlat     : strong slip
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_mpp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_mpp	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_mpp      !   Massively Parallel Processing                         ("key_mpp_mpi)
-!-----------------------------------------------------------------------
-   c_mpi_send =  'S'       !  mpi send/recieve type   ='S', 'B', or 'I' for standard send,
-                           !  buffer blocking send or immediate non-blocking sends, resp.
-   nn_buffer  =   0        !  size in bytes of exported buffer ('B' case), 0 no exportation
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_mpp_dyndist
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_mpp_dyndist	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_mpp_dyndist !   Massively Parallel Distribution                    ("key_agrif" && "key_mpp_dyndist")
-!-----------------------------------------------------------------------
-   jpni       =    1       !  jpni   number of processors following i
-   jpnj       =    1       !  jpnj   number of processors following j
-   jpnij      =    1       !  jpnij  number of local domains
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_traadv
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_traadv	(revision 1830)
+++ 	(revision )
@@ -1,9 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_traadv    !   advection scheme for tracer 
-!-----------------------------------------------------------------------
-   ln_traadv_cen2   =  .true.   !  2nd order centered scheme   
-   ln_traadv_tvd    =  .false.  !  TVD scheme                
-   ln_traadv_muscl  =  .false.  !  MUSCL scheme             
-   ln_traadv_muscl2 =  .false.  !  MUSCL2 scheme + cen2 at boundaries  
-   ln_traadv_ubs    =  .false.  !  UBS scheme                 
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_traldf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_traldf	(revision 1830)
+++ 	(revision )
@@ -1,15 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_traldf    !   lateral diffusion scheme for tracer 
-!-----------------------------------------------------------------------
-!                               !  Type of the operator : 
-   ln_traldf_lap    =  .true.   !     laplacian operator       
-   ln_traldf_bilap  =  .false.  !     bilaplacian operator     
-                                !  Direction of action  :
-   ln_traldf_level  =  .false.  !     iso-level                
-   ln_traldf_hor    =  .false.  !     horizontal (geopotential)         (require "key_ldfslp" when ln_sco=T)
-   ln_traldf_iso    =  .true.   !     iso-neutral                       (require "key_ldfslp")
-!                               !  Coefficient
-   aht0        =  2000.         !     horizontal eddy diffusivity for tracers [m2/s]
-   ahtb0       =     0.         !     background eddy diffusivity for ldf_iso [m2/s]
-   aeiv0       =  2000.         !     eddy induced velocity coefficient [m2/s]    (require "key_traldf_eiv")
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_zgr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_zgr	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_zgr       !   vertical coordinate
-!-----------------------------------------------------------------------
-   ln_zco      = .false.   !  z-coordinate - full    steps   (T/F)      ("key_zco" may also be defined)
-   ln_zps      = .true.    !  z-coordinate - partial steps   (T/F)
-   ln_sco      = .false.   !  s- or hybrid z-s-coordinate    (T/F)
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_zgr_sco
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nam_zgr_sco	(revision 1830)
+++ 	(revision )
@@ -1,9 +1,0 @@
-!-----------------------------------------------------------------------
-&nam_zgr_sco   !   s-coordinate or hybrid z-s-coordinate
-!-----------------------------------------------------------------------
-   sbot_min    =  300.     !  minimum depth of s-bottom surface (>0) (m)
-   sbot_max    = 5250.     !  maximum depth of s-bottom surface (= ocean depth) (>0) (m)
-   theta       =    6.0    !  surface control parameter (0<=theta<=20)
-   thetb       =    0.75   !  bottom control parameter  (0<=thetb<= 1)
-   r_max       =    0.15   !  maximum cut-off r-value allowed (0<r_max<1)
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namagrif
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namagrif	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namagrif	(revision 1831)
@@ -1,8 +1,8 @@
 !-----------------------------------------------------------------------
-&namagrif      !                                                        ("key_agrif")
+&namagrif      !  AGRIF zoom                                            ("key_agrif")
 !-----------------------------------------------------------------------
-    nbclineupdate = 3      !  baroclinic update frequency
+    nn_cln_update = 3      !  baroclinic update frequency
     ln_spc_dyn    = .true. !  use 0 as special value for dynamics
-    visc_tra      = 2880.  !  viscosity coeeficient for tracers sponge layer
-    visc_dyn      = 2880.  !  viscosity coeeficient for dynamics sponge layer
+    rn_sponge_tra = 2880.  !  coefficient for tracer   sponge layer [s]
+    rn_sponge_dyn = 2880.  !  coefficient for dynamics sponge layer [s]
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbc
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbc	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbc	(revision 1831)
@@ -2,7 +2,7 @@
 &nambbc        !   bottom temperature boundary condition
 !-----------------------------------------------------------------------
-   ngeo_flux   =    2      !  geothermal heat flux = 0 no flux considered 
-                           !                       = 1 constant flux
-                           !                       = 2 variable flux (read in geothermal_heating.nc in mW/m2) 
-   ngeo_flux_const = 86.4e-3  !  Constant value of geothermal heat flux [W/m2]
+   nn_geoflx   =    2      !  geothermal heat flux: = 0 no flux 
+                           !     = 1 constant flux
+                           !     = 2 variable flux (read in geothermal_heating.nc in mW/m2) 
+   rn_geoflx_cst = 86.4e-3 !  Constant value of geothermal heat flux [W/m2]
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbl
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbl	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambbl	(revision 1831)
@@ -4,4 +4,4 @@
 !                          !  diffusive bbl                             ("key_trabbl")
 !                          !  advective bbl                             ("key_trabbl_adv")
-   atrbbl      =  10000.   !  lateral mixing coefficient in the bbl  [m2/s]
+   rn_ahtbbl   =  10000.   !  lateral mixing coefficient in the bbl  [m2/s]
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy	(revision 1831)
@@ -1,4 +1,4 @@
 !-----------------------------------------------------------------------
-&nambdy        !  unstructured open boundaries parameters               ("key_bdy")
+&nambdy        !  unstructured open boundaries                          ("key_bdy")
 !-----------------------------------------------------------------------
     filbdy_mask    =  ''                  !  name of mask file (if ln_bdy_mask=.TRUE.)
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy_tide
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy_tide	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambdy_tide	(revision 1831)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&nambdy_tide     ! tidal forcing at unstructured boundaries              
+!-----------------------------------------------------------------------
+    filtide      = 'bdytide_'           !  file name root of tidal forcing files
+    tide_cpt     = 'M2','S1'            !  names of tidal components used
+    tide_speed   = 28.984106, 15.000001 !  phase speeds of tidal components (deg/hour)
+    ln_tide_date = .false.              !  adjust tidal harmonics for start date of run
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambfr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambfr	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambfr	(revision 1831)
@@ -2,8 +2,8 @@
 &nambfr        !   bottom friction
 !-----------------------------------------------------------------------
-   nbotfr      =    1      !  type of bottom friction :   = 0 : no   slip,  = 2 : nonlinear friction
+   nn_bfr      =    1      !  type of bottom friction :   = 0 : no   slip,  = 2 : nonlinear friction
                            !                              = 3 : free slip,  = 1 :    linear friction
-   bfri1       =    4.e-4  !  bottom drag coefficient (linear case)
-   bfri2       =    1.e-3  !  bottom drag coefficient (non linear case)
-   bfeb2       =    2.5e-3 !  bottom turbulent kinetic energy background  (m^2/s^2)
+   rn_bfri1    =    4.e-4  !  bottom drag coefficient (linear case)
+   rn_bfri2    =    1.e-3  !  bottom drag coefficient (non linear case)
+   rn_bfeb2    =    2.5e-3 !  bottom turbulent kinetic energy background  (m^2/s^2)
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namcla
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namcla	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namcla	(revision 1831)
@@ -2,4 +2,4 @@
 &namcla        !   cross land advection
 !-----------------------------------------------------------------------
-   n_cla       =    1      !  advection between 2 ocean pts separates by land
+   nn_cla      =    0      !  advection between 2 ocean pts separates by land
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namctl
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namctl	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namctl	(revision 1831)
@@ -1,16 +1,16 @@
 !-----------------------------------------------------------------------
-&namctl       !   Control prints & Benchmark
+&namctl        !   Control prints & Benchmark
 !-----------------------------------------------------------------------
-   ln_ctl     = .false.    !  trends control print (expensive!)
