Index: /branches/nemo_v3_3_beta/DOC/NEMO_book.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/NEMO_book.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/NEMO_book.tex	(revision 2282)
@@ -21,4 +21,19 @@
 %hyperref  ===>>>  don't work.....
 %\usepackage[a4paper=true,ps2pdf=true,pagebackref=true,breaklinks=true]{hyperref}
+%gn
+\usepackage[%
+  pdftitle={NEMO ocean engine},%
+  pdfauthor={Gurvan Madec},% pdfsubject={The preprint document class
+                           % elsart},% pdfkeywords={diapycnal diffusion,numerical mixing,z-level models},%
+  pdfstartview=FitH,%
+  bookmarks=true,%
+  bookmarksopen=true,%
+  breaklinks=true,%
+  colorlinks=true,%
+  linkcolor=blue,anchorcolor=blue,%
+  citecolor=blue,filecolor=blue,%
+  menucolor=blue,pagecolor=blue,%
+  urlcolor=blue]{hyperref}
+%gn end
 
 %\usepackage{amssymb}
@@ -61,6 +76,7 @@
 \usepackage{times}					% use vector fonts instead of bitmap
 \usepackage{fancybox}
-\usepackage{color}
-\usepackage{colortbl}
+\usepackage{graphicx}
+%\usepackage{color}
+%\usepackage{colortbl}
 
 %%%% add by Word2Tex
@@ -70,4 +86,5 @@
 \usepackage{amsmath}
 \usepackage{graphicx}
+\allowdisplaybreaks[1]				% allow page breaks in the middle of equations
 %%%%
 
@@ -116,22 +133,28 @@
 %%%% end style chapter %%%
 
-%%% eqution shortcut:
-\newcommand{\vect} [1] { \rm{\textbf{#1}} }    % vector style: non-italic bold
+%%% use maths shortcuts :
+%\newcommand{\vect} [1] { \rm{\textbf{#1}} }    % vector style: non-italic bold
+\usepackage{./TexFiles/math_abbrev}
+
 
 %%% essai commande
 \newcommand{\nl} [1] {\texttt{\small {\textcolor{blue}{#1}} } } 
 
-%essai commande  
+%%% index commands :   
 \newcommand{\mdl} [1] {\textit{#1.F90}\index{Modules!#1}}			%module (mdl)
-\newcommand{\rou} [1] {\textit{#1}\index{Routines!#1}}			%module (routine)
-\newcommand{\hf} [1] {\textit{#1.h90}\index{h90 file!#1}}			%module (h90 files)
-\newcommand{\np} [1] {\textit{#1}\index{Namelist parameters!#1}}%namelist parameter (nampar)
-\newcommand{\jp} [1] {\textit{#1}\index{Model parameters!#1}}%model parameter (jp)
-\newcommand{\pp} [1] {\textit{#1}\index{Model parameters!#1}}%namelist parameter (pp)
+\newcommand{\rou} [1] {\textit{#1}\index{Routines!#1}}				%module (routine)
+\newcommand{\hf} [1] {\textit{#1.h90}\index{h90 file!#1}}				%module (h90 files)
+\newcommand{\np} [1] {\textit{#1}\index{Namelist parameters!#1}}		%namelist parameter (nampar)
+\newcommand{\jp} [1] {\textit{#1}\index{Model parameters!#1}}			%model parameter (jp)
+\newcommand{\pp} [1] {\textit{#1}\index{Model parameters!#1}}		 	%namelist parameter (pp)
+\newcommand{\ifile} [1] {\textit{#1.nc}\index{Input NetCDF files!#1.nc}}	%input NetCDF files (.nc)
 \newcommand{\key} [1] {\textbf{key\_#1}\index{CPP keys!key\_#1}}	%key_cpp (key)
 \newcommand{\NEMO} {\textit{NEMO }}						%NEMO (nemo)
-\newcommand{\amtcomment}[1]{#1}   	% command to allow "commented out" portions of 
-\newcommand{\gmcomment}[1]{}   	% command to allow "commented out" portions of 
+% command to "commented out" portions of text ({} argument) or not ({#1} argument)
+\newcommand{\amtcomment}[1]{}   	% command to "commented out" portions of text or not (#1 in argument)
+\newcommand{\sgacomment}[1]{}   	% command to "commented out" portions of 
+\newcommand{\gmcomment}[1]{}   	% command to "commented out" portions of 
 %                                    				% text that span line breaks
+\newcommand{\alpbet} {\left(\alpha / \beta \right)} 	% alpha/beta  for slp computation
 
 %--------------------------------------------namlist---------------------------------------------------------
@@ -165,24 +188,25 @@
 
 \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 3.3 --}   }
 %{ -- Draft --}   }
 %\date{\today}
 \date{
-May 2008  \\
+October 2010  \\
+~  \\
 \textit{\small Note du P\^ole de mod\'{e}lisation de l'Institut Pierre-Simon Laplace No 27 }\\
 \vspace{0.45cm}
@@ -199,7 +223,9 @@
 %%        }  }%
 
-\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 }
 }
 
@@ -208,12 +234,12 @@
 
 % ================================================================
-% ================================================================
 %      Include ONLY order
 % ================================================================
 
 %\includeonly{./TexFiles/Chapters/Chap_MISC}
-%\includeonly{./TexFiles/Chapters/Annex_E}
+%\includeonly{./TexFiles/Chapters/Chap_ZDF}
+%\includeonly{./TexFiles/Chapters/Chap_STP,./TexFiles/Chapters/Chap_SBC,./TexFiles/Chapters/Chap_TRA}
 %\includeonly{./TexFiles/Chapters/Chap_LBC,./TexFiles/Chapters/Chap_MISC}
-%\includeonly{./TexFiles/Chapters/Chap_Model_Basics_zstar}
+%\includeonly{./TexFiles/Chapters/Chap_Model_Basics}
 %\includeonly{./TexFiles/Chapters/Annex_A,./TexFiles/Chapters/Annex_B,./TexFiles/Chapters/Annex_C,./TexFiles/Chapters/Annex_D}
 
@@ -250,21 +276,21 @@
 \include{./TexFiles/Chapters/Chap_Model_Basics}
 
-%\include{./TexFiles/Chapters/Chap_Model_Basics_zstar}	% attent to describe z*-s*
-
-\include{./TexFiles/Chapters/Chap_DOM}
-
-\include{./TexFiles/Chapters/Chap_TRA}
-
-\include{./TexFiles/Chapters/Chap_DYN}
-
-\include{./TexFiles/Chapters/Chap_SBC}
-
-\include{./TexFiles/Chapters/Chap_LBC}
-
-\include{./TexFiles/Chapters/Chap_LDF}
-
-\include{./TexFiles/Chapters/Chap_ZDF}
-
-\include{./TexFiles/Chapters/Chap_MISC}
+\include{./TexFiles/Chapters/Chap_STP}			% Time discretisation (time stepping strategy)
+
+\include{./TexFiles/Chapters/Chap_DOM}			% Space discretisation 
+
+\include{./TexFiles/Chapters/Chap_TRA}			% Tracer advection/diffusion equation
+
+\include{./TexFiles/Chapters/Chap_DYN}			% Dynamics : momentum equation
+
+\include{./TexFiles/Chapters/Chap_SBC}			% Surface Boundary Conditions
+
+\include{./TexFiles/Chapters/Chap_LBC}			% Lateral Boundary Conditions
+
+\include{./TexFiles/Chapters/Chap_LDF}			% Lateral diffusion
+
+\include{./TexFiles/Chapters/Chap_ZDF}			% Vertical diffusion
+
+\include{./TexFiles/Chapters/Chap_MISC}			% Miscellaneous topics
 
 % ================================================================
@@ -275,9 +301,10 @@
 
 %\include{./TexFiles/Chapters/Chap_Conservation}
-\include{./TexFiles/Chapters/Annex_A}
-\include{./TexFiles/Chapters/Annex_B}
-\include{./TexFiles/Chapters/Annex_C}
-%\include{./TexFiles/Chapters/Annex_D}
-%\include{./TexFiles/Chapters/Annex_E}
+\include{./TexFiles/Chapters/Annex_A}			% generalised vertical coordinate
+\include{./TexFiles/Chapters/Annex_B}			% diffusive operator
+\include{./TexFiles/Chapters/Annex_C}			% Discrete invariants of the eqs.
+\include{./TexFiles/Chapters/Annex_D}			% Coding rules
+\include{./TexFiles/Chapters/Annex_ISO}	                  % Notes on some on going staff (no included in the DOC) 
+%\include{./TexFiles/Chapters/Annex_E}	                  % Notes on some on going staff (no included in the DOC) 
 
 % ================================================================
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Biblio/Biblio.bib
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Biblio/Biblio.bib	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Biblio/Biblio.bib	(revision 2282)
@@ -12,10 +12,12 @@
 @STRING{AW = {Addison-Wesley}}
 
+@STRING{CP = {Clarendon Press}}
+
 @STRING{CD = {Clim. Dyn.}}
 
-@STRING{CP = {Clarendon Press}}
-
 @STRING{CUP = {Cambridge University Press}}
 
+@STRING{CSR = {Cont. Shelf Res.}}
+
 @STRING{D = {Dover Publications}}
 
@@ -64,4 +66,8 @@
 @STRING{Ocean = {Oceanology}}
 
+@STRING{OD = {Ocean Dynamics}}
+
+@STRING{OM = {Ocean Modelling}}
+
 @STRING{OS = {Ocean Science}}
 
@@ -76,7 +82,9 @@
 @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}}
+
+@STRING{RGSP = {Rev. Geophys. Space Phys.}}
 
 @STRING{Science = {Science}}
@@ -88,26 +96,20 @@
 @ARTICLE{Adcroft_Campin_OM04,
   author = {A. Adcroft and J.-M. Campin},
-  title = {Re-scaled height coordinates for accurate representation of free-surface
-	flows in ocean circulation models},
-  journal = {Ocean Modelling},
-  year = {2004},
+  title = {Re-scaled height coordinates for accurate representation of free-surface flows in ocean circulation models},
+  journal = OM,
   volume = {7},
-  owner = {gm},
-  timestamp = {2008.01.27}
+  year = {2004}
 }
 
 @ARTICLE{Arakawa1966,
   author = {A. Arakawa},
-  title = {Computational design for long term numerical integration of the equations
-	of fluid motion, two-dimensional incompressible flow, Part. I.},
+  title = {Computational design for long term numerical integration of the equations of fluid motion, two-dimensional incompressible flow, Part. I.},
   journal = JCP,
   year = {1966},
   volume = {I},
-  pages = {119-149},
-  owner = {gm},
-  timestamp = {2007.08.04}
-}
-
-@ARTICLE{Arakawa1990,
+  pages = {119--149}
+}
+
+@ARTICLE{Arakawa_Hsu_MWR90,
   author = {A. Arakawa and Y.-J. G. Hsu},
   title = {Energy Conserving and Potential-Enstrophy Dissipating Schemes for
@@ -116,7 +118,5 @@
   year = {1990},
   volume = {118},
-  pages = {1960--1969
-	
-	},
+  pages = {1960--1969},
   number = {10},
   abstract = {To incorporate potential enstrophy dissipation into discrete shallow
@@ -134,5 +134,5 @@
 }
 
-@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 +141,5 @@
   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
@@ -171,8 +169,8 @@
   year = {2006},
   volume = { 36},
-  pages = {1502-1522}
-}
-
-@ARTICLE{ASSELIN1972,
+  pages = {1502--1522}
+}
+
+@ARTICLE{ASSELIN_MWR72,
   author = {R. Asselin},
   title = {Frequency Filter for Time Integrations},
@@ -180,5 +178,5 @@
   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 +196,66 @@
 }
 
+@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 = {10.1029/2001JC000922},
+  owner = {gm},
+  timestamp = {2009.01.16},
+  url = {http://dx.doi.org/10.1029/2001JC000922}
+}
+
+@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.
@@ -206,9 +266,12 @@
   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.},
-  owner = {gm},
-  timestamp = {2008.01.25}
+  volume = {56},
+  pages = {543--567},
+  doi = {10.1007/s10236-006-0082-1},
+  owner = {gm},
+  timestamp = {2008.01.25},
+  url = {http://dx.doi.org/10.1007/s10236-006-0082-1}
 }
 
@@ -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}
 }
 
@@ -246,12 +326,12 @@
 }
 
-@ARTICLE{BeckDos1998,
+@ARTICLE{Beckmann_Doscher1997,
   author = {A. Beckmann and R. D\"{o}scher},
   title = {A method for improved representation of dense water spreading over
 	topography in geopotential-coordinate models},
   journal = JPO,
-  year = {1998},
+  year = {1997},
   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 +382,73 @@
 }
 
+@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},
+  doi = {10.1007/s00382-008-0429-z},
+  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{Bernie_al_JC05,
+  author = {D. J. Bernie and S. J. Woolnough and J. M. Slingo and E. Guilyardi},
+  title = {Modelling diurnal and intraseasonal variability of the ocean mixed
+	layer},
+  journal = JC,
+  year = {2005},
+  volume = {18 (8)},
+  pages = {1190--1200},
+  owner = {gm},
+  timestamp = {2010.09.26}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1029/2007GL032644}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1175/2008JCLI2404.1}
+}
+
 @ARTICLE{Blanke_al_JPO99,
   author = {B. Blanke and M. Arhan and G. Madec and S. Roche},
@@ -323,5 +470,5 @@
   year = {1993},
   volume = {23},
-  pages = {1363-1388}
+  pages = {1363--1388}
 }
 
@@ -332,5 +479,5 @@
   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 +550,29 @@
   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 +581,20 @@
   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 +606,5 @@
   year = {1989},
   volume = {117},
-  pages = {1872-1890},
+  pages = {1872--1890},
   number = {8},
   abstract = {The possibility of extending existing techniques for turbulence parameterization
@@ -442,5 +623,45 @@
 }
 
-@ARTICLE{Brown1978,
+@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},
+  url = {http://dx.doi.org/10.1029/2004JC002378}
+}
+
+@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{Brown_Campana_MWR78,
   author = {J. A. Brown and K. A. Campana},
   title = {An Economical Time-Differencing System for Numerical Weather Prediction},
@@ -448,5 +669,5 @@
   year = {1978},
   volume = {106},
-  pages = {1125-1136},
+  pages = {1125--1136},
   number = {8},
   month = aug,
@@ -495,7 +716,22 @@
   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 = {10.1016/S1463-5003(02)00009-4},
+  owner = {gm},
+  timestamp = {2008.11.28},
+  url = {http://dx.doi.org/10.1016/S1463-5003(02)00009-4}
 }
 
@@ -503,10 +739,24 @@
   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 = {10.1016/j.ocemod.2008.05.005},
+  issn = {1463-5003},
+  timestamp = {2010.01.20},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2008.05.005}
 }
 
@@ -518,7 +768,68 @@
   year = {1999},
   volume = {51},
-  pages = {412-430},
+  pages = {412--430},
   owner = {gm},
   timestamp = {2008.01.20}
+}
+
+@ARTICLE{Carrere_Lyard_GRL03,
+  author = {L. Carr\`{e}re and F. Lyard},
+  title = {Modelling the barotropic response of the global ocean to atmospheric
+	wind and pressure forcing - comparisons with observations},
+  journal = GRL,
+  year = {2003},
+  volume = {30, (6)},
+  doi = {10.1029/2002GL016473},
+  owner = {gm},
+  timestamp = {2010.05.21},
+  url = {http://dx.doi.org/10.1029/2002GL016473}
+}
+
+@TECHREPORT{Chanut2005,
+ author = {J. Chanut},
+ title = {Nesting code for NEMO},
+ year = {2005},
+ institution = {European Union: Marine Environment and Security for the European Area (MERSEA) Integrated Project},
+note = {MERSEA-WP09-MERCA-TASK-9.1.1}
+} 
+
+@ARTICLE{Chassignet_al_JPO03,
+  author = {Eric P. Chassignet and Linda T. Smith and George R. Halliwell},
+  title = {North Atlantic Simulations with the Hybrid Coordinate Ocean Model
+	(HYCOM): Impact of the Vertical Coordinate Choice, Reference Pressure,
+	and Thermobaricity},
+  journal = {Journal of Physical Oceanography},
+  year = {2003},
+  volume = {33},
+  pages = {2504-2526},
+  timestamp = {2010.02.01}
+}
+
+@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}
+}
+
+@ARTICLE{Canuto_2001,
+  author = {V. M. Canuto and A. Howard and Y. Cheng and M. S. Dubovikov},
+  title = {Ocean turbulence. PartI: One-point closure model-momentum and heat vertical diffusivities},
+  journal = JPO,
+  year = {2001},
+  volume = {24, 12},
+  pages = {2546--2559},
+  owner = {gr},
+  timestamp = {2010.09.09}
 }
 
@@ -526,10 +837,33 @@
   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{Craig_Banner_1994,
+  author = {P. D. Banner and M. L. Banner},
+  title = {Modeling wave-enhanced turbulence in the ocean surface layer},
+  journal = JPO,
+  year = {1994},
+  volume = {24, 12},
+  pages = {2546--2559},
+  owner = {g5},
+  timestamp = {2010.09.09}
+}
+
+@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 +875,28 @@
   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}
+}
+
+@ARTICLE{Davies_QJRMS76,
+ author = {H.C. Davies},
+ title = {A lateral boundary formulation for multi-level prediction models},
+ year = {1976},
+ journal = {QJRMS},
+ volume = {102},
+ pages = {405--418}
 }
 
@@ -552,5 +907,5 @@
   year = {2008},
   volume = {34},
-  pages = {8-13},
+  pages = {8--13},
   owner = {gm},
   timestamp = {2008.02.03}
@@ -562,10 +917,29 @@
   journal = {In \textit{Modeling the Earth's Climate and its Variability}, Les
 	Houches, Session, LXVII 1997,
-	
-	Eds. W. R. Holland, S. Joussaume and F. David, Elsevier Science,},
+	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},
+  url = {http://dx.doi.org/10.1029/2003GB002150}
 }
 
@@ -576,7 +950,35 @@
   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},
+  url = {http://dx.doi.org/10.1029/2003GL018906}
 }
 
@@ -590,4 +992,34 @@
 }
 
+@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 +1029,7 @@
 	
 	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 +1082,43 @@
 }
 
+@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},
+  url = {http://dx.doi.org/10.1029/2005GL022463}
+}
+
+@ARTICLE{Egbert_Ray_JGR01,
+  author = {G.B. Egbert and R.D. Ray},
+  title = {Estimates of M2 tidal energy dissipation from TOPEX/POSEIDON altimeter
+	data},
+  journal = JGR,
+  year = {2001},
+  volume = {106},
+  pages = {22475--22502},
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
+@ARTICLE{Egbert_Ray_Nat00,
+  author = {G.B. Egbert and R.D Ray},
+  title = {Significant dissipation of tidal energy in the deep ocean inferred
+	from satellite altimeter data},
+  journal = {Nature},
+  year = {2000},
+  volume = {405},
+  pages = {775--778},
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
 @ARTICLE{Eiseman1980,
   author = {P. R. Eiseman and A. P. Stone},
@@ -656,18 +1127,57 @@
   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{Engerdahl_Tel95,
+ author = {H. Engerdahl},
+ year = {1995},
+ title = {Use of the flow relaxation scheme in a three-dimensional baroclinic ocean model with realistic topography},
+ journal = {Tellus},
+ volume = {47A},
+ pages = {365--382},
+}
+
+@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}
 }
 
@@ -689,7 +1199,25 @@
   year = {1995},
   volume = {25},
-  pages = {1731-1741.},
+  pages = {1731--1741.},
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@ARTICLE{Flather1976,
+ author = {R.A. Flather},
+ year = {1976},
+ title = {A tidal model of the north-west European continental shelf},
+ journal = {Memoires de la Societ\'{e} Royale des Sciences de Li\`{e}ge},
+ volume = {6},
+ pages = {141--164}
+}
+
+@ARTICLE{Flather_JPO94,
+ author = {R.A. Flather},
+ year = {1994},
+ title = {A storm surge prediction model for the northern Bay of Bengal with application to the cyclone disaster in April 1991},
+ journal = {JPO},
+ volume = {24},
+ pages = {172--190}
 }
 
@@ -701,8 +1229,19 @@
   year = {1991},
   volume = {96},
-  pages = {759-774},
+  pages = {759--774},
   month = jan,
   owner = {gm},
   timestamp = {2007.08.04}
+}
+
+@ARTICLE{Galperin_al_JAS88,
+  author = {B. Galperin and L. H. Kantha and S. Hassid and A. Rosati},
+  title = {A quasi-equilibrium turbulent energy model for geophysical flows},
+  journal = JAS,
+  year = {1988},
+  volume = {45},
+  pages = {55--62},
+  owner = {gr},
+  timestamp = {2010.09.09}
 }
 
@@ -735,5 +1274,5 @@
   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 +1305,5 @@
   year = {1993},
   volume = {98},
-  pages = {14703-14726},
+  pages = {14703--14726},
   owner = {gm},
   timestamp = {2007.08.03}
@@ -794,17 +1333,83 @@
   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},
+  url = {http://dx.doi.org/10.1029/2006GL028210}
+}
+
+@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},
+  url = {http://dx.doi.org/}
+}
+
+@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 +1470,12 @@
 }
 
-@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 +1495,6 @@
 }
 
-@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 +1502,5 @@
   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 +1524,31 @@
 }
 
-@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 +1557,20 @@
   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 +1581,5 @@
   year = {1999},
   volume = {Euro-Par'99},
-  pages = {603-607},
+  pages = {603--607},
   owner = {gm},
   timestamp = {2008.05.27}
@@ -969,5 +1593,5 @@
   year = {1999},
   volume = {11, 4},
-  pages = {499-517},
+  pages = {499--517},
   owner = {gm},
   timestamp = {2008.05.27}
@@ -990,7 +1614,5 @@
   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 +1641,45 @@
 }
 
+@ARTICLE{Hirt_al_JCP74,
+  author = {C. W. Hirt and A. A. Amsden and J. L. Cook},
+  title = {An Arbitrary Lagrangian--Eulerian Computing Method for All Flow Speeds},
+  journal = JCP,
+  year = {1974},
+  volume = {14},
+  pages = {227-253},
+  owner = {gm},
+  timestamp = {2010.04.14}
+}
+
+@ARTICLE{Hofmeister_al_OM09,
+  author = {Richard Hofmeister and Hans Burchard and Jean-Marie Beckers},
+  title = {Non-uniform adaptive vertical grids for 3D numerical ocean models},
+  journal = {Ocean Modelling},
+  year = {2009},
+  volume = {33},
+  pages = {70 - 86},
+  doi = {10.1016/j.ocemod.2009.12.003},
+  file = {:Users/mlelod/Documents/Biblio/vertical_coordinates/Hofmeister_et_al_OM_2010.pdf:PDF},
+  issn = {1463-5003},
+  keywords = {Adaptive non-uniform grid},
+  owner = {mlelod},
+  timestamp = {2010.01.21},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2009.12.003}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1007/s00382-008-0416-4}
+}
+
 @ARTICLE{Hsu1990,
   author = {Hsu, Yueh-Jiuan G. and Arakawa, Akio},
@@ -1026,7 +1689,5 @@
   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 +1717,74 @@
 }
 