-   nprint     =    0       !  level of print (0 no extra print)
-   nictls     =    0       !  start i indice of control sum (use to compare mono versus
-   nictle     =    0       !  end   i indice of control sum        multi processor runs
-   njctls     =    0       !  start j indice of control               over a subdomain)
-   njctle     =    0       !  end   j indice of control 
-   isplt      =    1       !  number of processors in i-direction
-   jsplt      =    1       !  number of processors in j-direction
-   nbench     =    0       !  Bench mode (1/0): CAUTION use zero except for bench
+   nn_print    =    0      !  level of print (0 no extra print)
+   nn_bench    =    0      !  Bench mode (1/0): CAUTION use zero except for bench
                            !     (no physical validity of the results)
-   nbit_cmp   =    0       !  bit comparison mode (1/0): CAUTION use zero except for test
+   nn_bit_cmp  =    0      !  bit comparison mode (1/0): CAUTION use zero except for test
                            !     of comparison between single and multiple processor runs
+   ln_ctl      = .false.   !  trends control print (expensive!)
+   nn_ictls    =    0           !  start i indice of control sum (use to compare mono versus
+   nn_ictle    =    0           !  end   i indice of control sum        multi processor runs
+   nn_jctls    =    0           !  start j indice of control               over a subdomain)
+   nn_jctle    =    0           !  end   j indice of control
+   nn_isplt    =    1           !  number of processors in i-direction
+   nn_jsplt    =    1           !  number of processors in j-direction
 /
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namddm
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namddm	(revision 1830)
+++ 	(revision )
@@ -1,6 +1,0 @@
-!-----------------------------------------------------------------------
-&namddm        !   double diffusive mixing parameterization                  ("key_zdfddm")
-!-----------------------------------------------------------------------
-      avts     = 1.e-4     !  maximum avs (vertical mixing on salinity)
-      hsbfr    = 1.6       !  heat/salt buoyancy flux ratio
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdom
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdom	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdom	(revision 1831)
@@ -2,17 +2,17 @@
 &namdom        !   space and time domain (bathymetry, mesh, timestep)
 !-----------------------------------------------------------------------
-   ntopo       =    1      !  compute (=0) or read(=1) the bathymetry file
-   e3zps_min   =    5.     !  the thickness of the partial step is set larger than the minimum
-   e3zps_rat   =    0.1    !  of e3zps_min and e3zps_rat * e3t   (N.B. 0<e3zps_rat<1)
-   nmsh        =    0      !  create (=1) a mesh file (coordinates, scale factors, masks) or not (=0)
-   nacc        =    0      !  =1 acceleration of convergence method used, rdt < rdttra(k)
-                           !  =0, no acceleration, rdt = rdttra
-   atfp        =    0.1    !  asselin time filter parameter
-   rdt         = 5760.     !  time step for the dynamics (and tracer if nacc=0)
-   rdtmin      = 5760.     !  minimum time step on tracers (used if nacc=1)
-   rdtmax      = 5760.     !  maximum time step on tracers (used if nacc=1)
-   rdth        =  800.     !  depth variation of tracer time step  (used if nacc=1)
-   rdtbt       =   90.     !  barotropic time step (for the split explicit algorithm) ("key_dynspg_ts")
-   nclosea     =    0      !  = 0 no closed sea in the model domain
-                           !  = 1 closed sea (Black Sea, Caspian Sea, Great US Lakes...) 
+   nn_bathy    =    1      !  compute (=0) or read(=1) the bathymetry file
+   nn_closea   =    0      !  closed seas and lakes are removed (=0) or kept (=1) from the ORCA domain
+   nn_msh      =    0      !  create (=1) a mesh file (coordinates, scale factors, masks) or not (=0)
+   rn_e3zps_min=   20.     !  the thickness of the partial step is set larger than the minimum
+   rn_e3zps_rat=    0.1    !  of e3zps_min and e3zps_rat * e3t   (N.B. 0<e3zps_rat<1)
+                           !
+   rn_rdt      = 5760.     !  time step for the dynamics (and tracer if nacc=0)   ==> 5760
+   nn_baro     =   64      !  number of barotropic time step (for the split explicit algorithm) ("key_dynspg_ts")
+   rn_atfp     =    0.1    !  asselin time filter parameter
+   nn_acc      =    0      !  acceleration of convergence : =1      used, rdt < rdttra(k)
+                                 !                          =0, not used, rdt = rdttra
+   rn_rdtmin   = 28800.          !  minimum time step on tracers (used if nacc=1)
+   rn_rdtmax   = 28800.          !  maximum time step on tracers (used if nacc=1)
+   rn_rdth     =  800.           !  depth variation of tracer time step  (used if nacc=1)
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_adv
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_adv	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_adv	(revision 1831)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&namdyn_adv    !   formulation of the momentum advection
+!-----------------------------------------------------------------------
+   ln_dynadv_vec = .true.  !  vector form (T) or flux form (F)  
+   ln_dynadv_cen2= .false. !  flux form - 2nd order centered scheme
+   ln_dynadv_ubs = .false. !  flux form - 3rd order UBS      scheme 
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_hpg
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_hpg	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_hpg	(revision 1831)
@@ -0,0 +1,15 @@
+!-----------------------------------------------------------------------
+&namdyn_hpg    !   Hydrostatic pressure gradient option
+!-----------------------------------------------------------------------
+   ln_hpg_zco  = .false.   !  z-coordinate - full steps                   
+   ln_hpg_zps  = .true.    !  z-coordinate - partial steps (interpolation)
+   ln_hpg_sco  = .false.   !  s-coordinate (standard jacobian formulation)
+   ln_hpg_hel  = .false.   !  s-coordinate (helsinki modification)
+   ln_hpg_wdj  = .false.   !  s-coordinate (weighted density jacobian)
+   ln_hpg_djc  = .false.   !  s-coordinate (Density Jacobian with Cubic polynomial)
+   ln_hpg_rot  = .false.   !  s-coordinate (ROTated axes scheme)
+   rn_gamma    = 0.e0      !  weighting coefficient (wdj scheme)
+   ln_dynhpg_imp = .false. !  time stepping: semi-implicit time scheme  (T)
+                                 !           centered      time scheme  (F)
+   nn_dynhpg_rst =  0            ! =1 dynhpg restartable restart or not (=0)
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_ldf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_ldf	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_ldf	(revision 1831)
@@ -0,0 +1,14 @@
+!-----------------------------------------------------------------------
+&namdyn_ldf    !   lateral diffusion on momentum
+!-----------------------------------------------------------------------
+                           !  Type of the operator : 
+   ln_dynldf_lap    =  .true.   !  laplacian operator         
+   ln_dynldf_bilap  =  .false.  !  bilaplacian operator    
+                           !  Direction of action  : 
+   ln_dynldf_level  =  .false.  !  iso-level               
+   ln_dynldf_hor    =  .true.   !  horizontal (geopotential)            (require "key_ldfslp" in s-coord.)