+@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 +1793,22 @@
   year = {1995},
   volume = {12},
-  pages = {381-389},
+  pages = {381--389},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@BOOK{Jerlov1968,
+@ARTICLE{Jayne_St_Laurent_GRL01,
+  author = {S.R. Jayne and L.C. {St. Laurent}},
+  title = {Parameterizing tidal disspiation over rough topography
+	
+	.},
+  journal = GRL,
+  pages = {811--814},
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
+@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 +1819,39 @@
 }
 
-@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{Kantha_Clayson_1994,
+  author = {L. H. Kantha and C. A. Clayson},
+  title = {An improved mixed layer model for geophysical applications},
+  journal = JGR,
+  year = {1994},
+  volume = {99},
+  pages = {25,235--25,266},
+  owner = {gr},
+  timestamp = {2010.09.09}
+}
+
+@ARTICLE{Kantha_Carniel_CSR05,
+  author = {L. Kantha and S. Carniel},
+  title = {Comment on ''Generic length-scale equation for geophysical turbulence models'' by L. Umlauf and H. Burchard},
+  journal = {Journal of Marine Systems}, 
+  year = {2005},
+  volume = {61},
+  pages = {693--702},
+  owner = {gm},
+  timestamp = {2010.09.09}
+}
+
+@ARTICLE{Kasahara_MWR74,
+  author = {A. Kasahara},
+  title = {Various vertical coordinate systems used for numerical weather prediction},
+  journal = MWR,
+  year = {1974},
+  volume = {102},
+  pages = {509--522},
+  owner = {gm},
+  timestamp = {2010.04.14}
+}
+
+@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 +1878,84 @@
 }
 
+@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_CD10,
+  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 = {2010},
+  volume = {34,6},
+  pages = {891--904},
+  doi = {10.1007/s00382-009-0642-4},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1007/s00382-009-0642-4}
+}
+
+@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},
+  doi = {10.1007/s10236-008-0155-4},
+  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},
+  doi = {10.1029/2006GL028405},
+  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},
+  doi = {10.1007/s10236-008-0154-5},
+  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 +1964,10 @@
   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 +1979,5 @@
 }
 
-@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 +1989,53 @@
 }
 
-@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},
@@ -1195,11 +2055,24 @@
   year = {1994},
   volume = {32},
-  pages = {363-404},
+  pages = {363--404},
   doi = {10.1029/94RG01872},
   owner = {gm},
-  timestamp = {2007.08.03}
-}
-
-@PHDTHESIS{Lazar1997,
+  timestamp = {2007.08.03},
+  url = {http://dx.doi.org/10.1029/94RG01872}
+}
+
+@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 +2085,5 @@
 }
 
-@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 +2092,5 @@
   year = {1999},
   volume = {29},
-  pages = {2945-2961},
+  pages = {2945--2961},
   number = {11},
   abstract = {Numerous numerical simulations of basin-scale ocean circulation display
@@ -1256,4 +2129,67 @@
 }
 
+@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},
+  doi = {10.1016/j.ocemod.2008.11.007},
+  owner = {gm},
+  timestamp = {2009.08.16},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2008.11.007}
+}
+
+@PHDTHESIS{Leclair_PhD2010,
+  author = {M. Leclair},
+  title = {introduction d'une coordonn\'{e}e verticale arbitrairement Lagrangienne
+	Eul\'{e}rienne dans le code NEMO, 180pp.},
+  school = {Universit\'{e} Pierre and Marie Curie},
+  year = {2010},
+  owner = {gm},
+  timestamp = {2010.09.25}
+}
+
+@ARTICLE{Leclair_Madec_OM09,
+  author = {M. Leclair and G. Madec},
+  title = {A conservative leap-frog time stepping method},
+  journal = OM,
+  year = {2009},
+  volume = {30, 2-3},
+  pages = {88-94},
+  doi = {10.1016/j.ocemod.2009.06.006},
+  owner = {gm},
+  timestamp = {2009.08.15},
+  url = {http://dx.doi.org/}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1029/2001JC000841}
+}
+
 @ARTICLE{Lengaigne_al_JGR03,
   author = {M. Lengaigne and G. Madec and G. Alory and C. Menkes},
@@ -1263,7 +2199,34 @@
   year = {2003},
   volume = {108 (C11)},
-  pages = {3345, doi:10.1029/2002JC001704},
-  owner = {gm},
-  timestamp = {2008.01.26}
+  pages = {3345},
+  doi = {10.1029/2002JC001704},
+  owner = {gm},
+  timestamp = {2008.01.26},
+  url = {http://dx.doi.org/10.1029/2002JC001704}
+}
+
+@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 +2237,10 @@
   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 +2259,5 @@
   year = {1979},
   volume = {19},
-  pages = {59-98},
+  pages = {59--98},
   month = jun,
   owner = {gm},
@@ -1331,20 +2294,31 @@
 }
 
-@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 +2327,28 @@
 }
 
-@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 +2357,11 @@
   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 +2372,15 @@
   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 +2395,5 @@
   year = {1997},
   volume = {26},
-  pages = {12-15},
+  pages = {12--15},
   owner = {gm},
   timestamp = {2007.08.06}
@@ -1416,5 +2407,5 @@
 }
 
-@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 +2413,8 @@
   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 +2423,5 @@
   year = {1996},
   volume = {26},
-  pages = {1393-1408},
+  pages = {1393--1408},
   number = {8},
   month = aug,
@@ -1467,13 +2458,37 @@
 }
 
-@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 +2500,5 @@
   year = {1998},
   volume = {103(C13)},
-  pages = {30,825-30,854},
+  pages = {30,825--30,854},
   owner = {gm},
   timestamp = {2007.08.05}
@@ -1494,8 +2509,24 @@
   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},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2007.07.005}
 }
 
@@ -1512,5 +2543,5 @@
 }
 
-@PHDTHESIS{MartiTh1992,
+@PHDTHESIS{Marti_PhD92,
   author = {O. Marti},
   title = {Etude de l'oc\'{e}an mondial : mod\'{e}lisation de la circulation
@@ -1522,5 +2553,22 @@
 }
 
-@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 +2577,40 @@
   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},
+  url = {http://dx.doi.org/10.1007/978-3-540-74384-2}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1029/2004GL021980}
 }
 
@@ -1538,8 +2618,8 @@
   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 +2656,45 @@
   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{Mellor_Yamada_1982,
+  author = {G. L. Mellor and T. Yamada},
+  title = {Development of a turbulence closure model for geophysical fluid problems},
+  journal = RGSP,
+  year = {1982},
+  volume = {20},
+  pages = {851-875},
+  owner = {gr},
+  timestamp = {2010.09.09}
+}
+
+@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 +2705,5 @@
   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 +2732,36 @@
 }
 
-@ARTICLE{Murray1996,
+@ARTICLE{Morel_JGR88,
+  author = {A. Morel},
+  title = {Optical modeling of the upper ocean in relation to its biogenous
+	matter content (Case I waters)},
+  journal = JGR,
+  year = {1988},
+  volume = {93},
+  pages = {10,749--10,768},
+  owner = {gm},
+  timestamp = {2010.10.07}
+}
+
+@ARTICLE{Morel_Maritorena_JGR01,
+  author = {A. Morel and S. Maritorena},
+  title = {Bio-optical properties of oceanic waters: a reappraisal},
+  journal = JGR,
+  year = {2001},
+  owner = {gm},
+  timestamp = {2010.10.07}
+}
+
+@ARTICLE{Moun_al_JPO02,
+  author = {J.N. Moum and D.R. Caldwell and J.D. Nash and G.D. Gunderson},
+  title = {Observations of boundary mixing over the 
+	
+	continental slope},
+  journal = JPO,
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
+@ARTICLE{Murray_JCP96,
   author = {R. J. Murray},
   title = {Explicit Generation of Orthogonal Grids for Ocean Models},
@@ -1623,5 +2769,5 @@
   year = {1996},
   volume = {126},
-  pages = {251-273},
+  pages = {251--273},
   number = {2},
   month = {July},
@@ -1630,5 +2776,5 @@
 }
 
-@PHDTHESIS{OlivierPh2001,
+@PHDTHESIS{Olivier_PhD01,
   author = {F. Olivier},
   title = {Etude de l'activit\'{e} biologique et de la circulation oc\'{e}anique
@@ -1640,5 +2786,16 @@
 }
 
-@ARTICLE{PacPhil1981,
+@ARTICLE{Osborn_JPO80,
+  author = {T.R. Osborn},
+  title = {Estimates of the local rate of vertical diﬀusion from dissipation
+	measurements},
+  journal = JPO,
+  volume = {10},
+  pages = {83--89},
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
+@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 +2804,5 @@
   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 +2839,19 @@
 }
 
+@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},
+  url = {http://dx.doi.org/10.1175/2008JCLI2261.1}
+}
+
 @ARTICLE{Paulson1977,
   author = {C. A. Paulson and J. J. Simpson},
@@ -1688,5 +2860,5 @@
   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 +2887,35 @@
 }
 
-@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 +2926,35 @@
   journal = {Transactions of the American Mathematical Society},
   year = {1959},
-  volume = {90(2)},
-  pages = {193-254},
-  doi = {doi:10.2307/1993202},
-  owner = {gm},
-  timestamp = {2007.08.10}
+  volume = {90 (2)},
+  pages = {193--254},
+  doi = {10.2307/1993202},
+  owner = {gm},
+  timestamp = {2007.08.10},
+  url = {http://dx.doi.org/10.2307/1993202}
+}
+
+@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 +2965,5 @@
   year = {1982},
   volume = {13},
-  pages = {1154-1158},
+  pages = {1154--1158},
   owner = {gm},
   timestamp = {2008.02.02}
@@ -1782,13 +2992,50 @@
 }
 
-@ARTICLE{Robert1966,
+@ARTICLE{Robert_JMSJ66,
   author = {A. J. Robert},
   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},
+  url = {http://dx.doi.org/10.1029/2004GL019764}
+}
+
+@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},
+  url = {http://dx.doi.org/10.1029/2002GL016003}
+}
+
+@ARTICLE{Rodi_1987,
+  author = {W. Rodi},
+  title = {Examples of calculation methods for flow and mixing in stratified fluids},
+  journal = JGR,
+  year = {1987},
+  volume = {92, C5},
+  pages = {5305--5328},
+  owner = {gr},
+  timestamp = {2010.09.09}
 }
 
@@ -1802,5 +3049,5 @@
 }
 
-@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 +3056,6 @@
   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 +3068,5 @@
   year = {1975},
   volume = {32},
-  pages = {680-689},
+  pages = {680--689},
   number = {4},
   abstract = {Two simple numerical models of the shallow-water equations identical
@@ -1847,10 +3094,10 @@
   year = {1982},
   volume = {87},
-  pages = {394-409},
+  pages = {394--409},
   owner = {gm},
   timestamp = {2007.08.04}
 }
 
-@ARTICLE{Sacha2005,
+@ARTICLE{Shchepetkin_McWilliams_OM05,
   author = {A. F. Shchepetkin and J. C. McWilliams},
   title = {The regional oceanic modeling system (ROMS) - a split-explicit, free-surface,
@@ -1859,5 +3106,5 @@
   year = {2005},
   volume = {9, 4},
-  pages = {347-404},
+  pages = {347--404},
   owner = {gm},
   timestamp = {2007.08.04}
@@ -1873,7 +3120,9 @@
   year = {2003},
   volume = {108(C3)},
-  pages = {3090, doi:10.1029/2001JC001047},
-  owner = {gm},
-  timestamp = {2007.08.05}
+  pages = {3090},
+  doi = {10.1029/2001JC001047},
+  owner = {gm},
+  timestamp = {2007.08.05},
+  url = {http://dx.doi.org/10.1029/2001JC001047}
 }
 
@@ -1885,5 +3134,5 @@
   year = {1996},
   volume = {124},
-  pages = {1285-1300},
+  pages = {1285--1300},
   number = {6},
   abstract = {Dissipation in numerical ocean models has two purposes: to simulate
@@ -1916,12 +3165,10 @@
 }
 
-@ARTICLE{Simmons2003,
-  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
-	
-	ocean general circulation},
+@ARTICLE{Simmons_al_OM04,
+  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 ocean general circulation},
   journal = OM,
-  year = {2003},
-  pages = {1-19},
+  year = {2004},
+  pages = {245--263},
   abstract = {Astronomical data reveals that approximately 3.5 terawatts (TW) of
 	tidal energy is dissipated in the
@@ -1965,18 +3212,17 @@
 	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{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 +3236,5 @@
   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 +3282,5 @@
   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 +3317,17 @@
 }
 
-@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
@@ -2081,12 +3339,60 @@
 }
 
-@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}
+}
+
+@ARTICLE{St_Laurent_Garrett_JPO02,
+  author = {L.C. {St. Laurent} and C. Garrett},
+  title = {The role of internal tides in mixing the deep ocean},
+  journal = JPO,
+  pages = {2882--2899},
+  owner = {gm},
+  timestamp = {2010.05.20}
+}
+
+@ARTICLE{St_Laurent_Nash_DSR04,
+  author = {L.C. {St. Laurent} and J. D. Nash},
+  title = {An examination of the radiative and dissipative properties of deep
+	ocean internal tides},
+  journal = DSR,
+  year = {2004},
+  volume = {II, 51},
+  pages = {3029--3042},
+  doi = {10.1016/j.dsr2.2004.09.008},
+  owner = {gm},
+  timestamp = {2010.05.21},
+  url = {http://dx.doi.org/10.1016/j.dsr2.2004.09.008}
+}
+
+@ARTICLE{St_Laurent_al_GRL02,
+  author = {L.C. {St. Laurent} and H.L. Simmons and S.R. Jayne},
+  title = {Estimating tidally driven mixing in the deep ocean},
+  journal = GRL,
+  year = {2002},
+  volume = {29},
+  pages = {2106},
+  doi = {10.1029/2002GL015633},
+  owner = {gm},
+  timestamp = {2010.05.20},
+  url = {http://dx.doi.org/10.1029/2002GL015633}
 }
 
@@ -2094,5 +3400,5 @@
   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 +3422,5 @@
 }
 
-@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 +3429,14 @@
   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 +3452,5 @@
 	
 	sea ice-ocean model.},
-  journal = {Ocean Modelling},
+  journal = OM,
   year = {2005},
   volume = {8},
@@ -2154,5 +3460,5 @@
 }
 
-@ARTICLE{Treguier1992,
+@ARTICLE{Treguier_JGR92,
   author = {A.M. Tr\'{e}guier},
   title = {Kinetic energy analysis of an eddy resolving, primitive equation
@@ -2164,5 +3470,5 @@
 }
 
-@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 +3479,15 @@
   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 +3499,36 @@
   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 = {10.1007/s10236-006-0069-y},
+  owner = {gm},
+  timestamp = {2009.08.19},
+  url = {http://dx.doi.org/10.1007/s10236-006-0069-y}
 }
 
@@ -2195,5 +3540,5 @@
   year = {1997},
   volume = {27},
-  pages = {567-580},
+  pages = {567--580},
   number = {4},
   abstract = {A parameterization of mesoscale eddy fluxes in the ocean should be
@@ -2214,4 +3559,27 @@
 }
 
+@ARTICLE{Umlauf_Burchard_JMS03,
+  author = {L. Umlauf and H. Burchard},
+  title = {A generic length-scale equation for geophysical turbulence models},
+  journal = {JMS}, 
+  year = {2003},
+  volume = {61},
+  pages = {235--265},
+  number = {2},
+  owner = {gr},
+  timestamp = {2010.09.09}
+}
+
+@ARTICLE{Umlauf_Burchard_CSR05,
+  author = {L. Umlauf and H. Burchard},
+  title = {Second-order turbulence closure models for geophysical boundary layers. A review of recent work},
+  journal = {Journal of Marine Systems}, 
+  year = {2005},
+  volume = {25},
+  pages = {795--827},
+  owner = {gm},
+  timestamp = {2010.09.09}
+}
+
 @BOOK{UNESCO1983,
   title = {Algorithms for computation of fundamental property of sea water},
@@ -2233,25 +3601,54 @@
 }
 
-@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},
-  owner = {gm},
-  timestamp = {2008.07.05}
-}
-
-@ARTICLE{Weatherly1984,
+  journal = OM,
+  year = {2009},
+  volume = {27},
+  pages = {33--53},
+  doi = {10.1016/j.ocemod.2008.10.005},
+  owner = {gm},
+  timestamp = {2008.07.05},
+  url = {http://dx.doi.org/10.1016/j.ocemod.2008.10.005}
+}
+
+@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 +3661,5 @@
 }
 
-@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 +3674,5 @@
 }
 
-@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 +3707,28 @@
 }
 
-@ARTICLE{Willebrand2001,
+@ARTICLE{White_al_JCP09,
+  author = {L. White and A. Adcroft and R. Hallberg},
+  title = {High-order regridding-remapping schemes for continuous isopycnal
+	and generalized coordinates in ocean models},
+  journal = JC,
+  year = {2009},
+  volume = {228},
+  pages = {8665--8692},
+  owner = {gm},
+  timestamp = {2010.04.14}
+}
+
+@ARTICLE{Wilcox_1988,
+  author = {D. C. Wilcox},
+  title = {Reassessment of the scale-determining equation for advanced turbulence models},
+  journal = {AIAA journal},
+  year = {1988},
+  volume = {26, 11},
+  pages = {1299--1310},
+  owner = {gr},
+  timestamp = {2010.09.09}
+}
+
+@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 +3736,51 @@
   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},
+  url = {http://dx.doi.org/10.1029/2007GL029275}
+}
+
+@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 +3792,5 @@
 }
 
-@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 +3799,5 @@
   year = {1992},
   volume = {97},
-  pages = {11237-11255},
+  pages = {11,237--11,255},
   month = jul,
   owner = {gm}
@@ -2361,5 +3819,5 @@
 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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Abstracts_Foreword.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Abstracts_Foreword.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Abstracts_Foreword.tex	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_A.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_A.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_A.tex	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_B.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_B.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_B.tex	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_C.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_C.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_C.tex	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_D.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_D.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_D.tex	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_E.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_E.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_E.tex	(revision 2282)
@@ -1,9 +1,16 @@
 % ================================================================
-% Appendix E : UBS scheme
-% ================================================================
-\chapter{UBS scheme}
-\label{AN_E}
+% Appendix E : Note on some algorithms
+% ================================================================
+\chapter{Note on some algorithms}
+\label{Apdx_E}
 \minitoc
 
+\newpage
+$\ $\newline    % force a new ligne
+ 
+ This appendix some on going consideration on algorithms used or planned to be used
+in \NEMO. 
+
+$\ $\newline    % force a new ligne
 
 % -------------------------------------------------------------------------------------------------------------
@@ -39,5 +46,5 @@
 
 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation 
-error \citep{Sacha2005}. The overall performance of the 
+error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the 
 advection scheme is similar to that reported in \cite{Farrow1995}. 
 It is a relatively good compromise between accuracy and smoothness. It is 
@@ -56,7 +63,7 @@
 (centred in time) while the second term which is the diffusive part of the scheme, 
 is evaluated using the \textit{before} velocity (forward in time. This is discussed 
-by \citet{Webb1998} in the context of the Quick advection scheme. UBS and QUICK 
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme. UBS and QUICK 
 schemes only differ by one coefficient. Substituting 1/6 with 1/8 in 
-(\ref{Eq_tra_adv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
+(\ref{Eq_tra_adv_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 
@@ -74,5 +81,5 @@
 evaluated using either \textit{(a)} a centred $2^{nd}$ order scheme , 
 or  \textit{(b)} a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
-parabolic splines following \citet{Sacha2005} implementation of UBS in ROMS, 
+parabolic splines following \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, 
 or  \textit{(d)} an UBS. The $3^{rd}$ case has dispersion properties similar to an 
 eight-order accurate conventional scheme.
@@ -185,10 +192,10 @@
 \begin{subequations} \label{dt_mt}
 \begin{align}
- \delta _{t+\Delta t/2} [q]     &=  \  \ \,   q^{t+\Delta t}  - q^{t}		\\
- \overline q^{\,t+\Delta t/2} &= \left\{ q^{t+\Delta t} + q^{t} \right\} \; / \; 2
+ \delta _{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}		\\
+ \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2
 \end{align}
 \end{subequations}
 As for space operator, the adjoint of the derivation and averaging time operators are 
-$\delta_t^*=\delta_{t+\Delta t/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$
+$\delta_t^*=\delta_{t+\rdt/2}$ and $\overline{\cdot}^{\,t\,*}= \overline{\cdot}^{\,t+\Delta/2}$
 , respectively. 
 