+   ln_dynldf_iso    =  .false.  !  iso-neutral                          (require "key_ldfslp")
+                           !  Coefficient
+   rn_ahm_0    = 40000.         !  horizontal eddy viscosity   [m2/s]
+   rn_ahmb_0   =     0.         !  background eddy viscosity for ldf_iso [m2/s]
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_spg
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_spg	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_spg	(revision 1831)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+!namdyn_spg    !   surface pressure gradient   (CPP key only)
+!-----------------------------------------------------------------------
+!                          !  explicit free surface                     ("key_dynspg_exp")
+!                          !  filtered free surface                     ("key_dynspg_flt")
+!                          !  split-explicit free surface               ("key_dynspg_ts")
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_vor
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_vor	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namdyn_vor	(revision 1831)
@@ -0,0 +1,8 @@
+!-----------------------------------------------------------------------
+&namdyn_vor    !   option of physics/algorithm (not control by CPP keys)
+!-----------------------------------------------------------------------
+   ln_dynvor_ene = .false. !  enstrophy conserving scheme  
+   ln_dynvor_ens = .false. !  energy conserving scheme    
+   ln_dynvor_mix = .false. !  mixed scheme               
+   ln_dynvor_een = .true.  !  energy & enstrophy scheme  
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nameos
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nameos	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nameos	(revision 1831)
@@ -2,9 +2,9 @@
 &nameos        !   ocean physical parameters
 !-----------------------------------------------------------------------
-   neos        =    0      !  type of equation of state and Brunt-Vaisala frequency
+   nn_eos      =    0      !  type of equation of state and Brunt-Vaisala frequency
                            !     = 0, UNESCO (formulation of Jackett and McDougall (1994) and of McDougall (1987) )
                            !     = 1, linear: rho(T)   = rau0 * ( 1.028 - ralpha * T )
                            !     = 2, linear: rho(T,S) = rau0 * ( rbeta * S - ralpha * T )
-   ralpha      =    2.e-4  !  thermal expension coefficient (neos= 1 or 2)
-   rbeta       =    0.001  !  saline  expension coefficient (neos= 2)
+   rn_alpha    =    2.e-4  !  thermal expension coefficient (neos= 1 or 2)
+   rn_beta     =    0.001  !  saline  expension coefficient (neos= 2)
 /
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namflg
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namflg	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&namflg        !   algorithm flags (algorithm not control by CPP keys)
-!-----------------------------------------------------------------------
-   ln_dynhpg_imp = .false. !  hydrostatic pressure gradient: semi-implicit time scheme  (T)
-                           !                                 centered      time scheme  (F)
-   nn_dynhpg_rst =  0      !  add dynhpg implicit variables in restart ot not (1/0)
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namflo
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namflo	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namflo	(revision 1831)
@@ -3,6 +3,6 @@
 !-----------------------------------------------------------------------
     ln_rstflo = .false.    !  float restart (T) or not (F)
-    nwritefl  =      75    !  frequency of writing in float output file 
-    nstockfl  =    5475    !  frequency of creation of the float restart file 
+    nn_writefl=      75    !  frequency of writing in float output file 
+    nn_stockfl=    5475    !  frequency of creation of the float restart file 
     ln_argo   = .false.    !  Argo type floats (stay at the surface each 10 days)
     ln_flork4 = .false.    !  trajectories computed with a 4th order Runge-Kutta (T)
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namgap
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namgap	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namgap	(revision 1831)
@@ -2,5 +2,5 @@
 &namgap       !   level mean model-data gap                             ('key_diagap')
 !-----------------------------------------------------------------------
-   ngap       =  15        !  time-step frequency of model-data gap computation
-   nprg       =  10        !  time-step frequency of gap print in model output
+   nn_gap     =  15        !  time-step frequency of model-data gap computation
+   nn_prg     =  10        !  time-step frequency of gap print in model output
 /
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namkpp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namkpp	(revision 1830)
+++ 	(revision )
@@ -1,14 +1,0 @@
-!-----------------------------------------------------------------------
-&namkpp        !   K-Profile Parameterization dependent vertical mixing  ("key_zdfkpp", and optionnally:
-!                                                                         "key_kppcustom" or "key_kpplktb")
-!------------------------------------------------------------------------
-   ln_kpprimix = .true.    !  shear instability mixing 
-   difmiw      =  1.0e-04  !  constant internal wave viscosity [m2/s]
-   difsiw      =  0.1e-04  !  constant internal wave diffusivity [m2/s]
-   Riinfty     =  0.8      !  local Richardson Number limit for shear instability
-   difri       =  0.0050   !  maximum shear mixing at Rig = 0    [m2/s]
-   bvsqcon     = -0.01e-07 !  Brunt-Vaisala squared for maximum convection [1/s2] 
-   difcon      =  1.       !  maximum mixing in interior convection [m2/s]
-   navb        =  0        !  horizontal averaged (=1) or not (=0) on avt and amv
-   nave        =  1        !  constant (=0) or profile (=1) background on avt
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp	(revision 1831)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&nammpp        !   Massively Parallel Processing                        ("key_mpp_mpi)
+!-----------------------------------------------------------------------
+   cn_mpi_send =  'S'      !  mpi send/recieve type   ='S', 'B', or 'I' for standard send,
+                           !  buffer blocking send or immediate non-blocking sends, resp.
+   nn_buffer   =   0       !  size in bytes of exported buffer ('B' case), 0 no exportation
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp_dyndist
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp_dyndist	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nammpp_dyndist	(revision 1831)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&nammpp_dyndist !   Massively Parallel Distribution for AGRIF zoom      ("key_agrif" && "key_mpp_dyndist")
+!-----------------------------------------------------------------------
+   jpni        =    1      !  jpni   number of processors following i
+   jpnj        =    1      !  jpnj   number of processors following j
+   jpnij       =    1      !  jpnij  number of local domains
+/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namnpc
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namnpc	(revision 1830)
+++ 	(revision )
@@ -1,6 +1,0 @@
-!-----------------------------------------------------------------------
-&namnpc        !   non penetrative convection
-!-----------------------------------------------------------------------
-   nnpc1       =    1      !  non penetrative convective scheme computation frequency
-   nnpc2       =  365      !  non penetrative convective scheme print frequency
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namobc
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namobc	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namobc	(revision 1831)
@@ -2,20 +2,20 @@
 &namobc        !   open boundaries parameters                           ("key_obc")
 !-----------------------------------------------------------------------
-    nobc_dta   =    1      !  = 0 the obc data are equal to the initial state
-                           !  = 1 the obc data are read in 'obc.dta' files
-    cffile     = 'annual'  !  set to annual if obc datafile hold 1 year of data
-                           !  set to monthly if obc datafile hold 1 month of data
-    rdpein     =    1.     !  ???
-    rdpwin     =    1.     !  ???
-    rdpnin     =    1.     !  ???
-    rdpsin     =    1.     !  ???
-    rdpeob     = 3000.     !  time relaxation (days) for the east  open boundary
-    rdpwob     =   15.     !    "        "             "     west         "
-    rdpnob     = 3000.     !    "        "             "     north        "
-    rdpsob     =   15.     !    "        "             "     south        "
-    zbsic1     =  140.e+6  !   barotropic stream function on first  isolated coastline
-    zbsic2     =    1.e+6  !    "                   "        second       "
-    zbsic3     =    0.     !    "                   "        thrid        "
     ln_obc_clim= .false.   !  climatological obc data files (T) or not (F)
     ln_vol_cst = .true.    !  impose the total volume conservation (T) or not (F)
+    ln_obc_fla = .false.   !  Flather open boundary condition 
+    nn_obcdta  =    1      !  = 0 the obc data are equal to the initial state
+                           !  = 1 the obc data are read in 'obc.dta' files
+    cn_obcdta  = 'annual'  !  set to annual if obc datafile hold 1 year of data
+                           !  set to monthly if obc datafile hold 1 month of data
+    rn_dpein   =    1.     !  damping time scale for inflow at east  open boundary
+    rn_dpwin   =    1.     !     -           -         -       west    -      -
+    rn_dpnin   =    1.     !     -           -         -       north   -      -
+    rn_dpsin   =    1.     !     -           -         -       south   -      -
+    rn_dpeob   = 3000.     !  time relaxation (days) for the east  open boundary
+    rn_dpwob   =   15.     !     -           -         -     west    -      -
+    rn_dpnob   = 3000.     !     -           -         -     north   -      -
+    rn_dpsob   =   15.     !     -           -         -     south   -      -
+    rn_volemp  =  1.       !  = 0 the total volume change with the surface flux (E-P-R)
+                           !  = 1 the total volume remains constant
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namptr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namptr	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namptr	(revision 1831)
@@ -3,6 +3,8 @@
 !-----------------------------------------------------------------------
    ln_diaptr  = .false.    !  Poleward heat and salt transport (T) or not (F)
+   ln_diaznl  = .false.    !  Add zonal means and meridional stream functions
    ln_subbas  = .false.    !  Atlantic/Pacific/Indian basins computation (T) or not 
                            !  (orca configuration only, need input basins mask file named "subbasins.nc"
-   nf_ptr     =  15        !  Frequency of ptr computation [time step]
+   nf_ptr     =  1         !  Frequency of ptr computation [time step]
+   nf_ptr_wri =  15        !  Frequency of ptr outputs
 /
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namqsr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namqsr	(revision 1830)
+++ 	(revision )
@@ -1,8 +1,0 @@
-!-----------------------------------------------------------------------
-&namqsr        !   penetrative solar radiation
-!-----------------------------------------------------------------------
-   ln_traqsr   = .true.    !  penetrative solar radiation (T) or not (F)
-   rabs        =   0.58    !  fraction of qsr associated with xsi1
-   xsi1        =   0.35    !  first depth of extinction
-   xsi2        =   23.0    !  second depth of extinction
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namric
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namric	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&namric        !   richardson number dependent vertical diffusion       ("key_zdfric" )
-!-----------------------------------------------------------------------
-   avmri       = 100.e-4   !  maximum value of the vertical viscosity
-   alp         =   5.      !  coefficient of the parameterization
-   nric        =   2       !  coefficient of the parameterization
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namrun
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namrun	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namrun	(revision 1831)
@@ -2,17 +2,22 @@
 &namrun        !   parameters of the run
 !-----------------------------------------------------------------------
-   no          =       0   !  job number
-   cexper      =  "ORCA2"  !  experience name 
+   nn_no       =       0   !  job number
+   cn_exp      =  "ORCA2"  !  experience name 
+   nn_it000    =       1   !  first time step
+   nn_itend    =     315   !  last  time step (std 5475)
+   nn_date0    =  010101   !  initial calendar date yymmdd (used if nrstdt=1)
+   nn_leapy    =       0   !  Leap year calendar (1) or not (0)
+   nn_istate   =       0   !  output the initial state (1) or not (0)
+   nn_stock    =    5475   !  frequency of creation of a restart file (modulo referenced to 1)
+   nn_write    =    5475   !  frequency of write in the output file   (modulo referenced to nit000)
+   ln_dimgnnn  = .false.   !  DIMG file format: 1 file for all processors (F) or by processor (T)
+   ln_mskland  = .false.   !  mask land points in NetCDF outputs (costly: + ~15%)
+   ln_clobber  = .false.   !  clobber (overwrite) an existing file
+   nn_chunksz  =       0   !  chunksize (bytes) for NetCDF file (working only with iom_nf90 routines)
    ln_rstart   = .false.   !  start from rest (F) or from a restart file (T)
-   nrstdt      =       0   !  restart control = 0 nit000 is not compared to the restart file value
-                           !                  = 1 use ndate0 in namelist (not the value in the restart file)
-                           !                  = 2 calendar parameters read in the restart file
-   nit000      =       1   !  first time step
-   nitend      =    5475   !  last  time step
-   ndate0      =  010101   !  initial calendar date yymmdd (used if nrstdt=1)
-   nleapy      =       0   !  Leap year calendar (1) or not (0)
-   ninist      =       0   !  output the initial state (1) or not (0)
-   nstock      =    5475   !  frequency of creation of a restart file
-   nwrite      =    5475   !  frequency of write in the output file 
-   ln_dimgnnn  = .false.   !  DIMG file format: 1 file for all processors (F) or by processor (T)
+   nn_rstctl   =       0       !  restart control = 0 nit000 is not compared to the restart file value
+                               !                  = 1 use ndate0 in namelist (not the value in the restart file)
+                               !                  = 2 calendar parameters read in the restart file
+   cn_ocerst_in  = "restart"   !  suffix of ocean restart name (input)
+   cn_ocerst_out = "restart"   !  suffix of ocean restart name (output)
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc	(revision 1831)
@@ -13,11 +13,12 @@
                            !  =2 ice-model used                             ("key_lim3" or "key_lim2)
    nn_ico_cpl  = 0         !  ice-ocean coupling : =0 each nn_fsbc 
-                           !                       =1 stress recomputed each ocean time step ("key_lim3")
-                           !                       =2 combination of 0 and 1 cases           ("key_lim3")
+                           !                       =1 stresses recomputed each ocean time step ("key_lim3" only)
+                           !                       =2 combination of 0 and 1 cases             ("key_lim3" only)
    ln_dm2dc    = .false.   !  daily mean to diurnal cycle short wave (qsr)
    ln_rnf      = .true.    !  runoffs (T => fill namsbc_rnf)
    ln_ssr      = .true.    !  Sea Surface Restoring on T and/or S (T => fill namsbc_ssr)
-   nn_fwb      = 0         !  FreshWater Budget: =0 unchecked                              , 
-                           !                     =1 global mean of e-p-r set to zero at each nn_fsbc time step   ,
+   nn_fwb      = 3         !  FreshWater Budget: =0 unchecked 
+                           !                     =1 global mean of e-p-r set to zero at each time step 
                            !                     =2 annual global mean of e-p-r set to zero
+                           !                     =3 global emp set to zero and spread out over erp area
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_alb
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_alb	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_alb	(revision 1831)
@@ -1,9 +1,9 @@
 !-----------------------------------------------------------------------
-&namalb        !   albedo parameters
+&namsbc_alb    !   albedo parameters
 !-----------------------------------------------------------------------
-   cgren       =    0.06   !  correction of the snow or ice albedo to take into account the
-   albice      =    0.5    !  albedo of melting ice in the arctic and antarctic
-   alphd       =    0.80   !  coefficients for linear interpolation used to
-   alphc       =    0.65   !  compute albedo between two extremes values 
-   alphdi      =    0.72   !  (Pyane, 1972)
+   rn_cloud    =    0.06   !  cloud correction to snow and ice albedo 
+   rn_albice   =    0.53   !  albedo of melting ice in the arctic and antarctic
+   rn_alphd    =    0.80   !  coefficients for linear interpolation used to
+   rn_alphc    =    0.65   !  compute albedo between two extremes values 
+   rn_alphdi   =    0.72   !  (Pyane, 1972)
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_clio
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_clio	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_clio	(revision 1831)
@@ -1,15 +1,14 @@
 !-----------------------------------------------------------------------
-&namsbc_clio              CLIO bulk formulea
+&namsbc_clio   !   namsbc_clio  CLIO bulk formulea
 !-----------------------------------------------------------------------
-!              !      file name     ! frequency (hours) !  variable  ! time interpol. !  clim   ! 'yearly' or !
-!              !                    !  (if <0  months)  !    name    !    (logical)   !  (T/F)  !  'monthly'  !
-   sn_utau      =    'taux_1m'       ,      -12.         , 'sozotaux' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_vtau      =    'tauy_1m'       ,      -12.         , 'sometauy' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_wndm      =    'flx'           ,      -12.         , 'socliowi' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_tair      =    'flx'           ,      -12.         , 'socliot1' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_humi      =    'flx'           ,      -12.         , 'socliohu' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_ccov      =    'flx'           ,      -12.         , 'socliocl' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_prec      =    'flx'           ,      -12.         , 'socliopl' ,     .false.        ,    1   ,    0     , ''       , ''
-
+!              !   file name  ! frequency (hours) !  variable  ! time interpol. !  clim   ! 'yearly'/ ! weights  ! rotation !
+!              !              !  (if <0  months)  !    name    !    (logical)   !  (T/F)  ! 'monthly' ! filename ! pairing  !
+   sn_utau     = 'taux_1m'    ,       -1.         , 'sozotaux' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_vtau     = 'tauy_1m'    ,       -1.         , 'sometauy' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_wndm     = 'flx'        ,       -1.         , 'socliowi' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_tair     = 'flx'        ,       -1.         , 'socliot2' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_humi     = 'flx'        ,       -1.         , 'socliohu' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_ccov     = 'flx'        ,       -1.         , 'socliocl' ,    .false.     , .true.  , 'yearly'  , ''       , ''
+   sn_prec     = 'flx'        ,       -1.         , 'socliopl' ,    .false.     , .true.  , 'yearly'  , ''       , ''
 !
    cn_dir      = './'      !  root directory for the location of the bulk files are
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_core
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_core	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_core	(revision 1831)
@@ -2,17 +2,17 @@
 &namsbc_core   !   namsbc_core  CORE bulk formulea
 !-----------------------------------------------------------------------
-!              !   file name        ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly' or !