@@ -196,8 +203,8 @@
 \begin{equation} \label{LF}
    \frac{\partial q}{\partial t} 
-   		\equiv \frac{1}{\Delta t} \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} 
-		=         \frac{q^{t+\Delta t}-q^{t-\Delta t}}{2\Delta t}
+   		\equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} 
+		=         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt}
 \end{equation} 
-Note that \eqref{LF} shows that the leapfrog time step is $\Delta t$, not $2\Delta t$ 
+Note that \eqref{LF} shows that the leapfrog time step is $\rdt$, not $2\rdt$ 
 as it can be found sometime in literature. 
 The leap-Frog time stepping is a second order centered scheme. As such it respects 
@@ -212,9 +219,9 @@
 \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
 	&\equiv \sum\limits_{0}^{N} 
-			{\frac{1}{\Delta t} q^t \ \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} \ \Delta t} 
-	   \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta _{t+\Delta t/2}[q]}^{\,t} } \\
-	&\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\Delta t/2}[q]}}
-	   \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\Delta t/2}[q^2] }\\
-	&\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\Delta t/2}[q^2] }
+			{\frac{1}{\rdt} q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} \ \rdt} 
+	   \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta _{t+\rdt/2}[q]}^{\,t} } \\
+	&\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta _{t+\rdt/2}[q]}}
+	   \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }\\
+	&\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta _{t+\rdt/2}[q^2] }
 	   \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
 \end{split} \end{equation}
@@ -226,2 +233,574 @@
 