-!              !                    !  (if <0  months)  !   name     !    (logical)   !  (T/F)  !  'monthly'  !
-   sn_wndi      =    'u10'           ,       24.         , 'U_10_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_wndj      =    'v10'           ,       24.         , 'V_10_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_qsr       =    'rad'           ,       24.         , 'SWDN_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_qlw       =    'rad'           ,       24.         , 'LWDN_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_tair      =    't10'           ,       24.         , 'T_10_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_humi      =    'q10'           ,       24.         , 'Q_10_MOD' ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_prec      =    'precip'        ,      -12.         , 'RAIN'     ,     .false.        ,    1   ,    0     , ''       , ''
-   sn_snow      =    'precip'        ,      -12.         , 'SNOW'     ,     .false.        ,    1   ,    0     , ''       , ''
+!              !   file name  ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly'/ ! weights              ! rotation !
+!              !              !  (if <0  months)  !   name     !    (logical)   !  (T/F)  ! 'monthly' ! filename             ! pairing  !
+   sn_wndi     = 'u10_core'   ,       -1.         , 'u10'      ,    .true.      , .true.  , 'yearly'  ,'bicubic_weights_orca2.nc' , 'U1'
+   sn_wndj     = 'v10_core'   ,       -1.         , 'v10'      ,    .true.      , .true.  , 'yearly'  ,'bicubic_weights_orca2.nc' , 'V1'
+   sn_qsr      = 'qsw_core'   ,       -1.         , 'swdn'     ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
+   sn_qlw      = 'qlw_core'   ,       -1.         , 'lwdn'     ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
+   sn_tair     = 't2_core'    ,       -1.         , 't2'       ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
+   sn_humi     = 'q2_core'    ,       -1.         , 'q2'       ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
+   sn_prec     = 'precip_core',       -1.         , 'precip'   ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
+   sn_snow     = 'snow_core'  ,       -1.         , 'snow'     ,    .true.      , .true.  , 'yearly'  ,'bilinear_weights_orca2.nc', ''
 !
    cn_dir      = './'      !  root directory for the location of the bulk files
-   ln_2m       = .false.   !  air temperature and humidity referenced at 2m (T) instead 10m (F)
-   alpha_precip= 1.        !  multiplicative factor for precipitation (total & snow)
+   ln_2m       = .true.    !  air temperature and humidity referenced at 2m (T) instead 10m (F)
+   rn_pfac     = 1.        !  multiplicative factor for precipitation (total & snow)
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl	(revision 1831)
@@ -2,5 +2,5 @@
 &namsbc_cpl    !   coupled ocean/atmosphere model                        ("key_coupled")
 !-----------------------------------------------------------------------
-! SEND
+                                       ! send
 cn_snd_temperature= 'weighted oce and ice'  ! 'oce only' 'weighted oce and ice' 'mixed oce-ice'
 cn_snd_albedo     = 'weighted ice'          ! 'none' 'weighted ice' 'mixed oce-ice'
@@ -10,5 +10,5 @@
 cn_snd_crt_orient = 'eastward-northward'    ! 'eastward-northward' or 'local grid'
 cn_snd_crt_grid   = 'T'                     ! 'T'
-! RECEIVE
+                                       ! receive
 cn_rcv_w10m       = 'coupled'               ! 'none' 'coupled'
 cn_rcv_tau_nature = 'oce only'              ! 'oce only' 'oce and ice' 'mixed oce-ice'
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl_co2
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl_co2	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_cpl_co2	(revision 1831)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&namsbc_cpl_co2   !   coupled ocean/biogeo/atmosphere model            ("key_cpl_carbon_cycle")
+!-----------------------------------------------------------------------
+cn_snd_co2        = 'coupled'         ! send    :  'none' 'coupled'
+cn_rcv_co2        = 'coupled'         ! receive : 'none' 'coupled'
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_flx
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_flx	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_flx	(revision 1831)
@@ -2,11 +2,11 @@
 &namsbc_flx    !   surface boundary condition : flux formulation
 !-----------------------------------------------------------------------
-!              !      file name     ! frequency (hours) !  variable  ! time interpol. !  clim   ! 'yearly' or !
-!              !                    !  (if <0  months)  !    name    !    (logical)   !  (T/F)  !  'monthly'  !
-   sn_utau      =      'utau.nc'     ,        24.        ,  'utau'  ,      .false.       ,    0   ,     0    , ''       , ''
-   sn_vtau      =      'vtau.nc'     ,        24.        ,  'vtau'  ,      .false.       ,    0   ,     0    , ''       , ''
-   sn_qtot      =      'qtot.nc'     ,        24.        ,  'qtot'  ,      .false.       ,    0   ,     0    , ''       , ''
-   sn_qsr       =      'qsr.nc'      ,        24.        ,  'qsr'   ,      .false.       ,    0   ,     0    , ''       , ''
-   sn_emp       =      'emp.nc'      ,        24.        ,  'emp'   ,      .false.       ,    0   ,     0    , ''       , ''
+!              !   file name  ! frequency (hours) !  variable  ! time interpol. !  clim   ! 'yearly'/ ! weights  ! rotation !
+!              !              !  (if <0  months)  !    name    !    (logical)   !  (T/F)  ! 'monthly' ! filename ! pairing  !
+   sn_utau     = 'utau'       ,        24.        ,    'utau'  ,     .false.    , .false. , 'yearly'  , ''       , ''
+   sn_vtau     = 'vtau'       ,        24.        ,    'vtau'  ,     .false.    , .false. , 'yearly'  , ''       , ''
+   sn_qtot     = 'qtot'       ,        24.        ,    'qtot'  ,     .false.    , .false. , 'yearly'  , ''       , ''
+   sn_qsr      = 'qsr'        ,        24.        ,    'qsr'   ,     .false.    , .false. , 'yearly'  , ''       , ''
+   sn_emp      = 'emp'        ,        24.        ,    'emp'   ,     .false.    , .false. , 'yearly'  , ''       , ''
 !
    cn_dir      = './'      !  root directory for the location of the flux files
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_qsr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_qsr	(revision 1830)
+++ 	(revision )
@@ -1,8 +1,0 @@
-!-----------------------------------------------------------------------
-&namqsr        !   penetrative solar radiation
-!-----------------------------------------------------------------------
-   ln_traqsr   = .true.    !  penetrative solar radiation (T) or not (F)
-   rabs        =   0.58    !  fraction of qsr associated with xsi1
-   xsi1        =   0.35    !  first depth of extinction
-   xsi2        =   23.0    !  second depth of extinction
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_rnf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_rnf	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_rnf	(revision 1831)
@@ -2,13 +2,14 @@
 &namsbc_rnf    !   runoffs namelist surface boundary condition
 !-----------------------------------------------------------------------
-!               !   file name        ! frequency (hours) ! variable   ! time interpolation !  clim  ! starting !
-!               !                    !   (if <0  months) !   name     !     (logical)      !  (0/1) !  record  !
-   sn_rnf     = 'runoff_1m_nomask.nc' ,      -12.        ,  'sorunoff',      .true.        ,    1   ,    0     , ''       , ''
-   sn_cnf     = 'runoff_1m_nomask.nc' ,        0.        ,  'socoefr' ,      .false.       ,    1   ,    0     , ''       , ''
-!
-   cn_dir       = './'       ! directory in which the model is executed
+!              !   file name  ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly'/ ! weights  ! rotation !
+!              !              !  (if <0  months)  !   name     !    (logical)   !  (T/F)  ! 'monthly' ! filename ! pairing  !
+   sn_rnf      = 'runoff_1m_nomask' ,  -1.        , 'sorunoff' ,    .true.      , .true.  , 'yearly'  , ''       , ''
+   sn_cnf      = 'runoff_1m_nomask' ,   0.        , 'socoefr'  ,    .false.     , .true.  , 'yearly'  , ''       , ''
+ 
+   cn_dir       = './'      !  root directory for the location of the runoff files
    ln_rnf_emp   =   .false. !  runoffs included into precipitation field (T) or into a file (F)
    ln_rnf_mouth =   .false. !  specific treatment at rivers mouths
-   rn_hrnf      =   0.e0    !  depth over which enhanced vertical mixing is used
+   rn_hrnf      =  15.e0    !  depth over which enhanced vertical mixing is used
    rn_avt_rnf   =   1.e-3   !  value of the additional vertical mixing coef. [m2/s]
+   rn_rfact     =   1.e0    !  multiplicative factor for runoff
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_ssr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_ssr	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsbc_ssr	(revision 1831)
@@ -2,13 +2,16 @@
 &namsbc_ssr    !   surface boundary condition : sea surface restoring
 !-----------------------------------------------------------------------
-!              !     file name      ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly' or !
-!              !                    !  (if <0  months)  !   name     !    (logical)   !  (T/F)  !  'monthly'  !
-   sn_sst       =  'sst_data.nc'     ,         24.       ,  'sst'     ,      .false.       ,    0   ,    0
-   sn_sss       =  'sss_data.nc'     ,        -12.       ,  'sss'     ,      .true.        ,    0   ,    0
-!   
+!              !   file name  ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly'/ ! weights  ! rotation !
+!              !              !  (if <0  months)  !   name     !    (logical)   !  (T/F)  ! 'monthly' ! filename ! pairing  !
+   sn_sst      = 'sst_data'   ,        24.        ,  'sst'     ,     .false.    , .false. , 'yearly'  , ''       , ''
+   sn_sss      = 'sss_data'   ,        -1.        ,  'sss'     ,     .true.     , .true.  , 'yearly'  , ''       , ''
+    
    cn_dir      = './'      !  root directory for the location of the runoff files
    nn_sstr     =     0     !  add a retroaction term in the surface heat       flux (=1) or not (=0)
-   nn_sssr     =     1     !  add a damping     term in the surface freshwater flux (=1) or not (=0)
-   dqdt        =   -40.    !  magnitude of the retroaction on temperature   [W/m2/K]
-   deds        =   -27.7   !  magnitude of the damping on salinity   [mm/day/psu]
+   nn_sssr     =     2     !  add a damping     term in the surface freshwater flux (=2) 
+                           !  or to SSS only (=1) or no damping term (=0)
+   rn_dqdt     =   -40.    !  magnitude of the retroaction on temperature   [W/m2/K]
+   rn_deds     =   -27.7   !  magnitude of the damping on salinity   [mm/day/psu]
+   ln_sssr_bnd =   .true.  !  flag to bound erp term (associated with nn_sssr=2)
+   rn_sssr_bnd =   4.e0    !  ABS(Max/Min) value of the damping erp term [mm/day]
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsol
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsol	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namsol	(revision 1831)
@@ -2,17 +2,12 @@
 &namsol        !   elliptic solver / island / free surface 
 !-----------------------------------------------------------------------
-   nsolv       =      1    !  elliptic solver: =1 preconditioned conjugate gradient (pcg)
+   nn_solv     =      1    !  elliptic solver: =1 preconditioned conjugate gradient (pcg)
                            !                   =2 successive-over-relaxation (sor)
-                           !                   =3 FETI (fet)                               ("key_feti")
-                           !                   =4 sor with extra outer halo
-   nsol_arp    =      0    !  absolute/relative (0/1) precision convergence test
-   nmin        =    300    !  minimum of iterations for the SOR solver
-   nmax        =    800    !  maximum of iterations for the SOR solver
-   nmod        =     10    !  frequency of test for the SOR solver
-   eps         =  1.e-6    !  absolute precision of the solver
-   resmax      =  1.e-10   !  absolute precision for the SOR solver
-   sor         =  1.92     !  optimal coefficient for SOR solver (to be adjusted with the domain)
-   epsisl      =  1.e-10   !  absolute precision on stream function solver
-   nmisl       =   4000    !  maximum pcg iterations for island                            ("key_islands")
-   rnu         =      1.   !  strength of the additional force used in filtered free surface
+   nn_sol_arp  =      0    !  absolute/relative (0/1) precision convergence test
+   rn_eps      =  1.e-6    !  absolute precision of the solver
+   nn_nmin     =    300    !  minimum of iterations for the SOR solver
+   nn_nmax     =    800    !  maximum of iterations for the SOR solver
+   nn_nmod     =     10    !  frequency of test for the SOR solver
+   rn_resmax   =  1.e-10   !  absolute precision for the SOR solver
+   rn_sor      =  1.92     !  optimal coefficient for SOR solver (to be adjusted with the domain)
 /
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namspr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namspr	(revision 1830)
+++ 	(revision )
@@ -1,7 +1,0 @@
-!-----------------------------------------------------------------------
-&namspr       !   surface pressure diagnostic
-!-----------------------------------------------------------------------
-   nmaxp      =   1000     !  maximum of iterations for the solver
-   epsp       =  1.e-3     !  absolute precision of the solver
-   niterp     =    400     !  number of iteration done by the solver
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtdp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtdp	(revision 1830)
+++ 	(revision )
@@ -1,15 +1,0 @@
-!-----------------------------------------------------------------------
-&namtdp        !   tracer newtonian damping                             ('key_tradmp')
-!-----------------------------------------------------------------------
-   ndmp        =   -1      !  type of damping in temperature and salinity 
-                           !     ='latitude', damping poleward of 'ndmp' degrees and function 
-                           !                  of the distance-to-coast. Red and Med Seas as ndmp=-1
-                           !     =-1 damping only in Med and Red Seas
-   ndmpf       =    1      !  create a damping.coeff NetCDF file (=1) or not (=0)
-   nmldmp      =    1      !  type of damping: =0 damping throughout the water column
-                           !                   =1 no damping in the mixed layer defined by avt >5cm2/s )
-                           !                   =2 no damping in the mixed layer defined rho<rho(surf)+.01 )
-   sdmp        =   50.     !  surface time scale for internal damping (days)
-   bdmp        =  360.     !  bottom  time scale for internal damping (days)
-   hdmp        =  800.     !  depth of transition between sdmp and bdmp (meters)
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtide
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtide	(revision 1830)
+++ 	(revision )
@@ -1,8 +1,0 @@
-!-----------------------------------------------------------------------
-&namtide        ! tidal forcing at unstructured boundaries              
-!-----------------------------------------------------------------------
-    filtide      = 'bdytide_'           !  file name root of tidal forcing files
-    tide_cpt     = 'M2','S1'            !  names of tidal components used
-    tide_speed   = 28.984106, 15.000001 !  phase speeds of tidal components (deg/hour)
-    ln_tide_date = .false.              !  adjust tidal harmonics for start date of run
-/
Index: anches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtke
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtke	(revision 1830)
+++ 	(revision )
@@ -1,34 +1,0 @@
-!-----------------------------------------------------------------------
-&namtke        !   turbulent eddy kinetic dependent vertical diffusion  ("key_zdftke")
-!-----------------------------------------------------------------------
-   ln_rstke    = .false.   !  restart with tke from a run without tke (T) or not (F)
-   nn_itke     =  50       !  number of iterative loops if ln_rstke=T 
-   rn_ediff    =   0.1     !  coef. for vertical eddy coef. (avt=rn_ediff*mxl*sqrt(e) )
-   rn_ediss    =   0.7     !  coef. of the Kolmogoroff dissipation
-   rn_ebb      =   3.75    !  coef. of the surface input of tke
-   rn_efave    =   1.      !  boost of the tke diffusion ( avtke=rn_efave*avm )
-   rn_emin     =   1.e-6   !  minimum value of tke [m2/s2]
-   rn_emin0    =   1.e-4   !  surface minimum value of tke [m2/s2]
-   nn_mxl      =   2       !  mixing length: = 0 bounded by the distance to surface and bottom
-                           !                 = 1 bounded by the local vertical scale factor
-                           !                 = 2 first vertical derivative of mixing length bounded by 1
-                           !                 = 3 same criteria as case 2 but applied in a different way
-   nn_pdl      =   1       !  Prandtl number function of richarson number (=1, avt=pdl(Ri)*avm) or not (=0, avt=avm)
-   nn_avb      =   0       !  profile for constant background used on avt & avm (=1) or not (=0)
-   nn_ave      =   1       !  horizontal averaged on avt (=1) or not (=0) 
-   ln_mxl0     = .false.   !  mixing length scale surface value as function of wind stress (T) or not (F)
-   rn_lmin     =   0.4     !  interior buoyancy lenght scale minimum value
-   rn_lmin0    =   0.4     !  surface  buoyancy lenght scale minimum value
-   nn_etau     =   0       !  exponentially deceasing penetration of tke due to internal & intertial waves
-                           !        = 0 no penetration ( O(2 km) resolution)
-                           !        = 1 additional tke source 
-                           !        = 2 additional tke source applied only at the base of the mixed layer
-   nn_htau     =   2       !  type of exponential decrease of tke penetration
-                           !        = 0  constant 10 m length scale
-                           !        = 1  ???