 
+
+
+
+% ================================================================
+% Iso-neutral diffusion : 
+% ================================================================
+
+\section{Lateral diffusion operator}
+
+% ================================================================
+% Griffies' iso-neutral diffusion operator : 
+% ================================================================
+\subsection{Griffies' iso-neutral diffusion operator}
+
+Let try to define a scheme that get its inspiration from the \citet{Griffies_al_JPO98}
+scheme, but is formulated within the \NEMO framework ($i.e.$ using scale 
+factors rather than grid-size and having a position of $T$-points that is not 
+necessary in the middle of vertical velocity points, see Fig.~\ref{Fig_zgr_e3}).
+
+In the formulation \eqref{Eq_tra_ldf_iso} introduced in 1995 in OPA, the ancestor of \NEMO, 
+the off-diagonal terms of the small angle diffusion tensor contain several double 
+spatial averages of a gradient, for example $\overline{\overline{\delta_k \cdot}}^{\,i,k}$. 
+It is apparent that the combination of a $k$ average and a $k$ derivative of the tracer 
+allows for the presence of grid point oscillation structures that will be invisible 
+to the operator. These structures are \textit{computational modes}. They
+will not be damped by the iso-neutral operator, and even possibly amplified by it. 
+In other word, the operator applied to a tracer does not warranties the decrease of 
+its global average variance. To circumvent this, we have introduced a smoothing of 
+the slopes of the iso-neutral surfaces (see \S\ref{LDF}). Nevertheless, this technique 
+works fine for $T$ and $S$ as they are active tracers ($i.e.$ they enter the computation 
+of density), but it does not work for a passive tracer.   \citep{Griffies_al_JPO98} introduce 
+a different way to discretise the off-diagonal terms that nicely solve the problem. 
+The idea is to get rid of combinations of an averaged in one direction combined 
+with a derivative in the same direction by considering triads. For example in the 
+(\textbf{i},\textbf{k}) plane, the four triads are defined at the $(i,k)$ $T$-point as follows:
+\begin{equation} \label{Gf_triads}
+_i^k \mathbb{T}_{i_p}^{k_p} (T)
+= \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k  	\left(  
+                                                     \frac{ \delta_{i + i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} } 
+-\ {_i^k \mathbb{R}_{i_p}^{k_p}} \ \frac{ \delta_{k+k_p} [T^i] }{ {e_{3w}}_{\,i}^{\,k+k_p} } 
+									  \right)
+\end{equation}
+where the indices $i_p$ and $k_p$ define the four triads and take the following value: 
+$i_p = -1/2$ or $1/2$ and $k_p = -1/2$ or $1/2$, 
+$b_u= e_{1u}\,e_{2u}\,e_{3u}$ is the volume of $u$-cells, 
+$A_i^k$ is the lateral eddy diffusivity coefficient defined at $T$-point,
+and $_i^k \mathbb{R}_{i_p}^{k_p}$ is the slope associated with each triad :
+\begin{equation} \label{Gf_slopes}
+_i^k \mathbb{R}_{i_p}^{k_p} 
+=\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}} \ \frac 
+{\left(\alpha / \beta \right)_i^k  \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] }
+{\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }
+\end{equation}
+Note that in \eqref{Gf_slopes} we use the ratio $\alpha / \beta$ instead of 
+multiplying the temperature derivative by $\alpha$ and the salinity derivative 
+by $\beta$. This is more efficient as the ratio $\alpha / \beta$ can to be 
+evaluated directly.
+
+Note that in \eqref{Gf_triads}, we chose to use ${b_u}_{\,i+i_p}^{\,k}$ instead of 
+${b_{uw}}_{\,i+i_p}^{\,k+k_p}$. This choice has been motivated by the decrease 
+of tracer variance and the presence of partial cell at the ocean bottom 
+(see Appendix~\ref{Apdx_Gf_operator}).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht] \label{Fig_ISO_triad}
+\begin{center}
+\includegraphics[width=0.70\textwidth]{./TexFiles/Figures/Fig_ISO_triad.pdf}
+\caption{Triads used in the Griffies's like iso-neutral diffision scheme for 
+$u$-component (upper panel) and $w$-component (lower panel).}
+\end{center}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+The four iso-neutral fluxes associated with the triads are defined at $T$-point. 
+They take the following expression :
+\begin{flalign} \label{Gf_fluxes}
+\begin{split}
+{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 
+   &= \ \; \qquad  \quad    { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+i_p}^{\,k}}    \\
+{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T)
+   &=  -\; { _i^k \mathbb{R}_{i_p}^{k_p} }
+				 \ \; { _i^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+k_p}}
+\end{split}
+\end{flalign}
+
+The resulting iso-neutral fluxes at $u$- and $w$-points are then given by the 
+sum of the fluxes that cross the $u$- and $w$-face (Fig.~\ref{Fig_ISO_triad}):
+\begin{flalign} \label{Eq_iso_flux} 
+\textbf{F}_{iso}(T) 
+&\equiv  \sum_{\substack{i_p,\,k_p}} 
+   \begin{pmatrix} 
+      {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)      \\
+      \\
+      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)      \\   
+   \end{pmatrix}    \notag \\
+&  \notag \\
+&\equiv  \sum_{\substack{i_p,\,k_p}} 
+   \begin{pmatrix} 
+      && { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{1u}}_{\,i+1/2}^{\,k} }    \\
+      \\
+      & -\; { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }
+        & {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T) \;\ / \ { {e_{3w}}_{\,i}^{\,k+1/2} }   \\   
+   \end{pmatrix}      % \\
+% &\\
+% &\equiv  \sum_{\substack{i_p,\,k_p}} 
+%    \begin{pmatrix} 
+%       \qquad  \qquad  \qquad 
+%       \frac{1}{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \;  
+%        { _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} }(T)\\
+%       \\
+%       -\frac{1}{ {e_{3w}}_{\,i}^{\,k+1/2} }  \ \; 
+%        { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \;  
+%                  {_i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} }(T)\\   
+%    \end{pmatrix}      
+\end{flalign}
+resulting in a iso-neutral diffusion tendency on temperature given by the divergence 
+of the sum of all the four triad fluxes :
+\begin{equation} \label{Gf_operator}
+D_l^T = \frac{1}{b_T}  \sum_{\substack{i_p,\,k_p}} \left\{  
+		 \delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
+	     + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]   \right\}
+\end{equation}
+where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells. 
+
+This expression of the iso-neutral diffusion has been chosen in order to satisfy 
+the following six properties:
+\begin{description}
+\item[$\bullet$ horizontal diffusion] The discretization of the diffusion operator 
+recovers the traditional five-point Laplacian in the limit of flat iso-neutral direction :
+\begin{equation} \label{Gf_property1a}
+D_l^T = \frac{1}{b_T}  \ \delta_{i} 
+	\left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] 
+\qquad  \text{when} \quad 
+	{ _i^k \mathbb{R}_{i_p}^{k_p} }=0
+\end{equation}
+
+\item[$\bullet$ implicit treatment in the vertical]  In the diagonal term associated 
+with the vertical divergence of the iso-neutral fluxes (i.e. the term associated 
+with a second order vertical derivative) appears only tracer values associated 
+with a single water column. This is of paramount importance since it means
+that the implicit in time algorithm for solving the vertical diffusion equation can 
+be used to evaluate this term. It is a necessity since the vertical eddy diffusivity 
+associated with this term,  
+\begin{equation}
+	 \sum_{\substack{i_p, \,k_p}} \left\{  
+		A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+	\right\} 
+\end{equation}
+can be quite large.
+
+\item[$\bullet$ pure iso-neutral operator]  The iso-neutral flux of locally referenced 
+potential density is zero, $i.e.$
+\begin{align} \label{Gf_property2}
+\begin{matrix}
+&{_i^k {\mathbb{F}_u}_{i_p}^{k_p} (\rho)} 
+	&=    &\alpha_i^k   &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (T) 
+	&- \ \;  \beta _i^k    &{_i^k {\mathbb{F}_u}_{i_p}^{k_p} } (S) & = \ 0   \\
+&{_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)} 
+	&=    &\alpha_i^k   &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (T) 
+	&- \  \; \beta _i^k    &{_i^k {\mathbb{F}_w}_{i_p}^{k_p} } (S)  &= \ 0
+\end{matrix}
+\end{align}
+This result is trivially obtained using the \eqref{Gf_triads} applied to $T$ and $S$ 
+and the definition of the triads' slopes \eqref{Gf_slopes}.
+
+\item[$\bullet$ conservation of tracer] The iso-neutral diffusion term conserve the 
+total tracer content, $i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0
+\end{equation}
+This property is trivially satisfied since the iso-neutral diffusive operator 
+is written in flux form.
+
+\item[$\bullet$ decrease of tracer variance] The iso-neutral diffusion term does 
+not increase the total tracer variance, $i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0
+\end{equation}
+The property is demonstrated in the Appendix~\ref{Apdx_Gf_operator}. It is a 
+key property for a diffusion term. It means that the operator is also a dissipation 
+term, $i.e.$ it is a sink term for the square of the quantity on which it is applied. 
+It therfore ensure that, when the diffusivity coefficient is large enough, the field 
+on which it is applied become free of grid-point noise.
+
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion operator is self-adjoint, 
+$i.e.$
+\begin{equation} \label{Gf_property1}
+\sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\} 
+\end{equation}
+In other word, there is no needs to develop a specific routine from the adjoint of this 
+operator. We just have to apply the same routine. This properties can be demonstrated 
+quite easily in a similar way the "non increase of tracer variance" property has been 
+proved (see Appendix~\ref{Apdx_Gf_operator}).
+\end{description}
+
+
+$\ $\newline      %force an empty line
+% ================================================================
+% Skew flux formulation for Eddy Induced Velocity : 
+% ================================================================
+\subsection{ Eddy induced velocity and Skew flux formulation}
+
+When Gent and McWilliams [1990] diffusion is used (\key{traldf\_eiv} defined), 
+an additional advection term is added. The associated velocity is the so called 
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used 
+here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} 
+is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo}
++ \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. 
+
+The eddy induced velocity is given by: 
+\begin{equation} \label{Eq_eiv_v}
+\begin{split}
+ u^* & = - \frac{1}{e_2\,e_{3}}          \;\partial_k \left( e_2 \, A_e \; r_i  \right)   
+          = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_i  \right)            \\
+ v^* & = - \frac{1}{e_1\,e_3}\;             \partial_k \left( e_1 \, A_e \; r_j  \right)   
+          = - \frac{1}{e_3}                     \;\partial_k \left(           A_e \; r_j  \right)             \\
+w^* & =    \frac{1}{e_1\,e_2}\; \left\{   \partial_i  \left( e_2 \, A_e \; r_i  \right) 
+					              + \partial_j  \left( e_1 \, A_e \;r_j   \right) \right\}   \\
+\end{split}
+\end{equation}
+where $A_{e}$ is the eddy induced velocity coefficient, and $r_i$ and $r_j$ the 
+slopes between the iso-neutral and the geopotential surfaces.
+%%gm wrong: to be modified with 2 2D streamfunctions
+ In other words,
+the eddy induced velocity can be derived from a vector streamfuntion, $\phi$, which
+is given by $\phi = A_e\,\textbf{r}$ as $\textbf{U}^*  = \textbf{k} \times \nabla \phi$
+%%end gm
+
+A traditional way to implement this additional advection is to add it to the eulerian 
+velocity prior to compute the tracer advection. This allows us to take advantage of 
+all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just 
+a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers 
+where \emph{positivity} of the advection scheme is of paramount importance. 
+% give here the expression using the triads. It is different from the one given in \eqref{Eq_ldfeiv}
+% see just below a copy of this equation:
+%\begin{equation} \label{Eq_ldfeiv}
+%\begin{split}
+% u^* & = \frac{1}{e_{2u}e_{3u}}\; \delta_k \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right]\\
+% v^* & = \frac{1}{e_{1u}e_{3v}}\; \delta_k \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right]\\
+%w^* & = \frac{1}{e_{1w}e_{2w}}\; \left\{ \delta_i \left[e_{2u} \, A_{uw}^{eiv} \; \overline{r_{1w}}^{\,i+1/2} \right] + %\delta_j \left[e_{1v} \, A_{vw}^{eiv} \; \overline{r_{2w}}^{\,j+1/2} \right] \right\} \\
+%\end{split}
+%\end{equation}
+\begin{equation} \label{Eq_eiv_vd}  
+\textbf{F}_{eiv}^T   \equiv   \left( \begin{aligned}                                
+ \sum_{\substack{i_p,\,k_p}} &
+ +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k} 
+\ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}]      \\
+    \\
+ \sum_{\substack{i_p,\,k_p}} &
+ - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p} 
+\ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]    \\   
+\end{aligned}   \right)
+\end{equation}
+
+\ref{Griffies_JPO98} introduces another way to implement the eddy induced advection, 
+the so-called skew form. It is based on a transformation of the advective fluxes 
+using the non-divergent nature of the eddy induced velocity. 
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be 
+transformed as follows:
+\begin{flalign*}
+\begin{split}
+\textbf{F}_{eiv}^T = 
+\begin{pmatrix} 
+ 	        {e_{2}\,e_{3}\;  u^*} 	 	\\
+ 		{e_{1}\,e_{2}\; w^*}	 \\
+\end{pmatrix}   \;   T
+&=
+\begin{pmatrix} 
+ 	        { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} 	 	\\
+ 		{+ \partial_i  \left( e_{2} \, A_{e} \; r_i \right) \; T \;}	 \\
+\end{pmatrix} 			\\
+&=			
+\begin{pmatrix} 
+ 	        { - \partial_k \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}  \\
+ 		{+ \partial_i  \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}	 \\
+\end{pmatrix} 			
+ + 
+\begin{pmatrix} 
+ 	        {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}  \\
+ 		{ - e_{2} \, A_{e} \; r_i  \; \partial_i  T}	 \\
+\end{pmatrix} 	 
+\end{split}
+\end{flalign*}
+and since the eddy induces velocity field is no-divergent, we end up with the skew 
+form of the eddy induced advective fluxes:
+\begin{equation} \label{Eq_eiv_skew_continuous}
+\textbf{F}_{eiv}^T = \begin{pmatrix} 
+ 	        {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\
+ 		{ - e_{2} \, A_{e} \; r_i  \; \partial_i  T}	 \\
+                                 \end{pmatrix}
+\end{equation}
+The tendency associated with eddy induced velocity is then simply the divergence 
+of the \eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer 
+content, as it is expressed in flux form and, as the advective form, it preserve the 
+tracer variance. Another interesting property of \eqref{Eq_eiv_skew_continuous} 
+form is that when $A=A_e$, a simplification occurs in the sum of the iso-neutral 
+diffusion and eddy induced velocity terms:
+\begin{flalign} \label{Eq_eiv_skew+eiv_continuous}
+\textbf{F}_{iso}^T + \textbf{F}_{eiv}^T &= 
+\begin{pmatrix} 
+ 	        + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T -  e_2 \, A \; r_i                              \;\partial_k T   \\
+ 		-  e_2 \, A_{e} \; r_i           \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T   \\
+\end{pmatrix}
++
+\begin{pmatrix} 
+ 	        {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\
+ 		{ - e_{2} \, A_{e} \; r_i  \; \partial_i  T}	 \\
+\end{pmatrix}      \\
+&= \begin{pmatrix} 
+ 	        + \frac{e_2\,e_3\,}{e_1} A \;\partial_i T    \\
+ 		-  2\; e_2 \, A_{e} \; r_i      \;\partial_i T + \frac{e_1\,e_2}{e_3} \, A \; r_i^2 \;\partial_k T   \\
+\end{pmatrix}
+\end{flalign}
+The horizontal component reduces to the one use for an horizontal laplacian 
+operator and the vertical one keep the same complexity, but not more. This property
+has been used to reduce the computational time \citep{Griffies_JPO98}, but it is 
+not of practical use as usually $A \neq A_e$. Nevertheless this property can be used to 
+choose a discret form of  \eqref{Eq_eiv_skew_continuous} which is consistent with the 
+iso-neutral operator \eqref{Gf_operator}. Using the slopes \eqref{Gf_slopes} 
+and defining $A_e$ at $T$-point($i.e.$ as $A$, the eddy diffusivity coefficient),
+the resulting discret form is given by:
+\begin{equation} \label{Eq_eiv_skew}  
+\textbf{F}_{eiv}^T   \equiv   \frac{1}{4} \left( \begin{aligned}                                
+ \sum_{\substack{i_p,\,k_p}} &
+ +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k} 
+\ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}]      \\
+    \\
+ \sum_{\substack{i_p,\,k_p}} &
+ - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p} 
+\ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]    \\   
+\end{aligned}   \right)
+\end{equation}
+Note that \eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells. 
+In $z^*$ or $s$-coordinate, the slope between the level and the geopotential surfaces 
+must be added to $\mathbb{R}$ for the discret form to be exact. 
+
+Such a choice of discretisation is consistent with the iso-neutral operator as it uses the 
+same definition for the slopes. It also ensures the conservation of the tracer variance 
+(see Appendix \ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component 
+but is a "pure" advection term.
+
+
+
+
+$\ $\newpage      %force an empty line
+% ================================================================
+% Discrete Invariants of the iso-neutral diffrusion 
+% ================================================================
+\subsection{Discrete Invariants of the iso-neutral diffrusion}
+\label{Apdx_Gf_operator}
+
+Demonstration of the decrease of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
+
+This part will be moved in an Appendix.
+
+The continuous property to be demonstrated is :
+\begin{align*}
+\int_D  D_l^T \; T \;dv   \leq 0
+\end{align*}
+The discrete form of its left hand side is obtained using \eqref{Eq_iso_flux}
+
+\begin{align*}
+&\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\
+&\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+		\delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
+	     + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\
+&\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+                {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
+             + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\
+&\equiv -\sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+	\frac{ _{i+1/2-i_p}^k \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{1u}}_{\,i+1/2}^{\,k} }  \ \delta_{i+1/2} [T]
+ - { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} } \ \; 
+	\frac{ _i^{k+1/2-k_p} \mathbb{T}_{i_p}^{k_p} (T) }{ {e_{3w}}_{\,i}^{\,k+1/2}  } \ \delta_{k+1/2} [T]   
+	                                                                    \right\}      \\
+%
+\allowdisplaybreaks
+\intertext{ Expending the summation on $i_p$ and $k_p$, it becomes:}
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}  
+&\ \ \Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{-1/2} (T) } 
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_{i}^{k+1} \mathbb{R}_{-1/2}^{-1/2}} 
+&      {_{i}^{k+1} \mathbb{T}_{-1/2}^{-1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)
+& \\
+&+\Bigl( \ \;\; { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }  
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_i^{k+1} \mathbb{R}_{+1/2}^{-1/2}}
+& { _i^{k+1} \mathbb{T}_{+1/2}^{-1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}      \Bigr)
+& \\
+&+\Bigl(  { _{i+1}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) } 
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{-1/2}^{+1/2}} 
+&      \ \;\;{_{i}^{k} \mathbb{T}_{-1/2}^{+1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)
+& \\
+&+\Bigl( \ \;\; { _{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) } 
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ \ \;\;{_{i}^{k} \mathbb{R}_{+1/2}^{+1/2}} 
+&      \ \;\;{_{i}^{k} \mathbb{T}_{+1/2}^{+1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\
+\end{Bmatrix}
+%
+\allowdisplaybreaks
+\intertext{The summation is done over all $i$ and $k$ indices, it is therefore possible to introduce a shift of $-1$ either in $i$ or $k$ direction in order to regroup all the terms of the summation by triad at a ($i$,$k$) point. In other words, we regroup all the terms in the neighbourhood  that contain a triad at the same ($i$,$k$) indices. It becomes: }
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}  
+&\ \ \Bigl(  {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) } 
+&\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 
+&      {_i^k \mathbb{T}_{-1/2}^{-1/2} (T) }   
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr)
+& \\
+&+\Bigl(  { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }  
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}}
+&      { _i^k \mathbb{T}_{+1/2}^{-1/2} (T) }   
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr)
+& \\
+&+\Bigl(  {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) } 
+&\frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 
+&      {_i^k \mathbb{T}_{-1/2}^{+1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)
+& \\
+&+\Bigl( { _i^k \mathbb{T}_{+1/2}^{+1/2} (T) } 
+&\frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 
+&      {_i^k \mathbb{T}_{+1/2}^{+1/2} (T) }   
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)   \\
+\end{Bmatrix}   \\
+%
+\allowdisplaybreaks
+\intertext{Then outing in factor the triad in each of the four terms of the summation and substituting the triads by their expression given in \eqref{Gf_triads}. It becomes: }
+%
+&\equiv -\sum_{i,k}
+\begin{Bmatrix}  
+&\ \ \Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{-1/2}} 
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}     \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k
+& \\
+&+\Bigl(  \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{-1/2}}
+&\frac{ \delta_{k-1/2} [T] }{{e_{3w}}_{\,i}^{\,k-1/2}}      \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k
+& \\
+&+\Bigl(  \frac{ \delta_{i -1/2} [T] }{{e_{1u} }_{\,i-1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{-1/2}^{+1/2}} 
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i-1/2}^{\,k}  \  A_i^k
+& \\
+&+\Bigl( \frac{ \delta_{i +1/2} [T] }{{e_{1u} }_{\,i+1/2}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{+1/2}^{+1/2}} 
+&\frac{ \delta_{k+1/2} [T] }{{e_{3w}}_{\,i}^{\,k+1/2}}     \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+1/2}^{\,k}  \  A_i^k      \\
+\end{Bmatrix}   \\
+& \\
+%
+&\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+\begin{matrix}  
+&\Bigl( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
+& -\ \ {_i^k \mathbb{R}_{i_p}^{k_p}} 
+&\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \Bigr)^2
+& \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \ \ 
+\end{matrix}
+ \right\}   
+\quad   \leq 0
+\end{align*} 
+The last inequality is obviously obtained as we succeed in obtaining a negative summation of square quantities.
+
+Note that, if instead of multiplying $D_l^T$ by $T$, we were using another tracer field, let say $S$, then the previous demonstration would have let to:
+\begin{align*}
+\int_D  S \; D_l^T  \;dv &\equiv  \sum_{i,k} \left\{ S \ D_l^T \ b_T \right\}    \\
+&\equiv - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+\left( \frac{ \delta_{i +i_p} [S] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
+ - {_i^k \mathbb{R}_{i_p}^{k_p}} 
+\frac{ \delta_{k+k_p} [S] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right)  \right.    
+\\   & \qquad \qquad \qquad \ \left.
+\left( \frac{ \delta_{i +i_p} [T] }{{e_{1u} }_{\,i+i_p}^{\,k}} 
+ - {_i^k \mathbb{R}_{i_p}^{k_p}} 
+\frac{ \delta_{k+k_p} [T] }{{e_{3w}}_{\,i}^{\,k+k_p}}     \right) 
+ \frac{1}{4} \ {b_u}_{\,i+i_p}^{\,k}  \  A_i^k   \
+ \right\}   
+%
+\allowdisplaybreaks
+\intertext{which, by applying the same operation as before but in reverse order, leads to: }
+%
+&\equiv  \sum_{i,k} \left\{ D_l^S \ T \ b_T \right\}   
+\end{align*} 
+This means that the iso-neutral operator is self-adjoint. There is no need to develop a specific to obtain it.
+
+
+
+$\ $\newpage      %force an empty line
+% ================================================================
+% Discrete Invariants of the skew flux formulation
+% ================================================================
+\subsection{Discrete Invariants of the skew flux formulation}
+\label{Apdx_eiv_skew}
+
+
+Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
+
+This have to be moved in an Appendix.
+
+The continuous property to be demonstrated is :
+\begin{align*}
+\int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0
+\end{align*}
+The discrete form of its left hand side is obtained using \eqref{Eq_eiv_skew}
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\;
+ \delta_i  &\left[                                                    
+{e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]         
+   \right] \; T_i^k      \\
+- \delta_k &\left[ 
+{e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]   
+   \right] \; T_i^k      \         \Biggr\}   
+\end{align*}
+apply the adjoint of delta operator, it becomes
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\;
+  &\left(                                                    
+{e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]         
+   \right) \; \delta_{i+1/2}[T^{k}]      \\
+- &\left( 
+{e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]   
+     \right) \; \delta_{k+1/2}[T_{i}]       \         \Biggr\}       
+\end{align*}
+Expending the summation on $i_p$ and $k_p$, it becomes:
+\begin{align*}
+ \begin{matrix}  
+&\sum\limits_{i,k}   \Bigl\{ 
+  &+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
+  &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}]    &\delta_{i+1/2}[T^{k}]   &\\
+&&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}      
+  &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}}  &\delta_{k-1/2}[T_{i\ \ \ \;}]  &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
+  &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}]     &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}       
+    &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\
+%
+&&-{e_{2u}}_i^{k+1}                                &{A_{e}}_i^{k+1} 
+  &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}}   &\delta_{i-1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\   
+&&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
+  &{\ \ \;_i^k  \mathbb{R}_{-1/2}^{+1/2}}   &\delta_{i-1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] &\\
+&&-{e_{2u}}_i^{k+1    }                             &{A_{e}}_i^{k+1} 
+  &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}}   &\delta_{i+1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\   
+&&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
+  &{\ \ \;_i^k  \mathbb{R}_{+1/2}^{+1/2}}   &\delta_{i+1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] 
+&\Bigr\}  \\
+\end{matrix}   
+\end{align*}
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the 
+same but of opposite signs, they cancel out. 
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. 
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the 
+same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$ 
+they cancel out with the neighbouring grid points. 
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the 
+$i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the 
+tracer is preserved by the discretisation of the skew fluxes.
+
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_ISO.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_ISO.tex	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Annex_ISO.tex	(revision 2282)
@@ -0,0 +1,677 @@
+% ================================================================
+% Iso-neutral diffusion : 
+% ================================================================
+\chapter{Griffies's iso-neutral diffusion and \\
+  eddy-induced advection}
+\label{Apdx_C}
+\minitoc
+
+\section{Griffies's formulation of isoneutral diffusion}
+
+\subsection{Introduction}
+ We define a scheme that get its inspiration from the scheme of
+\citet{Griffies_al_JPO98}, but is formulated within the \NEMO
+framework, using scale factors rather than grid-sizes.
+
+The off-diagonal terms of the small angle diffusion tensor
+\eqref{Eq_PE_iso_tensor} produce skew-fluxes along the
+i- and j-directions resulting from the vertical tracer gradient:
+\begin{align}
+  \label{eq:i13c}
+  &+\kappa r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad+\kappa r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\
+\intertext{and in the k-direction resulting from the lateral tracer gradients}
+  \label{eq:i31c}
+ & \kappa r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\kappa r_2\frac{1}{e_1}\frac{\partial T}{\partial i}
+\end{align}
+where \eqref{Eq_PE_iso_slopes}
+\begin{align*}
+  r_1 &=-\frac{e_3 }{e_1 } \left( \frac{\partial \rho }{\partial i}
+  \right)
+  \left( {\frac{\partial \rho }{\partial k}} \right)^{-1} \\
+  &=-\frac{e_3 }{e_1 } \left( -\alpha\frac{\partial T }{\partial i} +
+    \beta\frac{\partial S }{\partial i} \right) \left(
+    -\alpha\frac{\partial T }{\partial k} + \beta\frac{\partial S
+    }{\partial k} \right)^{-1}
+\end{align*}
+is the i-component of the slope of the isoneutral surface relative to the computational
+surface, and $r_2$ is the j-component.
+
+The extra vertical diffusive flux associated with the $_{33}$
+component of the small angle diffusion tensor is
+\begin{equation}
+  \label{eq:i33c}
+  -\kappa(r_1^2 +r_2^2) \frac{1}{e_3}\frac{\partial T}{\partial k}.
+\end{equation}
+
+Since there are no cross terms involving $r_1$ and $r_2$ in the above, we can
+consider the isoneutral diffusive fluxes separately in the i-k and j-k
+planes, just adding together the vertical components from each
+plane. The following description will describe the fluxes on the i-k
+plane.
+
+There is no natural discretization for the i-component of the
+skew-flux, \eqref{eq:i13c}, as
+although it must be evaluated at u-points, it involves vertical
+gradients (both for the tracer and the slope $r_1$), defined at
+w-points. Similarly, the vertical skew flux, \eqref{eq:i31c}, is evaluated at
+w-points but involves horizontal gradients defined at u-points.
+
+\subsection{The standard discretization}
+The straightforward approach to discretize the lateral skew flux
+\eqref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995
+into OPA, \eqref{Eq_tra_ldf_iso}, is to calculate a mean vertical
+gradient at the u-point from the average of the four surrounding
+vertical tracer gradients, and multiply this by a mean slope at the
+u-point, calculated from the averaged surrounding vertical density
+gradients. The total area-integrated skew-flux from tracer cell $i,k$
+to $i+1,k$, noting that the $e_{{3u}_{i+1/2}^k}$ in the area
+$e_{{3u}_{i+1/2}^k}e_{{2u}_{i+1/2}^k}$ at the u-point cancels out with
+the $1/e_{{3u}_{i+1/2}^k}$ associated with the vertical tracer
+gradient, is then \eqref{Eq_tra_ldf_iso}
+\begin{equation*}
+  \left(F_u^{\mathrm{skew}} \right)_{i+\hhalf}^k = \kappa _{i+\hhalf}^k
+  e_{{2u}_{i+1/2}^k} \overline{\overline
+    r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k},
+\end{equation*}
+where
+\begin{equation*}
+  \overline{\overline
+    r_1} ^{\,i,k} = -\frac{e_{{3u}_{i+1/2}^k}}{e_{{1u}_{i+1/2}^k}}
+  \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}}.
+\end{equation*}
+
+Unfortunately the resulting combination $\overline{\overline{\delta_k
+    \bullet}}^{\,i,k}$ of a $k$ average and a $k$ difference %of the tracer
+reduces to $\bullet_{k+1}-\bullet_{k-1}$, so two-grid-point oscillations are
+invisible to this discretization of the iso-neutral operator. These
+\emph{computational modes} will not be damped by this operator, and
+may even possibly be amplified by it.  Consequently, applying this
+operator to a tracer does not guarantee the decrease of its
+global-average variance. To correct this, we introduced a smoothing of
+the slopes of the iso-neutral surfaces (see \S\ref{LDF}). This
+technique works fine for $T$ and $S$ as they are active tracers
+($i.e.$ they enter the computation of density), but it does not work
+for a passive tracer.
+\subsection{Expression of the skew-flux in terms of triad slopes}
+\citep{Griffies_al_JPO98} introduce a different discretization of the
+off-diagonal terms that nicely solves the problem.
+% Instead of multiplying the mean slope calculated at the u-point by
+% the mean vertical gradient at the u-point,
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h] \begin{center}
+    \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_triad_fluxes}
+    \caption{(a) Arrangement of triads $S_i$ and tracer gradients to
+      give lateral tracer flux from box $i,k$ to $i+1,k$ (b) Triads
+      $S'_i$ and tracer gradients to give vertical tracer flux from
+      box $i,k$ to $i,k+1$.}
+    \label{Fig_triad}
+  \end{center} \end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+They get the skew flux from the products of the vertical gradients at
+each w-point surrounding the u-point with the corresponding `triad'
+slope calculated from the lateral density gradient across the u-point
+divided by the vertical density gradient at the same w-point as the
+tracer gradient. See Fig.~\ref{Fig_triad}a, where the thick lines
+denote the tracer gradients, and the thin lines the corresponding
+triads, with slopes $s_1, \dotsc s_4$. The total area-integrated
+skew-flux from tracer cell $i,k$ to $i+1,k$
+\begin{multline}
+  \label{eq:i13}
+  \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = \kappa _{i+1}^k a_1 s_1
+  \delta _{k+\frac{1}{2}} \left[ T^{i+1}
+  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  + \kappa _i^k a_2 s_2 \delta
+  _{k+\frac{1}{2}} \left[ T^i
+  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\
+   +\kappa _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1}
+  \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  +\kappa _i^k a_4 s_4 \delta
+  _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
+\end{multline}
+where the contributions of the triad fluxes are weighted by areas
+$a_1, \dotsc a_4$, and $\kappa$ is now defined at the tracer points
+rather than the u-points. This discretization gives a much closer
+stencil, and disallows the two-point computational modes.
+
+ The vertical skew flux \eqref{eq:i31c} from tracer cell $i,k$ to $i,k+1$ at the
+w-point $i,k+\hhalf$ is constructed similarly (Fig.~\ref{Fig_triad}b)
+by multiplying lateral tracer gradients from each of the four
+surrounding u-points by the appropriate triad slope:
+\begin{multline}
+  \label{eq:i31}
+  \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  \kappa_i^{k+1} a_{1}'
+  s_{1}' \delta _{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1}
+   +\kappa_i^{k+1} a_{2}' s_{2}' \delta _{i+\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i+\frac{1}{2}}^{k+1}\\
+  + \kappa_i^k a_{3}' s_{3}' \delta _{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
+  +\kappa_i^k a_{4}' s_{4}' \delta _{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
+\end{multline}
+ 
+We notate the triad slopes in terms of the `anchor point' $i,k$
+(appearing in both the vertical and lateral gradient), and the u- and
+w-points $(i+i_p,k)$, $(i,k+k_p)$ at the centres of the `arms' of the
+triad as follows (see also Fig.~\ref{Fig_triad}):
+\begin{equation}
+  \label{Gf_slopes}
+  _i^k \mathbb{R}_{i_p}^{k_p} 
+  =\frac{ {e_{3w}}_{\,i}^{\,k+k_p}} { {e_{1u}}_{\,i+i_p}^{\,k}}
+  \
+  \frac 
+  {\left(\alpha / \beta \right)_i^k  \ \delta_{i + i_p}[T^k] - \delta_{i + i_p}[S^k] }
+  {\left(\alpha / \beta \right)_i^k  \ \delta_{k+k_p}[T^i ] - \delta_{k+k_p}[S^i ] }.
+\end{equation}
+In calculating the slopes of the local neutral
+surfaces, the expansion coefficients $\alpha$ and $\beta$ are
+evaluated at the anchor points of the triad \footnote{Note that in \eqref{Gf_slopes} we use the ratio $\alpha / \beta$
+instead of multiplying the temperature derivative by $\alpha$ and the
+salinity derivative by $\beta$. This is more efficient as the ratio
+$\alpha / \beta$ can to be evaluated directly}, while the metrics are
+calculated at the u- and w-points on the arms.
+
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[h] \begin{center}
+    \includegraphics[width=0.60\textwidth]{./TexFiles/Figures/Fig_qcells}
+    \caption{Triad notation for quarter cells.T-cells are inside
+      boxes, while the  $i+\half,k$ u-cell is shaded in green and the
+      $i,k+\half$ w-cell is shaded in pink.}
+    \label{qcells}
+  \end{center} \end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Each triad $\{_i^k\:_{i_p}^{k_p}\}$ is associated (Fig.~\ref{qcells}) with the quarter
+cell that is the intersection of the $i,k$ T-cell, the $i+i_p,k$
+u-cell and the $i,k+k_p$ w-cell. Expressing the slopes $s_i$ and
+$s'_i$ in \eqref{eq:i13} and \eqref{eq:i31} in this notation, we have
+e.g.\ $s_1=s'_1={\:}_i^k \mathbb{R}_{1/2}^{1/2}$. Each triad slope $_i^k
+\mathbb{R}_{i_p}^{k_p}$ is used once (as an $s$) to calculate the
+lateral flux along its u-arm, at $(i+i_p,k)$, and then again as an
+$s'$ to calculate the vertical flux along its w-arm at
+$(i,k+k_p)$. Each vertical area $a_i$ used to calculate the lateral
+flux and horizontal area $a'_i$ used to calculate the vertical flux
+can also be identified as the area across the u- and w-arms of a
+unique triad, and we can notate these areas, similarly to the triad
+slopes, as $_i^k{\mathbb{A}_u}_{i_p}^{k_p}$,
+$_i^k{\mathbb{A}_w}_{i_p}^{k_p}$, where e.g. in \eqref{eq:i13}
+$a_{1}={\:}_i^k{\mathbb{A}_u}_{1/2}^{1/2}$, and in \eqref{eq:i31}
+$a'_{1}={\:}_i^k{\mathbb{A}_w}_{1/2}^{1/2}$.
+
+\subsection{The full triad fluxes}
+A key property of isoneutral diffusion is that it should not affect
+the (locally referenced) density. In particular there should be no
+lateral or vertical density flux. The lateral density flux disappears so long as the
+area-integrated lateral diffusive flux from tracer cell $i,k$ to
+$i+1,k$ coming from the $_{11}$ term of the diffusion tensor takes the
+form
+\begin{equation}
+  \label{eq:i11}
+  \left( F_u^{11} \right) _{i+\frac{1}{2}} ^{k} =
+  - \left( \kappa_i^{k+1} a_{1} + \kappa_i^{k+1} a_{2} + \kappa_i^k
+    a_{3} + \kappa_i^k a_{4} \right)
+  \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
+\end{equation}
+where the areas $a_i$ are as in \eqref{eq:i13}. In this case,
+separating the total lateral flux, the sum of \eqref{eq:i13} and
+\eqref{eq:i11}, into triad components, a lateral tracer
+flux
+\begin{equation}
+  \label{eq:latflux-triad}
+  _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) = -A_i^k{ \:}_i^k{\mathbb{A}_u}_{i_p}^{k_p}
+  \left(
+    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -\ {_i^k\mathbb{R}_{i_p}^{k_p}} \
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right)
+\end{equation}
+can be identified with each triad. Then, because the
+same metric factors ${e_{3w}}_{\,i}^{\,k+k_p}$ and
+${e_{1u}}_{\,i+i_p}^{\,k}$ are employed for both the density gradients
+in $ _i^k \mathbb{R}_{i_p}^{k_p}$ and the tracer gradients, the lateral
+density flux associated with each triad separately disappears.
+\begin{equation}
+  \label{eq:latflux-rho}
+  {\mathbb{F}_u}_{i_p}^{k_p} (\rho)=-\alpha _i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{F}_u}_{i_p}^{k_p} (S)=0
+\end{equation}
+Thus the total flux $\left( F_u^{31} \right) ^i _{i,k+\frac{1}{2}} +
+\left( F_u^{11} \right) ^i _{i,k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i+1,k$ must also vanish since it is a sum of four such triad fluxes.
+
+The squared slope $r_1^2$ in the expression \eqref{eq:i33c} for the
+$_{33}$ component is also expressed in terms of area-weighted
+squared triad slopes, so the area-integrated vertical flux from tracer
+cell $i,k$ to $i,k+1$ resulting from the $r_1^2$ term is
+\begin{equation}
+  \label{eq:i33}
+  \left( F_w^{33} \right) _i^{k+\frac{1}{2}} =
+    - \left( \kappa_i^{k+1} a_{1}' s_{1}'^2
+    + \kappa_i^{k+1} a_{2}' s_{2}'^2
+    + \kappa_i^k a_{3}' s_{3}'^2
+    + \kappa_i^k a_{4}' s_{4}'^2 \right)\delta_{k+\frac{1}{2}} \left[ T^{i+1} \right],
+\end{equation}
+where the areas $a'$ and slopes $s'$ are the same as in
+\eqref{eq:i31}.
+Then, separating the total vertical flux, the sum of \eqref{eq:i31} and
+\eqref{eq:i33}, into triad components,  a vertical flux 
+\begin{align}
+  \label{eq:vertflux-triad}
+  _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
+  &= A_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p}
+  \left(
+    {_i^k\mathbb{R}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -\ \left({_i^k\mathbb{R}_{i_p}^{k_p}}\right)^2 \
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right) \\
+  &= - \left(\left.{ }_i^k{\mathbb{A}_w}_{i_p}^{k_p}\right/{ }_i^k{\mathbb{A}_u}_{i_p}^{k_p}\right)
+   {_i^k\mathbb{R}_{i_p}^{k_p}}{\: }_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \label{eq:vertflux-triad2}
+\end{align}
+may be associated with each triad. Each vertical density flux $_i^k {\mathbb{F}_w}_{i_p}^{k_p} (\rho)$
+associated with a triad then separately disappears (because the
+lateral flux $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (\rho)$
+disappears). Consequently the total vertical density flux $\left( F_w^{31} \right)_i ^{k+\frac{1}{2}} +
+\left( F_w^{33} \right)_i^{k+\frac{1}{2}}$ from tracer cell $i,k$
+to $i,k+1$ must also vanish since it is a sum of four such triad
+fluxes.
+
+We can explicitly identify (Fig.~\ref{qcells}) the triads associated with the $s_i$, $a_i$, and $s'_i$, $a'_i$ used in the definition of
+the u-fluxes and w-fluxes in
+\eqref{eq:i31}, \eqref{eq:i13}, \eqref{eq:i11} \eqref{eq:i33} and
+Fig.~\ref{Fig_triad} to  write out the iso-neutral fluxes at $u$- and
+$w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
+%(Fig.~\ref{Fig_ISO_triad}):
+\begin{flalign} \label{Eq_iso_flux} \textbf{F}_{iso}(T) &\equiv
+  \sum_{\substack{i_p,\,k_p}}
+  \begin{pmatrix}
+    {_{i+1/2-i_p}^k {\mathbb{F}_u}_{i_p}^{k_p} } (T)      \\
+    \\
+    {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p} } (T)      \\
+  \end{pmatrix}.
+\end{flalign}
+\subsection{Ensuring the scheme cannot increase tracer variance}
+\label{sec:variance}
+
+We now require that this operator cannot increase the
+globally-integrated tracer variance.
+%This changes according to
+% \begin{align*}
+% &\int_D  D_l^T \; T \;dv \equiv  \sum_{i,k} \left\{ T \ D_l^T \ b_T \right\}    \\
+% &\equiv + \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+% 		\delta_{i} \left[{_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \right] 
+% 	     + \delta_{k} \left[ {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right]  \ T \right\}    \\
+% &\equiv  - \sum_{i,k} \sum_{\substack{i_p,\,k_p}} \left\{  
+%                 {_{i+1/2-i_p}^k {\mathbb{F}_u }_{i_p}^{k_p}} \ \delta_{i+1/2} [T]
+%              + {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}}  \ \delta_{k+1/2} [T]   \right\}      \\
+% \end{align*}
+Each triad slope $_i^k\mathbb{R}_{i_p}^{k_p}$ drives a lateral flux
+$_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ across the u-point $i+i_p,k$ and
+a vertical flux $_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ across the
+w-point $i,k+k_p$.  The lateral flux drives a net rate of change of
+variance at points $i+i_p-\half,k$ and $i+i_p+\half,k$ of
+\begin{multline}
+  {b_T}_{i+i_p-1/2}^k\left(\frac{\partial T}{\partial t}T\right)_{i+i_p-1/2}^k+
+  \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial
+      t}T\right)_{i+i_p+1/2}^k \\
+ \begin{split}
+  &= -T_{i+i_p-1/2}^k{\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \quad + \quad  T_{i+i_p+1/2}^k
+  {\;}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T) \\
+  &={\;} _i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k], \label{locFdtdx}
+ \end{split}
+\end{multline}
+while the vertical flux similarly drives a net rate of change of variance at points $i,k+k_p-\half$ and
+$i,k+k_p+\half$ of
+\begin{equation}
+\label{locFdtdz}
+  _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
+\end{equation}
+The total variance tendency driven by the triad is the sum of these
+two. Expanding $_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)$ and
+$_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)$ with \eqref{eq:latflux-triad} and
+\eqref{eq:vertflux-triad}, it is
+\begin{multline*}
+-A_i^k\left \{
+{ } _i^k{\mathbb{A}_u}_{i_p}^{k_p}
+  \left(
+    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    - {_i^k\mathbb{R}_{i_p}^{k_p}} \
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }\right)\,\delta_{i+ i_p}[T^k] \right.\\
+- \left. { } _i^k{\mathbb{A}_w}_{i_p}^{k_p}
+  \left(
+    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right) {\,}_i^k\mathbb{R}_{i_p}^{k_p}\delta_{k+ k_p}[T^i]
+\right \}.
+\end{multline*}
+The key point is then that if we require
+$_i^k{\mathbb{A}_u}_{i_p}^{k_p}$ and $_i^k{\mathbb{A}_w}_{i_p}^{k_p}$
+to be related to a triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$ by
+\begin{equation}
+  \label{eq:V-A}
+  _i^k\mathbb{V}_{i_p}^{k_p}
+  ={\;}_i^k{\mathbb{A}_u}_{i_p}^{k_p}\,{e_{1u}}_{\,i + i_p}^{\,k}
+  ={\;}_i^k{\mathbb{A}_w}_{i_p}^{k_p}\,{e_{3w}}_{\,i}^{\,k + k_p},
+\end{equation}
+the variance tendency reduces to the perfect square
+\begin{equation}
+  \label{eq:perfect-square}
+  -A_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
+  \left(
+    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right)^2\leq 0.
+\end{equation}
+Thus, the constraint \eqref{eq:V-A} ensures that the fluxes associated
+with a given slope triad $_i^k\mathbb{R}_{i_p}^{k_p}$ do not increase
+the net variance. Since the total fluxes are sums of such fluxes from
+the various triads, this constraint, applied to all triads, is
+sufficient to ensure that the globally integrated variance does not
+increase.
+
+The expression \eqref{eq:V-A} can be interpreted as a discretization
+of the global integral
+\begin{equation}
+  \label{eq:cts-var}
+  \frac{\partial}{\partial t}\int\half T^2\, dV =
+  \int\mathbf{F}\cdot\nabla T\, dV,
+\end{equation}
+where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the
+lateral and vertical fluxes/unit area
+\[
+\mathbf{F}=\left(
+_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)/_i^k{\mathbb{A}_u}_{i_p}^{k_p},
+{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)/_i^k{\mathbb{A}_w}_{i_p}^{k_p}
+ \right)
+\]
+and the gradient
+ \[\nabla T = \left(
+\delta_{i+ i_p}[T^k] / {e_{1u}}_{\,i + i_p}^{\,k},
+\delta_{k+ k_p}[T^i] / {e_{3w}}_{\,i}^{\,k + k_p}
+\right)
+\]
+\subsection{Triad volumes in Griffes's scheme and in \NEMO}
+To complete the discretization we now need only specify the triad
+volumes $_i^k\mathbb{V}_{i_p}^{k_p}$. \citet{Griffies_al_JPO98} identify
+these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter
+cells, defined in terms of the distances between T, u,f and
+w-points. This is the natural discretization of
+\eqref{eq:cts-var}. The \NEMO model, however, operates with scale
+factors instead of grid sizes, and scale factors for the quarter
+cells are not defined. Instead, therefore we simply choose
+\begin{equation}
+  \label{eq:V-NEMO}
+  _i^k\mathbb{V}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k,
+\end{equation}
+as a quarter of the volume of the u-cell inside which the triad
+quarter-cell lies. This has the nice property that when the slopes
+$\mathbb{R}$ vanish, the lateral flux from tracer cell $i,k$ to
+$i+1,k$ reduces to the classical form
+\begin{equation}
+  \label{eq:lat-normal}
+-\overline{A}_{\,i+1/2}^k\;
+\frac{{b_u}_{i+1/2}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
+\;\frac{\delta_{i+ 1/2}[T^k] }{{e_{1u}}_{\,i + i_p}^{\,k}}
+ = -\overline{A}_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
+\end{equation}
+In fact if the diffusive coefficient is defined at u-points, so that
+we employ $A_{i+i_p}^k$ instead of $A_i^k$ in the definitions of the
+triad fluxes \eqref{eq:latflux-triad} and \eqref{eq:vertflux-triad},
+we can replace $\overline{A}_{\,i+1/2}^k$ by $A_{i+1/2}^k$ in the above.
+
+\subsection{Summary of the scheme}
+The divergence of the expression \eqref{Eq_iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
+each tracer point:
+\begin{equation} \label{Gf_operator} D_l^T = \frac{1}{b_T}
+  \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k
+        {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[
+      {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
+\end{equation}
+where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
+The diffusion scheme satisfies the following six properties:
+\begin{description}
+\item[$\bullet$ horizontal diffusion] The discretization of the
+  diffusion operator recovers \eqref{eq:lat-normal} the traditional five-point Laplacian in
+  the limit of flat iso-neutral direction :
+  \begin{equation} \label{Gf_property1a} D_l^T = \frac{1}{b_T} \
+    \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
+      \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right] \qquad
+    \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0
+  \end{equation}
+
+\item[$\bullet$ implicit treatment in the vertical] Only tracer values
+  associated with a single water column appear in the expression
+  \eqref{eq:i33} for the $_{33}$ fluxes, vertical fluxes driven by
+  vertical gradients. This is of paramount importance since it means
+  that an implicit in time algorithm can be used to solve the vertical
+  diffusion equation. This is a necessity since the vertical eddy
+  diffusivity associated with this term,
+  \begin{equation}
+    \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{  
+      {\:}_i^k\mathbb{V}_{i_p}^{k_p} \:A_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+    \right\}  = 
+    \frac{1}{4b_w}\sum_{\substack{i_p, \,k_p}} \left\{  
+      {b_u}_{i+i_p}^k\:A_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
+    \right\},
+ \end{equation}
+  (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
+
+\item[$\bullet$ pure iso-neutral operator] The iso-neutral flux of
+  locally referenced potential density is zero. See
+  \eqref{eq:latflux-rho} and \eqref{eq:vertflux-triad2}.
+
+\item[$\bullet$ conservation of tracer] The iso-neutral diffusion
+  conserves tracer content, $i.e.$
+  \begin{equation} \label{Gf_property1} \sum_{i,j,k} \left\{ D_l^T \
+      b_T \right\} = 0
+  \end{equation}
+  This property is trivially satisfied since the iso-neutral diffusive
+  operator is written in flux form.
+
+\item[$\bullet$ no increase of tracer variance] The iso-neutral diffusion
+  does not increase the tracer variance, $i.e.$
+  \begin{equation} \label{Gf_property1} \sum_{i,j,k} \left\{ T \ D_l^T
+      \ b_T \right\} \leq 0
+  \end{equation}
+  The property is demonstrated in
+  \ref{sec:variance} above. It is a key property for a diffusion
+  term. It means that it is also a dissipation term, $i.e.$ it is a
+  diffusion of the square of the quantity on which it is applied.  It
+  therefore ensures that, when the diffusivity coefficient is large
+  enough, the field on which it is applied become free of grid-point
+  noise.
+
+\item[$\bullet$ self-adjoint operator] The iso-neutral diffusion
+  operator is self-adjoint, $i.e.$
+  \begin{equation} \label{Gf_property1} \sum_{i,j,k} \left\{ S \ D_l^T
+      \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
+  \end{equation}
+  In other word, there is no need to develop a specific routine from
+  the adjoint of this operator. We just have to apply the same
+  routine. This property can be demonstrated similarly to the proof of
+  the `no increase of tracer variance' property. The contribution by a
+  single triad towards the left hand side of \eqref{Gf_property1}, can
+  be found by replacing $\delta[T]$ by $\delta[S]$ in \eqref{locFdtdx}
+  and \eqref{locFdtdx}. This results in a term similar to
+  \eqref{eq:perfect-square},
+\begin{equation}
+  \label{eq:TScovar}
+  -A_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
+  \left(
+    \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+    \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right)
+  \left(
+    \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+    -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+    \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+  \right).
+\end{equation}
+This is symmetrical in $T $ and $S$, so exactly the same term arises
+from the discretization of this triad's contribution towards the
+RHS of \eqref{Gf_property1}.
+\end{description}
+
+
+$\ $\newline %force an empty line
+% ================================================================
+% Skew flux formulation for Eddy Induced Velocity :
+% ================================================================
+\section{ Eddy induced velocity and Skew flux formulation}
+
+When Gent and McWilliams's [1990] diffusion is used (\key{traldf\_eiv} defined), 
+an additional advection term is added. The associated velocity is the so called 
+eddy induced velocity, the formulation of which depends on the slopes of iso-
+neutral surfaces. Contrary to the case of iso-neutral mixing, the slopes used 
+here are referenced to the geopotential surfaces, $i.e.$ \eqref{Eq_ldfslp_geo} 
+is used in $z$-coordinate, and the sum \eqref{Eq_ldfslp_geo}
++ \eqref{Eq_ldfslp_iso} in $z^*$ or $s$-coordinates. 
+
+The eddy induced velocity is given by: 
+\begin{equation} \label{Eq_eiv_v}
+\begin{split}
+ u^* & = - \frac{1}{e_{2}e_{3}}\;          \partial_k \left( e_{2} \, A_{e} \; r_i \right)   \\
+ v^* & = - \frac{1}{e_{1}e_{3}}\;          \partial_k \left( e_{1} \, A_{e} \; r_j \right)   \\
+w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i  \left( e_{2} \, A_{e} \; r_i \right) 
+							    + \partial_j  \left( e_{1} \, A_{e} \;r_j \right) \right\} \\
+\end{split}
+\end{equation}
+where $A_{e}$ is the eddy induced velocity coefficient, and $r_i$ and $r_j$ the slopes between the iso-neutral and the geopotential surfaces. 
+
+The traditional way to implement this additional advection is to add it to the Eulerian 
+velocity prior to computing the tracer advection. This allows us to take advantage of 
+all the advection schemes offered for the tracers (see \S\ref{TRA_adv}) and not just 
+a $2^{nd}$ order advection scheme. This is particularly useful for passive tracers 
+where \emph{positivity} of the advection scheme is of paramount importance. 
+
+\citet{Griffies_JPO98} introduces another way to implement the eddy induced advection, 
+the so-called skew form. It is based on a transformation of the advective fluxes 
+using the non-divergent nature of the eddy induced velocity. 
+For example in the (\textbf{i},\textbf{k}) plane, the tracer advective fluxes can be 
+transformed as follows:
+\begin{flalign*}
+\begin{split}
+\textbf{F}_{eiv}^T = 
+\begin{pmatrix} 
+ 	        {e_{2}\,e_{3}\;  u^*} 	 	\\
+ 		{e_{1}\,e_{2}\; w^*}	 \\
+\end{pmatrix}   \;   T
+&=
+\begin{pmatrix} 
+ 	        { - \partial_k \left( e_{2} \, A_{e} \; r_i \right) \; T \;} 	 	\\
+ 		{+ \partial_i  \left( e_{2} \, A_{e} \; r_i \right) \; T \;}	 \\
+\end{pmatrix} 			\\
+&=			
+\begin{pmatrix} 
+ 	        { - \partial_k \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}  \\
+ 		{+ \partial_i  \left( e_{2} \, A_{e} \; r_i  \; T \right) \;}	 \\
+\end{pmatrix} 			
+ + 
+\begin{pmatrix} 
+ 	        {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}  \\
+ 		{ - e_{2} \, A_{e} \; r_i  \; \partial_i  T}	 \\
+\end{pmatrix} 	 
+\end{split}
+\end{flalign*}
+and since the eddy induces velocity field is no-divergent, we end up with the skew 
+form of the eddy induced advective fluxes:
+\begin{equation} \label{Eq_eiv_skew_continuous}
+\textbf{F}_{eiv}^T = \begin{pmatrix} 
+ 	        {+ e_{2} \, A_{e} \; r_i  \; \partial_k T}   \\
+ 		{ - e_{2} \, A_{e} \; r_i  \; \partial_i  T}	 \\
+                                 \end{pmatrix}
+\end{equation}
+The tendency associated with eddy induced velocity is then simply the divergence 
+of the \eqref{Eq_eiv_skew_continuous} fluxes. It naturally conserves the tracer 
+content, as it is expressed in flux form. It also preserve the tracer variance.
+
+The discrete form of  \eqref{Eq_eiv_skew_continuous} using the slopes \eqref{Gf_slopes} and defining $A_e$ at $T$-point is then given by:
+\begin{flalign} \label{Eq_eiv_skew}   \begin{split}
+\textbf{F}_{eiv}^T   \equiv                                   
+                        \sum_{\substack{i_p,\,k_p}} \begin{pmatrix} 
++{e_{2u}}_{i+1/2-i_p}^{k}                                    \ {A_{e}}_{i+1/2-i_p}^{k} 
+\ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}]      \\
+    \\
+- {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p} 
+\ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]    \\   
+                                                                       \end{pmatrix}   
+\end{split}   \end{flalign}
+Note that \eqref{Eq_eiv_skew} is valid in $z$-coordinate with or without partial cells. In $z^*$ or $s$-coordinate, the the slope between the level and the geopotential surfaces must be added to $\mathbb{R}$ for the discret form to be exact. 
+
+Such a choice of discretisation is consistent with the iso-neutral operator as it uses the same definition for the slopes. It also ensures the conservation of the tracer variance (see Appendix \ref{Apdx_eiv_skew}), $i.e.$ it does not include a diffusive component but is a `pure' advection term.
+
+
+
+
+\newpage      %force an empty line
+% ================================================================
+% Discrete Invariants of the skew flux formulation
+% ================================================================
+\subsection{Discrete Invariants of the skew flux formulation}
+\label{Apdx_eiv_skew}
+
+
+Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
+
+This have to be moved in an Appendix.
+
+The continuous property to be demonstrated is :
+\begin{align*}
+\int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0
+\end{align*}
+The discrete form of its left hand side is obtained using \eqref{Eq_eiv_skew}
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\;
+ \delta_i  &\left[                                                    
+{e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]         
+   \right] \; T_i^k      \\
+- \delta_k &\left[ 
+{e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]   
+   \right] \; T_i^k      \         \Biggr\}   
+\end{align*}
+apply the adjoint of delta operator, it becomes
+\begin{align*}
+ \sum\limits_{i,k} \sum_{\substack{i_p,\,k_p}}  \Biggl\{   \;\;
+  &\left(                                                    
+{e_{2u}}_{i+i_p+1/2}^{k}                                  \;\ \ {A_{e}}_{i+i_p+1/2}^{k} 
+\ \ \ { _{i+i_p+1/2}^k \mathbb{R}_{-i_p}^{k_p} }   \quad \delta_{k+k_p}[T_{i+i_p+1/2}]         
+   \right) \; \delta_{i+1/2}[T^{k}]      \\
+- &\left( 
+{e_{2u}}_i^{k+k_p+1/2}                                     \ \ {A_{e}}_i^{k+k_p+1/2} 
+\ \ { _i^{k+k_p+1/2} \mathbb{R}_{i_p}^{-k_p} }   \ \ \delta_{i+i_p}[T^{k+k_p+1/2}]   
+     \right) \; \delta_{k+1/2}[T_{i}]       \         \Biggr\}       
+\end{align*}
+Expending the summation on $i_p$ and $k_p$, it becomes:
+\begin{align*}
+ \begin{matrix}  
+&\sum\limits_{i,k}   \Bigl\{ 
+  &+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
+  &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{-1/2}} &\delta_{k-1/2}[T_{i+1}]    &\delta_{i+1/2}[T^{k}]   &\\
+&&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}      
+  &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{-1/2}}  &\delta_{k-1/2}[T_{i\ \ \ \;}]  &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_{i+1}^{k}                             &{A_{e}}_{i+1    }^{k} 
+  &\ {_{i+1}^k \mathbb{R}_{- 1/2}^{+1/2}} &\delta_{k+1/2}[T_{i+1}]     &\delta_{i+1/2}[T^{k}] &\\
+&&+{e_{2u}}_i^{k\ \ \ \:}                            &{A_{e}}_{i}^{k\ \ \ \:}       
+    &\ {\ \ \;_i^k \mathbb{R}_{+1/2}^{+1/2}} &\delta_{k+1/2}[T_{i\ \ \ \;}] &\delta_{i+1/2}[T^{k}] &\\
+%
+&&-{e_{2u}}_i^{k+1}                                &{A_{e}}_i^{k+1} 
+  &{_i^{k+1} \mathbb{R}_{-1/2}^{- 1/2}}   &\delta_{i-1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\   
+&&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
+  &{\ \ \;_i^k  \mathbb{R}_{-1/2}^{+1/2}}   &\delta_{i-1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] &\\
+&&-{e_{2u}}_i^{k+1    }                             &{A_{e}}_i^{k+1} 
+  &{_i^{k+1} \mathbb{R}_{+1/2}^{- 1/2}}   &\delta_{i+1/2}[T^{k+1}]      &\delta_{k+1/2}[T_{i}] &\\   
+&&-{e_{2u}}_i^{k\ \ \ \:}                             &{A_{e}}_i^{k\ \ \ \:} 
+  &{\ \ \;_i^k  \mathbb{R}_{+1/2}^{+1/2}}   &\delta_{i+1/2}[T^{k\ \ \ \:}]  &\delta_{k+1/2}[T_{i}] 
+&\Bigr\}  \\
+\end{matrix}   
+\end{align*}
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{+1/2}}$ are the 
+same but of opposite signs, they cancel out. 
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{-1/2}}$. 
+The two terms associated with the triad ${_i^k \mathbb{R}_{+1/2}^{-1/2}}$ are the 
+same but both of opposite signs and shifted by 1 in $k$ direction. When summing over $k$ 
+they cancel out with the neighbouring grid points. 
+Exactly the same thing occurs for the triad ${_i^k \mathbb{R}_{-1/2}^{+1/2}}$ in the 
+$i$ direction. Therefore the sum over the domain is zero, $i.e.$ the variance of the 
+tracer is preserved by the discretisation of the skew fluxes.
+
+%%% Local Variables: 
+%%% TeX-master: "../../NEMO_book.tex"
+%%% End: 
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DOM.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DOM.tex	(revision 2282)
@@ -3,5 +3,5 @@
 % Chapter 2 Ñ Space and Time Domain (DOM)
 % ================================================================
-\chapter{Space and Time Domain (DOM) }
+\chapter{Space Domain (DOM) }
 \label{DOM}
 \minitoc
@@ -12,11 +12,6 @@
 %                  should be put outside of DOM routine (better with TRC staff and off-line
 %                  tracers)
-%	- daymod: definition of the time domain (nit000, nitend andd the calendar)
 % 	-geo2ocean:  how to switch from geographic to mesh coordinate
-%  - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
-
-\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction 
-which could be referred to here, would help  ==> to be added}
-%%%%
+%     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
 
 
@@ -24,10 +19,11 @@
 $\ $\newline    % force a new ligne
 
-
-Having defined the continuous equations in Chap.~\ref{PE}, we need to choose a 
-discretization on a grid, and numerical algorithms. In the present chapter, we 
-provide a general description of the staggered grid used in \NEMO, and other 
-information relevant to the main directory routines (time stepping, main program) 
-as well as the DOM (DOMain) directory. 
+Having defined the continuous equations in Chap.~\ref{PE} and chosen a time 
+discretization Chap.~\ref{STP}, we need to choose a discretization on a grid, 
+and numerical algorithms. In the present chapter, we provide a general description 
+of the staggered grid used in \NEMO, and other information relevant to the main 
+directory routines as well as the DOM (DOMain) directory. 
+
+$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -46,5 +42,5 @@
 \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 +53,5 @@
 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 +116,31 @@
 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 +150,7 @@
 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 +158,5 @@
 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 +175,5 @@
 
 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 +216,7 @@
 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 +228,8 @@
 \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 +243,10 @@
 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 +295,7 @@
 \S\ref{LBC_mpp}).
 
+
+$\ $\newline    % force a new ligne
+
 % ================================================================
 % Domain: Horizontal Grid (mesh) 
@@ -341,5 +330,5 @@
 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 +336,6 @@
 \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 +348,13 @@
 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 +369,5 @@
 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 
@@ -397,6 +386,6 @@
 \begin{description}
 \item[\jp{jphgr\_mesh}=0]  The most general curvilinear orthogonal grids.
-The coordinates and their first derivatives with respect to $i$ and $j$ are 
-provided in a file, read in \rou{hgr\_read} subroutine of the domhgr module.
+The coordinates and their first derivatives with respect to $i$ and $j$ are provided
+in a input file (\ifile{coordinates}), read in \rou{hgr\_read} subroutine of the domhgr module.
 \item[\jp{jphgr\_mesh}=1 to 5] A few simple analytical grids are provided (see below). 
 For other analytical grids, the \mdl{domhgr} module must be modified by the user. 
@@ -422,5 +411,5 @@
 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 +420,6 @@
 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, 
@@ -447,9 +436,9 @@
 %        Grid files
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Grid files}
+\subsection{Output Grid files}
 \label{DOM_hgr_files}
 
 All the arrays relating to a particular ocean model configuration (grid-point 
-position, scale factors, masks) can be saved in files if $\np{nmsh} \not= 0$ 
+position, scale factors, masks) can be saved in files if $\np{nn\_msh} \not= 0$ 
 (namelist parameter). This can be particularly useful for plots and off-line 
 diagnostics. In some cases, the user may choose to make a local modification 
@@ -458,5 +447,7 @@
 happens to be too wide due to insufficient model resolution). An example 
 is Gibraltar Strait in the ORCA2 configuration. When such modifications are done, 
-the output grid written when $\np{nmsh} \not=0$ is no more equal to the input grid.
+the output grid written when $\np{nn\_msh} \not=0$ is no more equal to the input grid.
+
+$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -467,5 +458,5 @@
 \label{DOM_zgr}
 %-----------------------------------------nam_zgr & namdom-------------------------------------------
-\namdisplay{nam_zgr} 
+\namdisplay{namzgr} 
 \namdisplay{namdom} 
 %-------------------------------------------------------------------------------------------------------------
@@ -510,8 +501,10 @@
 Contrary to the horizontal grid, the vertical grid is computed in the code and no 
 provision is made for reading it from a file. The only input file is the bathymetry 
-(in meters)\footnote{N.B. in full step $z$-coordinate, a \textit{bathy\_level} file can 
-replace the \textit{bathy\_meter} file, so that the computation of the number of 
-wet ocean point in each water column is by-passed}. After reading the bathymetry, 
-the algorithm for vertical grid definition differs between the different options:
+(in meters) (\ifile{bathy\_meter}) 
+\footnote{N.B. in full step $z$-coordinate, a \ifile{bathy\_level} file can replace the 
+\ifile{bathy\_meter} file, so that the computation of the number of wet ocean point 
+in each water column is by-passed}. 
+After reading the bathymetry, the algorithm for vertical grid definition differs 
+between the different options:
 \begin{description}
 \item[\textit{zco}] set a reference coordinate transformation $z_0 (k)$, and set $z(i,j,k,t)=z_0 (k)$.
@@ -535,5 +528,5 @@
 C-Pre-Processor (CPP) key. To improve the code readability while providing this 
 flexibility, the vertical coordinate and scale factors are defined as functions of 
-$(i,j,k)$ with "fs" as prefix (examples: \textit{fsdeptht, fse3t,} etc) that can be either 
+$(i,j,k)$ with "fs" as prefix (examples: \textit{fsdept, fse3t,} etc) that can be either 
 three-dimensional arrays, or a one dimensional array when \key{zco} is defined. 
 These functions are defined in the file \hf{domzgr\_substitute} of the DOM directory. 
@@ -548,13 +541,13 @@
 
 Three options are possible for defining the bathymetry, according to the 
-namelist variable \np{ntopo}: 
+namelist variable \np{nn\_bathy}: 
 \begin{description}
-\item[\np{ntopo} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
+\item[\np{nn\_bathy} = 0] a flat-bottom domain is defined. The total depth $z_w (jpk)$ 
 is given by the coordinate transformation. The domain can either be a closed 
 basin or a periodic channel depending on the parameter \jp{jperio}. 
-\item[\np{ntopo} = -1] a domain with a bump of topography one third of the 
+\item[\np{nn\_bathy} = -1] a domain with a bump of topography one third of the 
 domain width at the central latitude. This is meant for the "EEL-R5" configuration, 
 a periodic or open boundary channel with a seamount. 
-\item[\np{ntopo} = 1] read a bathymetry. The bathymetry file (Netcdf format) 
+\item[\np{nn\_bathy} = 1] read a bathymetry. The \ifile{bathy\_meter} file (Netcdf format) 
 provides the ocean depth (positive, in meters) at each grid point of the model grid. 
 The bathymetry is usually built by interpolating a standard bathymetry product 
@@ -563,8 +556,4 @@
 (all levels are masked).
 \end{description}
-
-When using the rigid lid approximation (\key{dynspg\_rl} is defined) isolated land 
-masses (islands) must be identified by negative integers in the input bathymetry file 
-(see \S\ref{MISC_solisl}).
 