-                           !        = 2  ???
-   rn_efr      =   0.05    !  fraction of surface tke value which penetrates inside the ocean
-   ln_lc       = .false.   !  Langmuir cell effect
-   rn_lc       =   0.15    !  coef. associated to Langmuir cells
-   nn_havtb    =   0       !  horizontal shape for avtb (=1) or not (=0)
-/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_adv
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_adv	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_adv	(revision 1831)
@@ -0,0 +1,9 @@
+!-----------------------------------------------------------------------
+&namtra_adv    !   advection scheme for tracer 
+!-----------------------------------------------------------------------
+   ln_traadv_cen2   =  .false.  !  2nd order centered scheme   
+   ln_traadv_tvd    =  .true.   !  TVD scheme                
+   ln_traadv_muscl  =  .false.  !  MUSCL scheme             
+   ln_traadv_muscl2 =  .false.  !  MUSCL2 scheme + cen2 at boundaries  
+   ln_traadv_ubs    =  .false.  !  UBS scheme                 
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_dmp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_dmp	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_dmp	(revision 1831)
@@ -0,0 +1,14 @@
+!-----------------------------------------------------------------------
+&namtra_dmp    !   tracer: T & S newtonian damping                      ('key_tradmp')
+!-----------------------------------------------------------------------
+   nn_hdmp     =   -1      !  horizontal shape =-1, damping in Med and Red Seas only
+                           !                   =XX, damping poleward of XX degrees (XX>0)
+                           !                      + F(distance-to-coast) + Red and Med Seas
+   nn_zdmp     =    1      !  vertical   shape =0    damping throughout the water column
+                           !                   =1 no damping in the mixing layer (kz  criteria)
+                           !                   =2 no damping in the mixed  layer (rho crieria)
+   rn_surf     =   50.     !  surface time scale of damping   [days]
+   rn_bot      =  360.     !  bottom  time scale of damping   [days]
+   rn_dep      =  800.     !  depth of transition between rn_surf and rn_bot [meters]
+   nn_file     =    1      !  create a damping.coeff NetCDF file (=1) or not (=0)
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_ldf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_ldf	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_ldf	(revision 1831)
@@ -0,0 +1,15 @@
+!-----------------------------------------------------------------------
+&namtra_ldf    !   lateral diffusion scheme for tracer 
+!-----------------------------------------------------------------------
+                           !  Type of the operator : 
+   ln_traldf_lap    =  .true.   !     laplacian operator       
+   ln_traldf_bilap  =  .false.  !     bilaplacian operator     
+                           !  Direction of action  :
+   ln_traldf_level  =  .false.  !     iso-level                
+   ln_traldf_hor    =  .false.  !     horizontal (geopotential)         (require "key_ldfslp" when ln_sco=T)
+   ln_traldf_iso    =  .true.   !     iso-neutral                       (require "key_ldfslp")
+                           !  Coefficient
+   rn_aht_0         =  2000.    !     horizontal eddy diffusivity for tracers [m2/s]
+   rn_ahtb_0        =     0.    !     background eddy diffusivity for ldf_iso [m2/s]
+   rn_aeiv_0        =  2000.    !     eddy induced velocity coefficient [m2/s]    (require "key_traldf_eiv")
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_qsr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_qsr	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtra_qsr	(revision 1831)
@@ -0,0 +1,18 @@
+!-----------------------------------------------------------------------
+&namtra_qsr    !   penetrative solar radiation
+!-----------------------------------------------------------------------
+!              !   file name  ! frequency (hours) ! variable   ! time interpol. !  clim   ! 'yearly'/ ! weights  ! rotation !
+!              !              !  (if <0  months)  !   name     !    (logical)   !  (T/F)  ! 'monthly' ! filename ! pairing  !
+   sn_chl      = 'chlorophyll',        -1.        , 'CHLA'     ,    .true.      , .true.  , 'yearly'  , ''       , ''
+ 
+   cn_dir      = './'      !  root directory for the location of the runoff files
+   ln_traqsr   = .true.    !  Light penetration (T) or not (F)
+   ln_qsr_rgb  = .true.    !  RGB (Red-Green-Blue) light penetration
+   ln_qsr_2bd  = .false.   !  2 bands              light penetration
+   ln_qsr_bio  = .false.   !  bio-model light penetration
+   nn_chldta   =      0    !  RGB : Chl data (=1) or cst value (=0)
+   rn_abs      =   0.58    !  RGB & 2 bands: fraction of light (rn_si1)
+   rn_si0      =   0.35    !  RGB & 2 bands: shortess depth of extinction
+   rn_si1      =   23.0    !  2 bands: longest depth of extinction
+   rn_si2      =   62.0    !  3 bands: longest depth of extinction (for blue waveband & 0.01 mg/m2 Chl)
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtrd
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtrd	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namtrd	(revision 1831)
@@ -1,11 +1,12 @@
 !-----------------------------------------------------------------------
-&namtrd       !   diagnostics on dynamics and/or tracer trends             ("key_trddyn" and/or "key_trdtra")
-!                          or mixed-layer trends                        ('key_trdmld')
-!                          or barotropic vorticity                      ("key_trdvor")
+&namtrd        !   diagnostics on dynamics and/or tracer trends         ("key_trddyn" and/or "key_trdtra")
+!              !       or mixed-layer trends or barotropic vorticity    ('key_trdmld' or  "key_trdvor")
 !-----------------------------------------------------------------------
-   ntrd       = 365        !  time step frequency dynamics and tracers trends
-   nctls      =   0        !  control surface type in mixed-layer trends (0,1 or n<jpk)
-   ucf        =   1.       !  unit conversion factor (=1 -> /seconds ; =86400. -> /day)
-   ln_trdmld_restart = .false.  !  restart for ML diagnostics
-   ln_trdmld_instant = .false.  !  flag to diagnose trends of instantantaneous or mean ML T/S
+   nn_trd      = 365       !  time step frequency dynamics and tracers trends
+   nn_ctls     =   0       !  control surface type in mixed-layer trends (0,1 or n<jpk)
+   rn_ucf      =   1.      !  unit conversion factor (=1 -> /seconds ; =86400. -> /day)
+   cn_trdrst_in      = "restart_mld"   ! suffix of ocean restart name (input)
+   cn_trdrst_out     = "restart_mld"   ! suffix of ocean restart name (output)
+   ln_trdmld_restart = .false.         !  restart for ML diagnostics
+   ln_trdmld_instant = .false.         !  flag to diagnose trends of instantantaneous or mean ML T/S
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf	(revision 1830)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf	(revision 1831)
@@ -2,11 +2,15 @@
 &namzdf        !   vertical physics
 !-----------------------------------------------------------------------
-   avm0        =   1.2e-4  !  vertical eddy viscosity   [m2/s]          (background Kz if not "key_zdfcst")
-   avt0        =   1.2e-5  !  vertical eddy diffusivity [m2/s]          (background Kz if not "key_zdfcst")
-   ln_zdfnpc   = .false.   !  convection: Non-Penetrative algorithm (T) or not (F)
-   ln_zdfevd   = .true.    !  convection: enhanced vertical diffusion (T) or not (F)    
-   avevd       = 100.      !  vertical coefficient for enhanced diffusion scheme [m2/s]
-   n_evdm      =   1       !  enhanced mixing apply on tracer (=0) or on tracer and momentum (=1)
-   ln_zdfexp   =  .false.  !  split explicit (T) or implicit (F) time stepping
-   n_zdfexp    =   3       !  number of sub-timestep for ln_zdfexp=T
+   rn_avm0     =   1.2e-4  !  vertical eddy viscosity   [m2/s]          (background Kz if not "key_zdfcst")