 When a global ocean is coupled to an atmospheric model it is better to represent 
@@ -593,9 +582,9 @@
 
 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 +652,5 @@
 \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 +717,6 @@
 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 +727,8 @@
 % -------------------------------------------------------------------------------------------------------------
 \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 
@@ -752,13 +741,13 @@
 \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 
 depth, since a mixed step-like and bottom-following representation of the 
 topography can be used (Fig.~\ref{Fig_z_zps_s_sps}d-e). In the example provided 
-(\hf{zgr\_s} file) $h$ is a smooth envelope bathymetry and steps are used to represent 
-sharp bathymetric gradients.
-
-A new flexible stretching function, modified from \citet{Song1994} is provided as an example:
+(\rou{zgr\_sco} routine, see \mdl{domzgr}) $h$ is a smooth envelope bathymetry 
+and steps are used to represent sharp bathymetric gradients.
+
+A new flexible stretching function, modified from \citet{Song_Haidvogel_JCP94} is provided as an example:
 \begin{equation} \label{DOM_sco_function}
 \begin{split}
@@ -801,27 +790,13 @@
 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 
-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 
-in meters: both will be allowed in future versions.
+$t$-point. mbathy is computed from the meter bathymetry using the definiton of 
+gdept as the number of $t$-points which gdept $\leq$ bathy. 
 
 Modifications of the model bathymetry are performed in the \textit{bat\_ctl} 
 routine (see \mdl{domzgr} module) after mbathy is computed. Isolated grid points 
 that do not communicate with another ocean point at the same level are eliminated.
-
-In the case of the rigid-lid approximation when islands occur in the computational 
-domain (\np{ln\_dynspg\_rl}=.true. and \key{island} is defined), the \textit{mbathy} 
-array must be provided and takes values from $-N$ to \jp{jpk}-1. It provides the 
-following information: $mbathy(i,j) = -n, \ n \in \left] 0,N \right]$, $T$-points are 
-land points on the $n^{th}$ island ; $mbathy(i,j) =0$, $T$-points are land points 
-on the main land (continent) ; $mbathy(i,j) =k$, the first $k$ $T$- and $w$-points 
-are ocean points, the others are points below the ocean floor. 
-
-This is used to compute the island barotropic stream function used in the rigid lid 
-computation (see \S\ref{MISC_solisl}).
 
 From the \textit{mbathy} array, the mask fields are defined as follows:
@@ -848,263 +823,2 @@
 \gmcomment{   \colorbox{yellow}{Add one word on tricky trick !} mbathy in further modified in zdfbfr{\ldots}.  }
 %%%
-
-% ================================================================
-% Time Discretisation
-% ================================================================
-\section{Time Discretisation}
-\label{DOM_nxt}
-
-The time stepping used in \NEMO is a three level scheme that can be 
-represented as follows:
-\begin{equation} \label{Eq_DOM_nxt}
-   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t-\Delta t,t,t+\Delta t}
-\end{equation} 
-where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the 
-corresponding time evolution equation; $\Delta t$ is the time step; and the 
-superscripts indicate the time at which a quantity is evaluated. Each term of the 
-RHS is evaluated at a specific time step(s) depending on the physics with which 
-it is associated. 
-
-The choice of the time step used for this evaluation is discussed below as 
-well as the implications in terms of starting or restarting a model 
-simulation. Note that the time stepping is generally performed in a one step 
-operation. With such a complex and nonlinear system of equations it would be 
-dangerous to let a prognostic variable evolve in time for each term separately.
-
-The three level scheme requires three arrays for each prognostic variables. 
-For each variable $x$ there is $x_b$ (before) and $x_n$ (now). The third array, 
-although referred to as $x_a$ (after) in the code, is usually not the variable at 
-the next time step; but rather it is used to store the time derivative (RHS in 
-\eqref{Eq_DOM_nxt}) prior to time-stepping the equation. Generally, the time 
-stepping is performed once at each time step in \mdl{tranxt} and \mdl{dynnxt} 
-modules, except for implicit vertical diffusion or sea surface height when 
-time-splitting options are used.
-
-% -------------------------------------------------------------------------------------------------------------
-%        Non-Diffusive Part---Leapfrog Scheme
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Non-Diffusive Part --- Leapfrog Scheme}
-\label{DOM_nxt_leap_frog}
-
-The time stepping used for non-diffusive processes is the well-known 
-leapfrog scheme. It is a time centred scheme, i.e. the RHS is evaluated at 
-time step $t$, the now time step. It is only used for non-diffusive terms, 
-that is momentum and tracer advection, pressure gradient, and Coriolis 
-terms. This scheme is widely used for advective processes in low-viscosity 
-fluids. It is an efficient method that achieves second-order accuracy with 
-just one right hand side evaluation per time step. Moreover, it does not 
-artificially damp linear oscillatory motion nor does it produce instability 
-by amplifying the oscillations. These advantages are somewhat diminished by 
-the large phase-speed error of the leapfrog scheme, and the unsuitability of 
-leapfrog differencing for the representation of diffusive and Rayleigh 
-damping processes. However, the most serious problem associated with the 
-leapfrog scheme is a high-frequency computational noise called 
-"time-splitting" \citep{Haltiner1980} that develops when the method 
-is used to model non linear fluid dynamics: the even and odd time steps tend 
-to diverge into a physical and a computational mode. Time splitting can 
-be controlled through the use of an Asselin time filter (first designed by 
-\citep{Robert1966} and more comprehensively studied by \citet{Asselin1972}), 
-or by periodically reinitialising the leapfrog solution through a single 
-integration step with a two-level scheme. In \NEMO we follow the first 
-strategy:
-\begin{equation} \label{Eq_DOM_nxt_asselin}
-x_F^t  = x^t + \gamma \, \left[ x_f^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
-\end{equation} 
-where the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
-coefficient. $\gamma$ is initialized as \np{atfp} (namelist parameter). 
-Its default value is \np{atfp}=0.1.  This default value causes a significant dissipation 
-of high frequency motions. Recommended values in idealized studies of shallow 
-water turbulence are two orders of magnitude smaller (\citep{Farge1987}). 
-Both strategies do, nevertheless, degrade the accuracy of the calculation from 
-second to first order. The leapfrog scheme combined with a Robert-Asselin 
-time filter has been preferred to other time differencing schemes such as 
-predictor corrector or trapezoidal schemes, because the user has an explicit 
-and simple control of the magnitude of the time diffusion of the scheme. 
-In association with the 2nd order centred space discretisation of the 
-advective terms in the momentum and tracer equations, it avoids implicit 
-numerical diffusion in both the time and space discretisations of the 
-advective term: they are both set explicitly by the user through the Robert-Asselin 
-filter parameter and the viscous and diffusive coefficients.
-
-\gmcomment{
-%gm - reflexion about leapfrog: ongoing work with Matthieu Leclair
-% to be updated latter with addition of new time stepping strategy
-\colorbox{yellow}{Note}: 
-The Robert-Asselin time filter slightly departs from a simple second order time 
-diffusive operator computed with a forward time stepping due to the presence of 
-$x_f^{t-\Delta t}$ in the right hand side of  \ref{Eq_DOM_nxt_asselin}. The original 
-willing of Robert1966 and Asselin1972 was to design a time filter that allow much 
-larger parameter than 0.5.   is due to computer saving consideration. In the original 
-asselin filter, $x^{t-\Delta t}$ is used instead:
- \begin{equation} \label{Eq_DOM_nxt_asselin_true}
-x_f^t  = x^t + \gamma \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right]
-\end{equation} 
-Applying a "true" Asselin time filter is nothing more than adding a harmonic 
-diffusive operator in time. Indeed, equations \ref{Eq_DOM_nxt} and 
-\ref{Eq_DOM_nxt_asselin_true} can be rewritten together as:
-\begin{equation} \label{Eq_DOM_nxt2}
-\begin{split}
-  \frac{ x^{t+\Delta t} - x^{t-\Delta t} } { 2 \,\Delta t } 
-  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} + \frac{ x_f^t  - x^t }{2 \,\Delta t} \\
-  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
-    + \gamma\ \frac{  \, \left[ x^{t-\Delta t} - 2 x^t + x^{t+\Delta t} \right] }{2 \,\Delta t}  \\
-  &=  \text{RHS}_x^{t-\Delta t,t,t+\Delta t} 
-  + 2 \Delta t \ \gamma \ \frac{1}{{2 \Delta t}^2}  
-  	\,\delta_{t-1}\,\left[ \delta_{t+1/2}\left[ x^t \right] \right] 
-  \end{split}
-\end{equation} 
-expressing this again in a continuous form, one finds that the Asselin filter leads to :
-\begin{equation} \label{Eq_DOM_nxt3}
-  \frac{ \partial x} { \partial t } =  \text{RHS} + 2 \,\Delta t \ \gamma \ \frac{ {\partial}^2 x}{ \partial t ^2 }
-\end{equation} 
-
-Equations  \ref{Eq_DOM_nxt2} and \ref{Eq_DOM_nxt3} suggest several remarks. 
-First the Asselin filter is definitively a second order time diffusive operator which is 
-evaluated at centered time step. The magnitude of this diffusion is proportional to 
-the time step (with $\gamma$ usually taken between $10^{-1}$ to $10^{-3}$). 
-Second, this term has to be taken into account in all budgets of the equations 
-(mass, heat content, salt content, kinetic energy...). Nevertheless,we stress here 
-that it is small and does not create systematic biases. Indeed let us evaluate how 
-it contributes to the time evolution of $x$ between $t_o$ and $t_1$:
-\begin{equation} \label{Eq_DOM_nxt4}
-\begin{split}
- t_1-t_o &= \int_{t_o}^{t_1} \frac{ \partial x} { \partial t } dt \\
- 		&= \int_{t_o}^{t_1} \text{RHS} dt + 2 \,\Delta t \ \gamma \left(
-		  \left. \frac{ \partial x}{ \partial t } \right|_1 
-		- \left. \frac{ \partial x}{ \partial t } \right|_o  \right)
- \end{split}
-\end{equation} 
-}
-
-Alternative time stepping schemes are currently under investigation.
-
-% -------------------------------------------------------------------------------------------------------------
-%        Diffusive Part---Forward or Backward Scheme
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Diffusive Part --- Forward or Backward Scheme}
-\label{DOM_nxt_forward_imp}
-
-The leapfrog differencing scheme is unsuitable for the representation of 
-diffusive and damping processes. For a tendancy $D_x$, representing a 
-diffusive term or a restoring term to a tracer climatology 
-(when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme
- is used :
-\begin{equation} \label{Eq_DOM_nxt_euler}
-   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \ {D_x}^{t-\Delta t}
-\end{equation} 
-
-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}:
-\begin{equation} \label{Eq_DOM_nxt_euler_stability}
-A^h < \left\{
-\begin{aligned}
-                    &\frac{e^2}{  8 \; \Delta t } 	&&\quad \text{laplacian diffusion} 	\\
-                    &\frac{e^4}{64 \; \Delta t } 	&&\quad \text{bilaplacian diffusion} 
-            \end{aligned}
-\right.
-\end{equation}
-where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is 
-the mixing coefficient. The linear constraint \eqref{Eq_DOM_nxt_euler_stability} 
-is a necessary condition, but not sufficient. If it is not satisfied, even mildly, 
-then the model soon becomes wildly unstable. The instability can be removed 
-by either reducing the length of the time steps or reducing the mixing coefficient.
-
-For the vertical diffusion terms, a forward time differencing scheme can be 
-used, but usually the numerical stability condition implies a strong 
-constraint on the time step. Two solutions are available in \NEMO to overcome 
-the stability constraint: $(a)$ a forward time differencing scheme using a 
-time splitting technique (\np{ln\_zdfexp}=.true.) or $(b)$ a backward (or implicit) 
-time differencing scheme by \np{ln\_zdfexp}=.false.). In $(a)$, the master 
-time step $\Delta $t is cut into $N$ fractional time steps so that the 
-stability criterion is reduced by a factor of $N$. The computation is done as 
-follows:
-\begin{equation} \label{Eq_DOM_nxt_ts}
-\begin{split}
-& u_\ast ^{t-\Delta t} = u^{t-\Delta t}   \\
-& u_\ast ^{t-\Delta t+L\frac{2\Delta t}{N}}=u_\ast ^{t-\Delta t+\left( {L-1} 
-\right)\frac{2\Delta t}{N}}+\frac{2\Delta t}{N}\;\text{DF}^{t-\Delta t+\left( {L-1} \right)\frac{2\Delta t}{N}}
-        \quad \text{for $L=1$ to $N$}      \\
-& u^{t+\Delta t} = u_\ast^{t+\Delta t}
-\end{split}
-\end{equation}
-with DF a vertical diffusion term. The number of fractional time steps, $N$, is given 
-by setting \np{n\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 
-stable but diffusive. It can be written as follows:
-\begin{equation} \label{Eq_DOM_nxt_imp}
-   x^{t+\Delta t} = x^{t-\Delta t} + 2 \, \Delta t \  \text{RHS}_x^{t+\Delta t}
-\end{equation} 
-
-This scheme is rather time consuming since it requires a matrix inversion, 
-but it becomes attractive since a splitting factor of 3 or more is needed 
-for the forward time differencing scheme. For example, the finite difference 
-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 
-_k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta _{k+1/2} \left[ {T^{t+1}} \right]} 
-\right]
-\end{equation}
-where RHS is the right hand side of the equation except for the vertical diffusion term. 
-We rewrite \eqref{Eq_DOM_nxt_imp} as:
-\begin{equation} \label{Eq_DOM_nxt_imp_mat}
--c(k+1)\;u^{t+1}(k+1)+d(k)\;u^{t+1}(k)-\;c(k)\;u^{t+1}(k-1) \equiv b(k)
-\end{equation}
-where 
-\begin{align*} 
- c(k) &= A_w^{vm} (k) \, / \, e_{3uw} (k)     \\
- d(k) &= e_{3u} (k)       \, / \, (2\Delta t) + c_k + c_{k+1}    \\
- b(k) &= e_{3u} (k) \; \left( u^{t-1}(k) \, / \, (2\Delta t) + \text{RHS} \right)    
-\end{align*}
-
-\eqref{Eq_DOM_nxt_imp_mat} is a linear system of equations which associated 
-matrix is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal 
-term is greater than the sum of the two extra-diagonal terms, therefore a special 
-adaptation of the Gauss elimination procedure is used to find the solution 
-(see for example \citet{Richtmyer1967}).
-
-% -------------------------------------------------------------------------------------------------------------
-%        Start/Restart strategy
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Start/Restart strategy}
-\label{DOM_nxt_rst}
-%--------------------------------------------namrun-------------------------------------------
-\namdisplay{namrun}         
-%--------------------------------------------------------------------------------------------------------------
-
-The first time step of this three level scheme when starting from initial conditions 
-is a forward step (Euler time integration): 
-\begin{equation} \label{Eq_DOM_euler}
-	x^1 = x^0 + \Delta t \ \text{RHS}^0
-\end{equation}
-
-It is also possible to restart from a previous computation, by using a 
-restart file. The restart strategy is designed to ensure perfect 
-restartability of the code: the user should obtain the same results to 
-machine precision either by running the model for $2N$ time steps in one go, 
-or by performing two consecutive experiments of $N$ steps with a restart. 
-This requires saving two time levels and many auxiliary data in the restart 
-files in machine precision. 
-
-Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure 
-gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
-added in the restart file to ensure an exact restartability. This is done only optionally 
-via the namelist parameter \np{nn\_dynhpg\_rst}, so that a reduction of the size of 
-restart file can be obtained when the restartability is not a key issue (operational 
-oceanography or ensemble simulation for seasonal forcast).
-%%%
-\gmcomment{add here how to force the restart to contain only one time step for operational purposes}
-%%%
-
-\gmcomment{       % add a subsection here  
-
-%-------------------------------------------------------------------------------------------------------------
-%        Time Domain
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Time domain}
-\label{DOM_nxt_time}
-
- \colorbox{yellow}{add here a few word on nit000 and nitend}
-
- \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}
-
-}     	%% end add
-
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DYN.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_DYN.tex	(revision 2282)
@@ -11,9 +11,9 @@
 $\ $\newline      %force an empty line
 
-Using the representation described in Chap.\ref{DOM}, several semi-discrete 
+Using the representation described in Chapter \ref{DOM}, several semi-discrete 
 space forms of the dynamical equations are available depending on the vertical 
 coordinate used and on the conservation properties of the vorticity term. In all 
 the equations presented here, the masking has been omitted for simplicity. 
-One must be aware that all the quantities are masked fields and that each time a 
+One must be aware that all the quantities are masked fields and that each time an
 average or difference operator is used, the resulting field is multiplied by a mask.
 
@@ -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 
+the rhs of the this equation 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, either a vertical advection part (ZAD) in the vector invariant formulation, or a Coriolis 
+and advection part (COR+ADV) in the flux formulation. The terms following these 
 are the pressure gradient contributions (HPG, Hydrostatic Pressure Gradient, 
 and SPG, Surface Pressure Gradient); and contributions from lateral diffusion 
@@ -40,9 +39,9 @@
 
 In the present chapter we also describe the diagnostic equations used to compute 
-the horizontal divergence and curl of the velocities (\emph{divcur} module) as well 
-as the vertical velocity (\emph{wzvmod} module).
+the horizontal divergence, curl of the velocities (\emph{divcur} module) and 
+the vertical velocity (\emph{wzvmod} module).
 
 The different options available to the user are managed by namelist variables. 
-For equation term \textit{ttt}, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 
+For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx}, 
 where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 
 If a CPP key is used for this term its name is \textbf{key\_ttt}. The corresponding 
@@ -50,8 +49,8 @@
 usually computed in the \textit{dyn\_ttt\_xxx} subroutine.
 
-The user has the option of extracting each tendency term of both the rhs of the 
-3D momentum equation (\key{trddyn} defined) for output, as described in 
-Chap.\ref{MISC}.  Furthermore, the tendency terms associated to the 2D 
-barotropic vorticity balance (\key{trdvor} defined) can be derived on-line from the 
+The user has the option of extracting and outputting each tendency term from the
+3D momentum equations (\key{trddyn} defined), as described in 
+Chap.\ref{MISC}.  Furthermore, the tendency terms associated with the 2D 
+barotropic vorticity balance (when \key{trdvor} is defined) can be derived from the 
 3D terms.
 %%%
@@ -60,4 +59,104 @@
 
 $\ $\newline    % force a new ligne
+
+% ================================================================
+% Sea Surface Height evolution & Diagnostics variables
+% ================================================================
+\section{Sea surface height and diagnostic variables ($\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 (when \key{zco} is defined), 
+$e_{3u}$=$e_{3v}$=$e_{3f}$ so that these metric terms cancel in \eqref{Eq_divcur_div}.
+
+Note also that although the vorticity has the same discrete expression in $z$- 
+and $s$-coordinates, 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 not necessarily 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 density 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 scheme 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. Replacing $T$ by the number $1$ in the tracer equation and summing
+over the water column must lead to the sea surface height equation otherwise tracer content
+will not be conserved \ref{Griffies_al_MWR01, LeclairMadec2009}.
+
+The vertical velocity is computed by an upward integration of the horizontal 
+divergence starting at the bottom, taking 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 \rdt}
+\end{aligned}   \right.
+\end{equation}
+\sgacomment{should e3t involve k in this equation?}
+
+In the case of a non-linear free surface (\key{vvl}), the top vertical velocity is $-\textit{emp}/\rho_w$, 
+as changes in the divergence of the barotropic transport are absorbed into the change 
+of the level thicknesses, re-orientated downward.
+In the case of a linear free surface, the time derivative in \eqref{Eq_wzv} disappears.
+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 $s$-coordinates, 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-orientated downwards in the \textsc{fortran} code compared 
+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
@@ -66,10 +165,10 @@
 \label{DYN_adv_cor_vect}
 %-----------------------------------------nam_dynadv----------------------------------------------------
-\namdisplay{nam_dynadv} 
+\namdisplay{namdyn_adv} 
 %-------------------------------------------------------------------------------------------------------------
 
 The vector invariant form of the momentum equations is the one most 
-often used in applications of \NEMO ocean model. The flux form option 
-(see next section) has been introduced since version $2$. 
+often used in applications of the \NEMO ocean model. The flux form option 
+(see next section) has been present since version $2$. 
 Coriolis and momentum advection terms are evaluated using a leapfrog 
 scheme, $i.e.$ the velocity appearing in these expressions is centred in 
@@ -85,20 +184,22 @@
 \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. The vorticity terms are all computed in dedicated routines that can be found in 
+the \mdl{dynvor} module.
 
 %-------------------------------------------------------------
 %                 enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=.true.)}
+\subsubsection{Enstrophy conserving scheme (\np{ln\_dynvor\_ens}=true)}
 \label{DYN_vor_ens}
 
@@ -106,12 +207,12 @@
 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.
@@ -121,5 +222,5 @@
 %                 energy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=.true.)}
+\subsubsection{Energy conserving scheme (\np{ln\_dynvor\_ene}=true)}
 \label{DYN_vor_ene}
 
@@ -129,7 +230,7 @@
 \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} 
@@ -138,24 +239,22 @@
 %                 mix energy/enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=.true.) }
+\subsubsection{Mixed energy/enstrophy conserving scheme (\np{ln\_dynvor\_mix}=true) }
 \label{DYN_vor_mix}
 
-The mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 
+For the mixed energy/enstrophy conserving scheme (MIX scheme), a mixture of the 
 two previous schemes is used. It consists of the ENS scheme (\ref{Eq_dynvor_ens}) 
-to the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 
+for the relative vorticity term, and of the ENE scheme (\ref{Eq_dynvor_ene}) applied 
 to the planetary vorticity term.
 \begin{equation} \label{Eq_dynvor_mix}
-\left\{ {
-\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} 
 
@@ -163,23 +262,30 @@
 %                 energy and enstrophy conserving scheme
 %-------------------------------------------------------------
-\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=.true.) }
+\subsubsection{Energy and enstrophy conserving scheme (\np{ln\_dynvor\_een}=true) }
 \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 be at least partly damped by the momentum diffusion operator ($i.e.$ the 
+subgrid-scale advection), but not by the resolved advection term. The ENS and ENE schemes
+therefore do not contribute to any grid point noise in the horizontal velocity field.
+Such noise would result in more noise in the vertical velocity field, an undesirable feature. This is a well-known 
+characteristic of $C$-grid discretization where $u$ and $v$ are located at different grid points,
+a price worth paying to avoid a double averaging in the pressure gradient term as in the $B$-grid. 
+\gmcomment{ To circumvent this, Adcroft (ADD REF HERE) 
+
+Nevertheless, this technique strongly distort the phase and group velocity of Rossby waves....}
+
+A very nice solution to the problem of double averaging 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 diffusion 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. 
+First consider 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 
@@ -199,52 +305,41 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-Note that a key point in \eqref{Eq_een_e3f} is that the averaging in \textbf{i}- and 
+Note that a key point in \eqref{Eq_een_e3f} is that the averaging in the \textbf{i}- and 
 \textbf{j}- directions uses the masked vertical scale factor but is always divided by 
-$4$, not by the sum of the mask at $T$-point. This preserves the continuity of 
-$e_{3f}$ when one or more of the neighbouring $e_{3T}$ tends to zero and 
-extends by continuity the value of $e_{3f}$ in the land areas. 
-%%%
-\gmcomment{this has to be further investigate in case of several step topography}
-%%%
-
-The vorticity terms are represented as: 
+$4$, not by the sum of the masks at the four $T$-points. This preserves the continuity of 
+$e_{3f}$ when one or more of the neighbouring $e_{3t}$ tends to zero and 
+extends by continuity the value of $e_{3f}$ into the land areas. This feature is essential for 
+the $z$-coordinate with partial steps.
+
+
+Next, the vorticity triads, $ {^i_j}\mathbb{Q}^{i_p}_{j_p}$ can be defined at a $T$-point as 
+the following triad combinations of the neighbouring potential vorticities defined at f-points 
+(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 values: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$. 
+
+Finally, the vorticity terms are represented as: 
 \begin{equation} \label{Eq_dynvor_een}
 \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 the 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 it
+leads to a significant reduction of the noise in the vertical velocity field \citep{Le_Sommer_al_OM09}. 
+Furthermore, used in combination with a partial steps 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}. 
 
 %--------------------------------------------------------------------------------------------------------------
@@ -255,13 +350,11 @@
 \label{DYN_keg}
 
-As demonstarted in Appendix~\ref{Apdx_C}, there is a single discrete formulation 
+As demonstrated in Appendix~\ref{Apdx_C}, there is a single discrete formulation 
 of the kinetic energy gradient term that, together with the formulation chosen for 
 the vertical advection (see below), conserves the total kinetic energy:
 \begin{equation} \label{Eq_dynkeg}
 \left\{ \begin{aligned}
- -\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.
@@ -352,7 +442,7 @@
 Two advection schemes are available: a $2^{nd}$ order centered finite 
 difference scheme, CEN2, or a $3^{rd}$ order upstream biased scheme, UBS. 
-The latter is described in \citet{Sacha2005}. The schemes are selected using 
-the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. In flux 
-form, the schemes differ by the choice of a space and time interpolation to 
+The latter is described in \citet{Shchepetkin_McWilliams_OM05}. The schemes are 
+selected using the namelist logicals \np{ln\_dynadv\_cen2} and \np{ln\_dynadv\_ubs}. 
+In flux form, the schemes differ by the choice of a space and time interpolation to 
 define the value of $u$ and $v$ at the centre of each face of $u$- and $v$-cells, 
 $i.e.$ at the $T$-, $f$-, and $uw$-points for $u$ and at the $f$-, $T$- and 
@@ -362,5 +452,5 @@
 %                 2nd order centred scheme
 %-------------------------------------------------------------
-\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=.true.)}
+\subsubsection{$2^{nd}$ order centred scheme (cen2) (\np{ln\_dynadv\_cen2}=true)}
 \label{DYN_adv_cen2}
 
@@ -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} 
 
@@ -387,5 +473,5 @@
 %                 UBS scheme
 %-------------------------------------------------------------
-\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=.true.)}
+\subsubsection{Upstream Biased Scheme (UBS) (\np{ln\_dynadv\_ubs}=true)}
 \label{DYN_adv_ubs}
 
@@ -400,10 +486,12 @@
 \end{equation}
 where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$. This results 
-in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Sacha2005}. 
+in a dissipatively dominant ($i.e.$ hyper-diffusive) truncation error \citep{Shchepetkin_McWilliams_OM05}. 
 The overall performance of the advection scheme is similar to that reported in 
 \citet{Farrow1995}. It is a relatively good compromise between accuracy and 
 smoothness. It is not a \emph{positive} scheme, meaning that false extrema are 
 permitted. But the amplitudes of the false extrema are significantly reduced over 
-those in the centred second order method. 
+those in the centred second order method. As the scheme already includes 
+a diffusion component, it can be used without explicit  lateral diffusion on momentum 
+($i.e.$ \np{ln\_dynldf\_lap}=\np{ln\_dynldf\_bilap}=false), and it is recommended to do so.
 
 The UBS scheme is not used in all directions. In the vertical, the centred $2^{nd}$ 
@@ -415,13 +503,13 @@
 For stability reasons,  the first term in (\ref{Eq_dynadv_ubs}), which corresponds 
 to a second order centred scheme, is evaluated using the \textit{now} velocity 
-(centred in time), while the second term, which is the diffusive part of the scheme, 
+(centred in time), while the second term, which is the diffusion part of the scheme, 
 is evaluated using the \textit{before} velocity (forward in time). This is discussed 
-by \citet{Webb1998} in the context of the Quick advection scheme.
-
-Note that the UBS and Quadratic Upstream Interpolation for Convective Kinematics 
-(QUICK) schemes only differ by one coefficient. Substituting $1/6$ with $1/8$ in 
-(\ref{Eq_dynadv_ubs}) leads to the QUICK advection scheme \citep{Webb1998}. 
+by \citet{Webb_al_JAOT98} in the context of the Quick advection scheme.
+
+Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) 
+schemes only differ by one coefficient. Replacing $1/6$ by $1/8$ in 
+(\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 
+is hard coded. Nevertheless it is quite easy to make the substitution in the
 \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
 
@@ -440,10 +528,6 @@
 \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 +538,5 @@
 
 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.
@@ -462,5 +546,5 @@
 %--------------------------------------------------------------------------------------------------------------
 \subsection   [$z$-coordinate with full step (\np{ln\_dynhpg\_zco}) ]
-			{$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=.true.)}
+			{$z$-coordinate with full step (\np{ln\_dynhpg\_zco}=true)}
 \label{DYN_hpg_zco}
 
@@ -495,5 +579,7 @@
 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 $e_{3w}$.
 
 %--------------------------------------------------------------------------------------------------------------
@@ -501,5 +587,5 @@
 %--------------------------------------------------------------------------------------------------------------
 \subsection   [$z$-coordinate with partial step (\np{ln\_dynhpg\_zps})]
-			{$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=.true.)}
+			{$z$-coordinate with partial step (\np{ln\_dynhpg\_zps}=true)}
 \label{DYN_hpg_zps}
 
@@ -524,16 +610,17 @@
 \label{DYN_hpg_sco}
 
-Pressure gradient formulations in $s$-coordinate have been the subject of a vast 
-literature ($e.g.$, \citet{Song1998, Sacha2003}). A number of different pressure 
-gradient options are coded, but they are not yet fully documented or tested. 
-
-$\bullet$ Traditional coding (see for example \citet{Madec1996}: (\np{ln\_dynhpg\_sco}=.true., 
-\np{ln\_dynhpg\_hel}=.true.)
+Pressure gradient formulations in an $s$-coordinate have been the subject of a vast 
+number of papers ($e.g.$, \citet{Song1998, Shchepetkin_McWilliams_OM05}). 
+A number of different pressure gradient options are coded, but they are not yet fully 
+documented or tested. 
+
+$\bullet$ Traditional coding (see for example \citet{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} 
@@ -542,14 +629,14 @@
 \eqref{Eq_dynhpg_zco_surf} - \eqref{Eq_dynhpg_zco}, and $z_T$ is the depth of 
 the $T$-point evaluated from the sum of the vertical scale factors at the $w$-point 
-($e_{3w}$). The version \np{ln\_dynhpg\_hel}=.true. has been added by Aike 
+($e_{3w}$). The version \np{ln\_dynhpg\_hel}=true has been added by Aike 
 Beckmann and involves a redefinition of the relative position of $T$-points relative 
 to $w$-points. 
 
-$\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=.true.)
-
-$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Sacha2003} 
-(\np{ln\_dynhpg\_djc}=.true.)
-
-$\bullet$ Rotated axes scheme (rot) \citep{Thiem2006} (\np{ln\_dynhpg\_rot}=.true.)
+$\bullet$ Weighted density Jacobian (WDJ) \citep{Song1998} (\np{ln\_dynhpg\_wdj}=true)
+
+$\bullet$ Density Jacobian with cubic polynomial scheme (DJC) \citep{Shchepetkin_McWilliams_OM05} 
+(\np{ln\_dynhpg\_djc}=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 
@@ -562,5 +649,5 @@
 %--------------------------------------------------------------------------------------------------------------
 \subsection   [Time-scheme (\np{ln\_dynhpg\_imp}) ]
-			{Time-scheme (\np{ln\_dynhpg\_imp}=.true./.false.)}
+			{Time-scheme (\np{ln\_dynhpg\_imp}= true/false)}
 \label{DYN_hpg_imp}
 
@@ -569,29 +656,28 @@
 above is the  \textit{now} density, computed from the \textit{now} temperature and 
 salinity. In some specific cases (usually high resolution simulations over an ocean 
-domain which includes weakly stratified regions) the physical phenomenum that 
+domain which includes weakly stratified regions) the physical phenomenon that 
 controls the time-step is internal gravity waves (IGWs). A semi-implicit scheme for 
-doubling the stability limit associated with IGWs can be used \citep{Brown1978, 
+doubling the stability limit associated with IGWs can be used \citep{Brown_Campana_MWR78, 
 Maltrud1998}. It involves the evaluation of the hydrostatic pressure gradient as an 
-average over the three time levels $t-\Delta t$, $t$, and $t+\Delta t$ ($i.e.$  
-\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at central 
+average over the three time levels $t-\rdt$, $t$, and $t+\rdt$ ($i.e.$  
+\textit{before},  \textit{now} and  \textit{after} time-steps), rather than at the central 
 time level $t$ only, as in the standard leapfrog scheme. 
 
-$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=.true.):
+$\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}=true):
 
 \begin{equation} \label{Eq_dynhpg_lf}
-\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
-=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
-\end{equation}
-
-$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=.true.):
+\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
+	-\frac{1}{\rho _o \,e_{1u} }\delta _{i+1/2} \left[ {p_h^t } \right]
+\end{equation}
+
+$\bullet$ semi-implicit scheme (\np{ln\_dynhpg\_imp}=true):
 \begin{equation} \label{Eq_dynhpg_imp}
-\frac{u^{t+\Delta t}-u^{t-\Delta t}}{2\Delta t}
-=\;\cdots \;-\frac{1}{\rho _o \,e_{1u} } \delta _{i+1/2} \left[ \frac{ p_h^{t+\Delta t} +2p_h^t 
-+p_h^{t-\Delta t} } { 4 }  \right]
+\frac{u^{t+\rdt}-u^{t-\rdt}}{2\rdt} = \;\cdots \;
+	-\frac{1}{4\,\rho _o \,e_{1u} } \delta_{i+1/2} \left[ p_h^{t+\rdt} +2\,p_h^t +p_h^{t-\rdt}  \right]
 \end{equation}
 
 The semi-implicit time scheme \eqref{Eq_dynhpg_imp} is made possible without 
 significant additional computation since the density can be updated to time level 
-$t+\Delta t$ before computing the horizontal hydrostatic pressure gradient. It can 
+$t+\rdt$ before computing the horizontal hydrostatic pressure gradient. It can 
 be easily shown that the stability limit associated with the hydrostatic pressure 
 gradient doubles using \eqref{Eq_dynhpg_imp} compared to that using the 
@@ -602,5 +688,5 @@
 the stability limits associated with advection or diffusion.
 
-In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=.true.. 
+In practice, the semi-implicit scheme is used when \np{ln\_dynhpg\_imp}=true. 
 In this case, we choose to apply the time filter to temperature and salinity used in 
 the equation of state, instead of applying it to the hydrostatic pressure or to the 
@@ -609,14 +695,12 @@
 as follows:
 \begin{equation} \label{Eq_rho_flt}
-	\rho^t = \rho( \widetilde{T},\widetilde {S},z_T)
+	\rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
  \quad	  \text{with}	\quad 
-	\widetilde{\,\cdot\,} = \frac{  \,\cdot\,^{t+\Delta t} +2 \,\,\cdot\,^t + \,\cdot\,^{t-\Delta t}  } {4}
-\end{equation}
-\gmcomment{STEVEN: bullets look odd in this, could use X}
+	\widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
+\end{equation}
 
 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.
 
 % ================================================================
@@ -625,87 +709,68 @@
 \section  [Surface pressure gradient (\textit{dynspg}) ]
 		{Surface pressure gradient (\mdl{dynspg})}
-\label{DYN_hpg_spg}
+\label{DYN_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 surface pressure gradient term is related to the representation of the free surface (\S\ref{PE_hor_pg}). The main distinction is between the fixed volume case (linear free surface) and the variable volume case (nonlinear free surface, \key{vvl} is defined). In the linear free surface case (\S\ref{PE_free_surface}) the vertical scale factors $e_{3}$ are fixed in time, while they are time-dependent in the nonlinear case (\S\ref{PE_free_surface}). With both linear and nonlinear free surface, external gravity waves are allowed in the equations, which imposes a very small time step when an explicit time stepping is used. Two methods are proposed to allow a longer time step for the three-dimensional equations: the filtered free surface, which is a modification of the continuous equations (see \eqref{Eq_PE_flt}), and the split-explicit free surface described below. The extra term introduced in the filtered method is calculated implicitly, so that the update of the next velocities is done in module \mdl{dynspg\_flt} and not in \mdl{dynnxt}.
+
+%%%
+
+
+The form of the surface pressure gradient term depends on how the user wants to handle 
+the fast external gravity waves that are a solution of the analytical equation (\S\ref{PE_hor_pg}). 
+Three formulations are available, all controlled by a CPP key (ln\_dynspg\_xxx):
+an explicit formulation which requires a small time step ;
+a filtered free surface formulation which allows a larger time step by adding a filtering 
+term into the momentum equation ; 
+and a split-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. As a consequence 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). 
-
-The surface pressure gradient, also evaluated using a leap-frog scheme, is 
-then simply given by :
+In the explicit free surface formulation (\key{dynspg\_exp} defined), the model time step 
+is chosen to be small enough to resolve the external gravity waves (typically a few tens of seconds). 
+The surface pressure gradient, evaluated using a leap-frog scheme ($i.e.$ centered in time),
+is thus simply given by :
 \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 
+allowed in the computation of 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} 
-%--------------------------------------------------------------------------------------------------------------
-
-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}). 
+
+The split-explicit free surface formulation used in \NEMO (\key{dynspg\_ts} defined),
+also called the time-splitting formulation, follows the one 
+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 $\rdt$, 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 the \np{nn\_baro} namelist parameter as: 
+$\Delta_e = \Delta / nn\_baro$.
+ 
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -716,19 +781,19 @@
 and internal modes. Time increases to the right. 
 Internal mode time steps (which are also the model time steps) are denoted 
-by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$. 
+by $t-\rdt$, $t, t+\rdt$, and $t+2\rdt$. 
 The curved line represents a leap-frog time step, and the smaller time 
-steps $N \Delta t_e=\frac{3}{2}\Delta t$ are denoted by the zig-zag line. 
+steps $N \rdt_e=\frac{3}{2}\rdt$ are denoted by the zig-zag line. 
 The vertically integrated forcing \textbf{M}(t) computed at the model time step $t$ 
 represents the interaction between the external and internal motions. 
 While keeping \textbf{M} and freshwater forcing field fixed, a leap-frog 
 integration carries the external mode variables (surface height and vertically 
-integrated velocity) from $t$ to $t+\frac{3}{2} \Delta t$ using N external time 
-steps of length $\Delta t_e$. Time averaging the external fields over the 
+integrated velocity) from $t$ to $t+\frac{3}{2} \rdt$ using N external time 
+steps of length $\rdt_e$. Time averaging the external fields over the 
 $\frac{2}{3}N+1$ time steps (endpoints included) centers the vertically integrated 
-velocity and the sea surface height at the model timestep $t+\Delta t$. 
+velocity and the sea surface height at the model timestep $t+\rdt$. 
 These averaged values are used to update \textbf{M}(t) with both the surface 
-pressure gradient and the Coriolis force, therefore providing the $t+\Delta t$
+pressure gradient and the Coriolis force, therefore providing the $t+\rdt$
 velocity.  The model time stepping scheme can then be achieved by a baroclinic 
-leap-frog time step that carries the surface height from $t-\Delta t$ to $t+\Delta t$.  }
+leap-frog time step that carries the surface height from $t-\rdt$ to $t+\rdt$.  }
 \end{center}
 \end{figure}
@@ -736,75 +801,118 @@
 
 The split-explicit formulation has a damping effect on external gravity waves, 
-which is weaker damping than for the filtered free surface but still significant as 
+which is weaker damping than that for the filtered free surface but still significant, as 
 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 
-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]
+%>>>>>===============
+\gmcomment{               %%% copy from griffies Book 
+
+\textbf{title: Time stepping the barotropic system }
+
+Assume knowledge of the full velocity and tracer fields at baroclinic time $\tau$. Hence, 
+we can update the surface height and vertically integrated velocity with a leap-frog 
+scheme using the small barotropic time step $\rdt$. We have 
+
+\begin{equation} \label{DYN_spg_ts_eta}
+\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+	= 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
+\end{equation}
+\begin{multline} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
+	= 2\rdt \left[ - f \textbf{k} \times \textbf{U}^{(b)}(\tau,t_{n}) 
+	- H(\tau) \nabla p_s^{(b)}(\tau,t_{n}) +\textbf{M}(\tau) \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.
-
+\
+
+In these equations, araised (b) denotes values of surface height and vertically integrated velocity updated with the barotropic time steps. The $\tau$ time label on $\eta^{(b)}$ 
+and $U^{(b)}$ denotes the baroclinic time at which the vertically integrated forcing $\textbf{M}(\tau)$ (note that this forcing includes the surface freshwater forcing), the tracer fields, the freshwater flux $\text{EMP}_w(\tau)$, and total depth of the ocean $H(\tau)$ are held for the duration of the barotropic time stepping over a single cycle. This is also the time 
+that sets the barotropic time steps via 
+\begin{equation} \label{DYN_spg_ts_t}
+t_n=\tau+n\rdt   
+\end{equation}
+with $n$ an integer. The density scaled surface pressure is evaluated via 
+\begin{equation} \label{DYN_spg_ts_ps}
+p_s^{(b)}(\tau,t_{n}) = \begin{cases}
+	g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
+	g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case} 
+	\end{cases}
+\end{equation}
+To get started, we assume the following initial conditions 
+\begin{equation} \label{DYN_spg_ts_eta}
+\begin{split}
+\eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}
+\\
+\eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0} 
+\end{split}
+\end{equation}
+with 
+\begin{equation} \label{DYN_spg_ts_etaF}
+ \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
+\end{equation}
+the time averaged surface height taken from the previous barotropic cycle. Likewise, 
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}	\\
+\\
+\textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}  	
+\end{equation}
+with 
+\begin{equation} \label{DYN_spg_ts_u}
+ \overline{\textbf{U}^{(b)}(\tau)} 
+ 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
+\end{equation}
+the time averaged vertically integrated transport. Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
+
+Upon reaching $t_{n=N} = \tau + 2\Delta \tau$ , the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at baroclinic time $\tau + \Delta \tau$ 
+\begin{equation} \label{DYN_spg_ts_u}
+\textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} 
+ 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\end{equation}
+The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using the following form 
+
+\begin{equation} \label{DYN_spg_ts_ssh}
+\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]  
+\end{equation}
+
+ The use of this "big-leap-frog" scheme for the surface height ensures compatibility between the mass/volume budgets and the tracer budgets. More discussion of this point is provided in Chapter 10 (see in particular Section 10.2). 
+ 
+In general, some form of time filter is needed to maintain integrity of the surface 
+height field due to the leap-frog splitting mode in equation \ref{DYN_spg_ts_ssh}. We 
+have tried various forms of such filtering, with the following method discussed in 
+\cite{Griffies_al_MWR01} chosen due to its stability and reasonably good maintenance of 
+tracer conservation properties (see Section ??) 
+
+\begin{equation} \label{DYN_spg_ts_sshf}
+\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
+\end{equation}
+Another approach tried was 
+
+\begin{equation} \label{DYN_spg_ts_sshf2}
+\eta^{F}(\tau-\Delta) = \eta(\tau) 
+	+ (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
+				    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
+\end{equation}
+
+which is useful since it isolates all the time filtering aspects into the term multiplied 
+by $\alpha$. This isolation allows for an easy check that tracer conservation is exact when 
+eliminating tracer and surface height time filtering (see Section ?? for more complete discussion). However, in the general case with a non-zero $\alpha$, the filter \ref{DYN_spg_ts_sshf} was found to be more conservative, and so is recommended. 
+
+}            %%end gm comment (copy of griffies book)
+
+%>>>>>===============
+
+
+%--------------------------------------------------------------------------------------------------------------
+% 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}.
+
+%% gm %%======>>>>   given here the discrete eqs provided to the solver
+
+Note that in the linear free surface formulation (\key{vvl} not defined), the ocean depth 
+is time-independent and so is the matrix to be inverted. It is computed once and for all and applies to all ocean time steps. 
 
 % ================================================================
@@ -815,12 +923,12 @@
 \label{DYN_ldf}
 %------------------------------------------nam_dynldf----------------------------------------------------
-\namdisplay{nam_dynldf} 
+\namdisplay{namdyn_ldf} 
 %-------------------------------------------------------------------------------------------------------------
 
-The options available for lateral diffusion are for the choice of  laplacian 
+The options available for lateral diffusion are to use either laplacian 
 (rotated or not) or biharmonic operators. The coefficients may be constant 
 or spatially variable; the description of the coefficients is found in the chapter 
-on lateralphysics (Chap.\ref{LDF}). The lateral diffusion of momentum is 
-evaluated using a forward scheme, i.e. the velocity appearing in its expression 
+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 
@@ -832,5 +940,5 @@
 % ================================================================
 \subsection   [Iso-level laplacian operator (\np{ln\_dynldf\_lap}) ]
-			{Iso-level laplacian operator (\np{ln\_dynldf\_lap}=.true.)}
+			{Iso-level laplacian operator (\np{ln\_dynldf\_lap}=true)}
 \label{DYN_ldf_lap}
 
@@ -850,7 +958,7 @@
 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}.
 