+   rn_avt0     =   1.2e-5  !  vertical eddy diffusivity [m2/s]          (background Kz if not "key_zdfcst")
+   nn_avb      =    0      !  profile for background avt & avm (=1) or not (=0)
+   nn_havtb    =    0      !  horizontal shape for avtb (=1) or not (=0)
+   ln_zdfevd   = .true.    !  enhanced vertical diffusion (evd) (T) or not (F)
+   nn_evdm     =    0      !  evd apply on tracer (=0) or on tracer and momentum (=1)
+   rn_avevd    =  100.     !  evd mixing coefficient [m2/s]
+   ln_zdfnpc   = .false.   !  Non-Penetrative algorithm (T) or not (F)
+   nn_npc      =    1            !  frequency of application of npc
+   nn_npcp     =  365            !  npc control print frequency
+   ln_zdfexp   = .false.   !  time-stepping: split-explicit (T) or implicit (F) time stepping
+   nn_zdfexp   =    3            !  number of sub-timestep for ln_zdfexp=T
 /
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ddm
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ddm	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ddm	(revision 1831)
@@ -0,0 +1,6 @@
+!-----------------------------------------------------------------------
+&namzdf_ddm    !   double diffusive mixing parameterization             ("key_zdfddm")
+!-----------------------------------------------------------------------
+   rn_avts     = 1.e-4     !  maximum avs (vertical mixing on salinity)
+   rn_hsbfr    = 1.6       !  heat/salt buoyancy flux ratio
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_kpp
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_kpp	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_kpp	(revision 1831)
@@ -0,0 +1,13 @@
+!------------------------------------------------------------------------
+&namzdf_kpp    !   K-Profile Parameterization dependent vertical mixing  ("key_zdfkpp", and optionnally:
+!------------------------------------------------------------------------ "key_kppcustom" or "key_kpplktb")
+   ln_kpprimix = .true.    !  shear instability mixing 
+   rn_difmiw   =  1.0e-04  !  constant internal wave viscosity [m2/s]
+   rn_difsiw   =  0.1e-04  !  constant internal wave diffusivity [m2/s]
+   rn_riinfty  =  0.8      !  local Richardson Number limit for shear instability
+   rn_difri    =  0.0050   !  maximum shear mixing at Rig = 0    [m2/s]
+   rn_bvsqcon  = -0.01e-07 !  Brunt-Vaisala squared for maximum convection [1/s2] 
+   rn_difcon   =  1.       !  maximum mixing in interior convection [m2/s] 
+   nn_avb      =  0        !  horizontal averaged (=1) or not (=0) on avt and amv
+   nn_ave      =  1        !  constant (=0) or profile (=1) background on avt
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ric
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ric	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_ric	(revision 1831)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&namzdf_ric    !   richardson number dependent vertical diffusion       ("key_zdfric" )
+!-----------------------------------------------------------------------
+   rn_avmri    = 100.e-4   !  maximum value of the vertical viscosity
+   rn_alp      =   5.      !  coefficient of the parameterization
+   nn_ric      =   2       !  coefficient of the parameterization
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tke
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tke	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tke	(revision 1831)
@@ -0,0 +1,28 @@
+!-----------------------------------------------------------------------
+&namzdf_tke    !   turbulent eddy kinetic dependent vertical diffusion  ("key_zdftke")
+!-----------------------------------------------------------------------
+   rn_ediff    =   0.1     !  coef. for vertical eddy coef. (avt=rn_ediff*mxl*sqrt(e) )
+   rn_ediss    =   0.7     !  coef. of the Kolmogoroff dissipation
+   rn_ebb      =  60.      !  coef. of the surface input of tke
+   rn_emin     =   1.e-6   !  minimum value of tke [m2/s2]
+   rn_emin0    =   1.e-4   !  surface minimum value of tke [m2/s2]
+   rn_bshear   =   1.e-20  !  background shear (>0)
+   nn_mxl      =   2       !  mixing length: = 0 bounded by the distance to surface and bottom
+                           !                 = 1 bounded by the local vertical scale factor
+                           !                 = 2 first vertical derivative of mixing length bounded by 1
+                           !                 = 3 same criteria as case 2 but applied in a different way
+   nn_pdl      =   1       !  Prandtl number function of richarson number (=1, avt=pdl(Ri)*avm) or not (=0, avt=avm)
+   ln_mxl0     = .false.   !  mixing length scale surface value as function of wind stress (T) or not (F)
+   rn_lmin     =   0.001   !  interior buoyancy lenght scale minimum value
+   rn_lmin0    =   0.01    !  surface  buoyancy lenght scale minimum value
+   nn_etau     =   0       !  exponentially deceasing penetration of tke due to internal & intertial waves
+                           !        = 0 no penetration ( O(2 km) resolution)
+                           !        = 1 additional tke source
+                           !        = 2 additional tke source applied only at the base of the mixed layer
+   nn_htau     =   1       !  type of exponential decrease of tke penetration
+                           !        = 0  constant 10 m length scale
+                           !        = 1  0.5m at the equator to 30m at high latitudes
+   rn_efr      =   0.05    !  fraction of surface tke value which penetrates inside the ocean
+   ln_lc       = .false.   !  Langmuir cell parameterisation
+   rn_lc       =   0.15    !  coef. associated to Langmuir cells
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tmx
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tmx	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzdf_tmx	(revision 1831)
@@ -0,0 +1,10 @@
+!-----------------------------------------------------------------------
+&namzdf_tmx    !   tidal mixing parameterization                        ("key_zdftmx")
+!-----------------------------------------------------------------------
+   rn_htmx     = 500.      !  vertical decay scale for turbulence (meters)
+   rn_n2min    = 1.e-8     !  threshold of the Brunt-Vaisala frequency (s-1)
+   rn_tfe      = 0.333     !  tidal dissipation efficiency
+   rn_me       = 0.2       !  mixing efficiency 
+   ln_tmx_itf  = .FALSE.   !  ITF specific parameterisation
+   rn_tfe_itf  = 1.        !  ITF tidal dissipation efficiency
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr	(revision 1831)
@@ -0,0 +1,7 @@
+!-----------------------------------------------------------------------
+&namzgr        !   vertical coordinate
+!-----------------------------------------------------------------------
+   ln_zco      = .false.   !  z-coordinate - full    steps   (T/F)      ("key_zco" may also be defined)
+   ln_zps      = .true.    !  z-coordinate - partial steps   (T/F)
+   ln_sco      = .false.   !  s- or hybrid z-s-coordinate    (T/F)
+/
Index: /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr_sco
===================================================================
--- /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr_sco	(revision 1831)
+++ /branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/namzgr_sco	(revision 1831)
@@ -0,0 +1,12 @@
+!-----------------------------------------------------------------------
+&namzgr_sco    !   s-coordinate or hybrid z-s-coordinate
+!-----------------------------------------------------------------------
+   rn_sbot_min =  300.     !  minimum depth of s-bottom surface (>0) (m)
+   rn_sbot_max = 5250.     !  maximum depth of s-bottom surface (= ocean depth) (>0) (m)
+   rn_theta    =    6.0    !  surface control parameter (0<=theta<=20)
+   rn_thetb    =    0.75   !  bottom control parameter  (0<=thetb<= 1)
+   rn_rmax     =    0.15   !  maximum cut-off r-value allowed (0<r_max<1)
+   ln_s_sigma  = .false.   !  hybrid s-sigma coordinates
+   rn_bb       =    0.8    !  stretching with s-sigma
+   rn_hc       =  150.0    !  critical depth with s-sigma 
+/