 %--------------------------------------------------------------------------------------------------------------
@@ -858,13 +966,13 @@
 %--------------------------------------------------------------------------------------------------------------
 \subsection   [Rotated laplacian operator (\np{ln\_dynldf\_iso}) ]
-			{Rotated laplacian operator (\np{ln\_dynldf\_iso}=.true.)}
+			{Rotated laplacian operator (\np{ln\_dynldf\_iso}=true)}
 \label{DYN_ldf_iso}
 
-A rotation of the lateral momentum diffusive operator is needed in several cases: 
-for iso-neutral diffusion in $z$-coordinate (\np{ln\_dynldf\_iso}=.true.) and for 
-either iso-neutral (\np{ln\_dynldf\_iso}=.true.) or geopotential 
-(\np{ln\_dynldf\_hor}=.true.) diffusion in $s$-coordinate. In the partial step 
-case, coordinates are horizontal excepted at the deepest level and no 
-rotation is performed when \np{ln\_dynldf\_hor}=.true.. The diffusive operator 
+A rotation of the lateral momentum diffusion operator is needed in several cases: 
+for iso-neutral diffusion in the $z$-coordinate (\np{ln\_dynldf\_iso}=true) and for 
+either iso-neutral (\np{ln\_dynldf\_iso}=true) or geopotential 
+(\np{ln\_dynldf\_hor}=true) diffusion in the $s$-coordinate. In the partial step 
+case, coordinates are horizontal except at the deepest level and no 
+rotation is performed when \np{ln\_dynldf\_hor}=true. The diffusion operator 
 is defined simply as the divergence of down gradient momentum fluxes on each 
 momentum component. It must be emphasized that this formulation ignores 
@@ -875,6 +983,6 @@
  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 +1010,6 @@
  \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] 
@@ -919,5 +1027,5 @@
 \end{equation}
 where $r_1$ and $r_2$ are the slopes between the surface along which the 
-diffusive operator acts and the surface of computation ($z$- or $s$-surfaces). 
+diffusion operator acts and the surface of computation ($z$- or $s$-surfaces). 
 The way these slopes are evaluated is given in the lateral physics chapter 
 (Chap.\ref{LDF}).
@@ -927,5 +1035,5 @@
 %--------------------------------------------------------------------------------------------------------------
 \subsection   [Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap})]
-			{Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=.true.)}
+			{Iso-level bilaplacian operator (\np{ln\_dynldf\_bilap}=true)}
 \label{DYN_ldf_bilap}
 
@@ -953,8 +1061,8 @@
 would be too restrictive a constraint on the time step. Two time stepping schemes 
 can be used for the vertical diffusion term : $(a)$ a forward time differencing 
-scheme (\np{ln\_zdfexp}=.true.) using a time splitting technique 
-(\np{n\_zdfexp} $>$ 1) or $(b)$ a backward (or implicit) time differencing scheme 
-(\np{ln\_zdfexp}=.false.) (see \S\ref{DOM_nxt}). Note that namelist variables 
-\np{ln\_zdfexp} and \np{n\_zdfexp} apply to both tracers and dynamics. 
+scheme (\np{ln\_zdfexp}=true) using a time splitting technique 
+(\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{nn\_zdfexp} apply to both tracers and dynamics. 
 
 The formulation of the vertical subgrid scale physics is the same whatever 
@@ -974,5 +1082,5 @@
 depends on the vertical physics used (see \S\ref{ZDF}).
 
-The surface boundary condition on momentum is given by the stress exerted by 
+The surface boundary condition on momentum is the stress exerted by 
 the wind. At the surface, the momentum fluxes are prescribed as the boundary 
 condition on the vertical turbulent momentum fluxes,
@@ -1002,9 +1110,7 @@
 enter the dynamical equations. 
 
-One is the effect of atmospheric pressure on the ocean dynamics (to be 
-introduced later). 
-
-Another forcing term is the tidal potential, which will be introduced in the 
-reference version soon. 
+One is the effect of atmospheric pressure on the ocean dynamics.
+Another forcing term is the tidal potential.
+Both of which will be introduced into the reference version soon. 
 
 % ================================================================
@@ -1021,94 +1127,36 @@
 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  	\\
-\\
-&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\Delta t} -2u^t+u^{t+\Delta t}} \right]
-\end{split}
+(cf. Chap.\ref{STP}). The scheme is applied to the velocity, except when using 
+the flux form of momentum advection (cf. \S\ref{DYN_adv_cor_flux}) in the variable 
+volume 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 ; \key{vvl} not defined):
+\begin{equation} \label{Eq_dynnxt_vec}
+\left\{   \begin{aligned}
+&u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t  	\\
+&u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
+\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+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; 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-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \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 
-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). 
-
-% ================================================================
+initialized as \np{nn\_atfp} (namelist parameter). Its default value is \np{nn\_atfp} = $10^{-3}$.
+In both cases, the modified Asselin filter is not applied since perfect conservation 
+is not an issue for the momentum equations.
+
+Note that with the filtered free surface, the update of the \textit{after} velocities 
+is done in the \mdl{dynsp\_flt} module, and only array swapping
+and Asselin filtering is done in \mdl{dynnxt}.
+
+
+
+% ================================================================
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LBC.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LBC.tex	(revision 2282)
@@ -15,8 +15,8 @@
 % Boundary Condition at the Coast
 % ================================================================
-\section{Boundary Condition at the Coast (\np{shlat})}
+\section{Boundary Condition at the Coast (\np{rn\_shlat})}
 \label{LBC_coast}
 %--------------------------------------------nam_lbc-------------------------------------------------------
-\namdisplay{nam_lbc} 
+\namdisplay{namlbc} 
 %--------------------------------------------------------------------------------------------------------------
 
@@ -69,5 +69,5 @@
 condition influences the relative vorticity and momentum diffusive trends, and is 
 required in order to compute the vorticity at the coast. Four different types of 
-lateral boundary condition are available, controlled by the value of the \np{shlat} 
+lateral boundary condition are available, controlled by the value of the \np{rn\_shlat} 
 namelist parameter. (The value of the mask$_{f}$ array along the coastline is set 
 equal to this parameter.) These are:
@@ -76,6 +76,6 @@
 \begin{figure}[!p] \label{Fig_LBC_shlat}  \begin{center}
 \includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_LBC_shlat.pdf}
-\caption {lateral boundary condition (a) free-slip ($shlat=0$) ; (b) no-slip ($shlat=2$) 
-; (c) "partial" free-slip ($0<shlat<2$) and (d) "strong" no-slip ($2<shlat$). 
+\caption {lateral boundary condition (a) free-slip ($rn\_shlat=0$) ; (b) no-slip ($rn\_shlat=2$) 
+; (c) "partial" free-slip ($0<rn\_shlat<2$) and (d) "strong" no-slip ($2<rn\_shlat$). 
 Implied "ghost" velocity inside land area is display in grey. }
 \end{center}   \end{figure}
@@ -84,10 +84,10 @@
 \begin{description}
 
-\item[free-slip boundary condition (\np{shlat}=0): ]  the tangential velocity at the 
+\item[free-slip boundary condition (\np{rn\_shlat}=0): ]  the tangential velocity at the 
 coastline is equal to the offshore velocity, $i.e.$ the normal derivative of the 
 tangential velocity is zero at the coast, so the vorticity: mask$_{f}$ array is set 
 to zero inside the land and just at the coast (Fig.~\ref{Fig_LBC_shlat}-a).
 
-\item[no-slip boundary condition (\np{shlat}=2): ] the tangential velocity vanishes 
+\item[no-slip boundary condition (\np{rn\_shlat}=2): ] the tangential velocity vanishes 
 at the coastline. Assuming that the tangential velocity decreases linearly from 
 the closest ocean velocity grid point to the coastline, the normal derivative is 
@@ -108,5 +108,5 @@
 \end{equation}
 
-\item["partial" free-slip boundary condition (0$<$\np{shlat}$<$2): ] the tangential 
+\item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2): ] the tangential 
 velocity at the coastline is smaller than the offshore velocity, $i.e.$ there is a lateral 
 friction but not strong enough to make the tangential velocity at the coast vanish 
@@ -114,5 +114,5 @@
 strictly inbetween $0$ and $2$.
 
-\item["strong" no-slip boundary condition (2$<$\np{shlat}): ] the viscous boundary 
+\item["strong" no-slip boundary condition (2$<$\np{rn\_shlat}): ] the viscous boundary 
 layer is assumed to be smaller than half the grid size (Fig.~\ref{Fig_LBC_shlat}-d). 
 The friction is thus larger than in the no-slip case.
@@ -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.
 
@@ -355,8 +355,10 @@
 ($e1t$, $e2t$, etc) and do not expect the grid size to be zero, even on land. It may be 
 best not to eliminate land processors when running the model especially to write the 
-mesh files as outputs (when \np{nmsh} namelist parameter differs from 0).
+mesh files as outputs (when \np{nn\_msh} namelist parameter differs from 0).
+%%
 \gmcomment{Steven : dont understand this, no land processor means no output file 
 covering this part of globe; its only when files are stitched together into one that you 
 can leave a hole}
+%%
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -380,15 +382,12 @@
 %-    nobc_dta    =    0     !  = 0 the obc data are equal to the initial state
 %-                           !  = 1 the obc data are read in 'obc   .dta' files
-%-    rdpein      =    1.    !  ???
-%-    rdpwin      =    1.    !  ???
-%-    rdpnin      =   30.    !  ???
-%-    rdpsin      =    1.    !  ???
-%-    rdpeob      = 1500.    !  time relaxation (days) for the east  open boundary
-%-    rdpwob      =   15.    !    "        "           for the west  open boundary
-%-    rdpnob      =  150.    !    "        "           for the north open boundary
-%-    rdpsob      =   15.    !    "        "           for the south open boundary
-%-    zbsic1      =  140.e+6 !  barotropic stream function on first  isolated coastline
-%-    zbsic2      =    1.e+6 !    "                   "    on second isolated coastline
-%-    zbsic3      =    0.    !    "                   "    on thrid  isolated coastline
+%-    rn_dpein      =    1.    !  ???
+%-    rn_dpwin      =    1.    !  ???
+%-    rn_dpnin      =   30.    !  ???
+%-    rn_dpsin      =    1.    !  ???
+%-    rn_dpeob      = 1500.    !  time relaxation (days) for the east  open boundary
+%-    rn_dpwob      =   15.    !    "        "           for the west  open boundary
+%-    rn_dpnob      =  150.    !    "        "           for the north open boundary
+%-    rn_dpsob      =   15.    !    "        "           for the south open boundary
 %-    ln_obc_clim = .true.   !  climatological obc data files (default T)
 %-    ln_vol_cst  = .true.   !  total volume conserved
@@ -428,5 +427,5 @@
 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 
@@ -520,6 +519,6 @@
 It is necessary to provide information at the boundaries. The simplest case is 
 when this information does not change in time and is equal to the initial conditions 
-(namelist variable \np{nobc\_dta}=0). This is the case for the standard configuration 
-EEL5 with open boundaries. When (\np{nobc\_dta}=1), open boundary information 
+(namelist variable \np{nn\_obcdta}=0). This is the case for the standard configuration 
+EEL5 with open boundaries. When (\np{nn\_obcdta}=1), open boundary information 
 is read from netcdf files. For convenience the input files are supposed to be similar 
 to the ''history'' NEMO output files, for dimension names and variable names. 
@@ -529,5 +528,5 @@
 
 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 +547,5 @@
 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.
 
@@ -596,5 +595,5 @@
 data is held fixed in time. If the files contain 12 values, it is assumed that the input 
 is a climatology for a repeated annual cycle (corresponding to the case \np{ln\_obc\_clim} 
-= .True.). The case of an arbitrary number of time frames is not yet implemented 
+=true). The case of an arbitrary number of time frames is not yet implemented 
 correctly; the user is required to write his own code in the module \mdl{obc\_dta} 
 to deal with this situation. 
@@ -608,8 +607,8 @@
 of energy. The constraints are specified separately at each boundary as time 
 scales for ''inflow'' and ''outflow'' as defined below. The time scales are set (in days) 
-by namelist parameters such as \np{rdpein}, \np{rdpeob} for the eastern open 
+by namelist parameters such as \np{rn\_dpein}, \np{rn\_dpeob} for the eastern open 
 boundary for example. When both time scales are zero for a given boundary 
-($e.g.$ for the western boundary, \jp{lp\_obc\_west}=.True., \np{rdpwob}=0 and 
-\np{rdpwin}=0) this means that the boundary in question is a ''fixed '' boundary 
+($e.g.$ for the western boundary, \jp{lp\_obc\_west}=true, \np{rn\_dpwob}=0 and 
+\np{rn\_dpwin}=0) this means that the boundary in question is a ''fixed '' boundary 
 where the solution is set exactly by the boundary data. This is not recommended, 
 except in combination with increased viscosity in a ''sponge'' layer next to the 
@@ -623,5 +622,5 @@
 $s$-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 +636,5 @@
 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 
@@ -665,5 +664,5 @@
 propagation), the radiation condition (\ref{Eq_obc_rado}) is used. 
 When  $C_{\phi x}$ is westward (inward propagation), (\ref{Eq_obc_radi}) is 
-used with a strong relaxation to climatology (usually $\tau_{i}=\np{rdpein}=$1~day).
+used with a strong relaxation to climatology (usually $\tau_{i}=\np{rn\_dpein}=$1~day).
 Equation (\ref{Eq_obc_radi}) is solved with a Euler time-stepping scheme. As a 
 consequence, setting $\tau_{i}$ smaller than, or equal to the time step is equivalent 
@@ -672,8 +671,8 @@
 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 
@@ -733,12 +732,214 @@
 \colorbox{yellow}{OBC rigid lid? {\ldots}}
 
-
-
-
 % ====================================================================
-% Flow Relaxation Scheme 
+% Unstructured open boundaries BDY 
 % ====================================================================
-\section{Flow Relaxation Scheme (???)}
+\section{Unstructured Open Boundary Conditions (\key{bdy})}
 \label{LBC_bdy}
 
-%gm% to be updated by Met Office
+%-----------------------------------------nambdy--------------------------------------------
+%-    filbdy_mask    =  ''                  !  name of mask file (if ln_bdy_mask=.TRUE.)
+%-    filbdy_data_T  = 'bdydata_grid_T.nc'  !  name of data file for FRS condition (T-points)
+%-    filbdy_data_U  = 'bdydata_grid_U.nc'  !  name of data file for FRS condition (U-points)
+%-    filbdy_data_V  = 'bdydata_grid_V.nc'  !  name of data file for FRS condition (V-points)
+%-    filbdy_data_bt_T  = 'bdydata_bt_grid_T.nc'  !  name of data file for Flather condition (T-points)
+%-    filbdy_data_bt_U  = 'bdydata_bt_grid_U.nc'  !  name of data file for Flather condition (U-points)
+%-    filbdy_data_bt_V  = 'bdydata_bt_grid_V.nc'  !  name of data file for Flather condition (V-points)
+%-    ln_bdy_clim    = .false.              !  contain 1 (T) or 12 (F) time dumps and be cyclic
+%-    ln_bdy_vol     = .true.               !  total volume correction (see volbdy parameter)
+%-    ln_bdy_mask    = .false.              !  boundary mask from filbdy_mask (T) or boundaries are on edges of domain (F)
+%-    ln_bdy_tides   = .true.               !  Apply tidal harmonic forcing with Flather condition
+%-    ln_bdy_dyn_fla = .true.               !  Apply Flather condition to velocities
+%-    ln_bdy_tra_frs = .false.              !  Apply FRS condition to temperature and salinity 
+%-    ln_bdy_dyn_frs = .false.              !  Apply FRS condition to velocities
+%-    nbdy_dta       =  1                   !  = 0, bdy data are equal to the initial state
+%-                                          !  = 1, bdy data are read in 'bdydata   .nc' files
+%-    nb_rimwidth    = 9                    !  width of the relaxation zone
+%-    volbdy         = 0                    !  = 0, the total water flux across open boundaries is zero
+\namdisplay{nambdy} 
+%-----------------------------------------------------------------------------------------------
+
+The BDY module is an alternative implementation of open boundary
+conditions for regional configurations. It implements the Flow
+Relaxation Scheme algorithm for temperature, salinity, velocities and
+ice fields, and the Flather radiation condition for the depth-mean
+transports. The specification of the location of the open boundary is
+completely flexible and allows for example the open boundary to follow
+an isobath or other irregular contour. 
+
+The BDY module was modelled on the OBC module and shares many features
+and a similar coding structure \citep{Chanut2005}.
+
+%----------------------------------------------
+\subsection{The Flow Relaxation Scheme}
+\label{BDY_FRS_scheme}
+
+The Flow Relaxation Scheme (FRS) \citep{Davies_QJRMS76,Engerdahl_Tel95},
+applies a simple relaxation of the model fields to
+externally-specified values over a zone next to the edge of the model
+domain. Given a model prognostic variable $\Phi$ 
+\begin{equation}  \label{Eq_bdy_frs1}
+\Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
+\end{equation}
+where $\Phi_{m}$ is the model solution and $\Phi_{e}$ is the specified
+external field, $d$ gives the discrete distance from the model
+boundary  and $\alpha$ is a parameter that varies from $1$ at $d=1$ to
+a small value at $d=N$. It can be shown that this scheme is equivalent
+to adding a relaxation term to the prognostic equation for $\Phi$ of
+the form:
+\begin{equation}  \label{Eq_bdy_frs2}
+-\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
+\end{equation}
+where the relaxation time scale $\tau$ is given by a function of
+$\alpha$ and the model time step $\Delta t$:
+\begin{equation}  \label{Eq_bdy_frs3}
+\tau = \frac{1-\alpha}{\alpha}  \,\rdt
+\end{equation}
+Thus the model solution is completely prescribed by the external
+conditions at the edge of the model domain and is relaxed towards the
+external conditions over the rest of the FRS zone. The application of
+a relaxation zone helps to prevent spurious reflection of outgoing
+signals from the model boundary. 
+
+The function $\alpha$ is specified as a $tanh$ function:
+\begin{equation}  \label{Eq_bdy_frs4}
+\alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
+\end{equation}
+The width of the FRS zone is specified in the namelist as 
+\np{nb\_rimwidth}. This is typically set to a value between 8 and 10. 
+
+%----------------------------------------------
+\subsection{The Flather radiation scheme}
+\label{BDY_flather_scheme}
+
+The \citet{Flather_JPO94} scheme is a radiation condition on the normal, depth-mean
+transport across the open boundary. It takes the form
+\begin{equation}  \label{Eq_bdy_fla1}
+U = U_{e} + \frac{c}{h}\left(\eta - \eta_{e}\right),
+\end{equation}
+where $U$ is the depth-mean velocity normal to the boundary and $\eta$
+is the sea surface height, both from the model. The subscript $e$
+indicates the same fields from external sources. The speed of external
+gravity waves is given by $c = \sqrt{gh}$, and $h$ is the depth of the
+water column. The depth-mean normal velocity along the edge of the
+model domain is set equal to the
+external depth-mean normal velocity, plus a correction term that
+allows gravity waves generated internally to exit the model boundary.
+Note that the sea-surface height gradient in \eqref{Eq_bdy_fla1}
+is a spatial gradient across the model boundary, so that $\eta_{e}$ is
+defined on the $T$ points with $nbrdta=1$ and $\eta$ is defined on the
+$T$ points with $nbrdta=2$. $U$ and $U_{e}$ are defined on the $U$ or
+$V$ points with $nbrdta=1$, $i.e.$ between the two $T$ grid points.
+
+%----------------------------------------------
+\subsection{Choice of schemes}
+\label{BDY_choice_of_schemes}
+
+The Flow Relaxation Scheme may be applied separately to the
+temperature and salinity (\np{ln\_bdy\_tra\_frs} = true) and
+the velocity fields (\np{ln\_bdy\_dyn\_frs} = true). Flather
+radiation conditions may be applied using externally defined
+barotropic velocities and sea-surface height (\np{ln\_bdy\_dyn\_fla} = true) 
+or using tidal harmonics fields (\np{ln\_bdy\_tides} = true) 
+or both. FRS and Flather conditions may be applied simultaneously. 
+A typical configuration where all possible conditions might be used is a tidal, 
+shelf-seas model, where the barotropic boundary conditions are fixed 
+with the Flather scheme using tidal harmonics and possibly output 
+from a large-scale model, and FRS conditions are applied to the tracers 
+and baroclinic velocity fields, using fields from a large-scale model.
+
+Note that FRS conditions will work with the filtered
+(\key{dynspg\_flt}) or time-split (\key{dynspg\_ts}) solutions for the
+surface pressure gradient. The Flather condition will only work for
+the time-split solution (\key{dynspg\_ts}). FRS conditions are applied
+at the end of the main model time step. Flather conditions are applied
+during the barotropic subcycle in the time-split solution. 
+
+%----------------------------------------------
+\subsection{Boundary geometry}
+\label{BDY_geometry}
+
+The definition of the open boundary is completely flexible. An example
+is shown in Fig.~\ref{Fig_LBC_bdy_geom}. The boundary zone is
+defined by a series of index arrays read in from the input boundary
+data files: $nbidta$, $nbjdta$, and $nbrdta$. The first two of these
+define the global $(i,j)$ indices of each point in the boundary zone
+and the $nbrdta$ array defines the discrete distance from the boundary
+with $nbrdta=1$ meaning that the point is next to the edge of the
+model domain and $nbrdta>1$ showing that the point is increasingly
+further away from the edge of the model domain. These arrays are
+defined separately for each of the $T$, $U$ and $V$ grids, but the
+relationship between the points is assumed to be as in Fig.
+\ref{Fig_LBC_bdy_geom} with the $T$ points forming the outermost row
+of the boundary and the first row of velocities normal to the boundary
+being inside the first row of $T$ points. The order in which the
+points are defined is unimportant. 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_LBC_bdy_geom}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_bdy_geom.pdf}
+\caption {Example of geometry of unstructured open boundary}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%----------------------------------------------
+\subsection{Input boundary data files}
+\label{BDY_data}
+
+The input data files for the FRS conditions are defined in the
+namelist as \np{filbdy\_data\_T}, \np{filbdy\_data\_U},
+\np{filbdy\_data\_V}. The input data files for the Flather conditions
+are defined in the namelist as \np{filbdy\_data\_bt\_T},
+\np{filbdy\_data\_bt\_U}, \np{filbdy\_data\_bt\_V}.
+
+The netcdf header of a typical input data file is shown in Figure
+\ref{Fig_LBC_nc_header}. The file contains the index arrays which
+define the boundary geometry as noted above and the data arrays for
+each field.  The data arrays are dimensioned on: a time
+dimension; $xb$ which is the index of the boundary data point in the
+horizontal; and $yb$ which is a degenerate dimension of 1 to enable
+the file to be read by the standard NEMO I/O routines. The 3D fields
+also have a depth dimension.
+
+If \np{ln\_bdy\_clim} is set to $false$, the model expects the
+units of the time axis to have the form shown in
+\ref{Fig_bdy_input_file}, $i.e.$ {\it ``seconds since yyyy-mm-dd
+hh:mm:ss''} The fields are then linearly interpolated to the model
+time at each timestep. Note that for this option, the time axis of the
+input files must completely span the time period of the model
+integration. If \np{ln\_bdy\_clim} is set to $.true.$ (climatological
+boundary forcing), the model will expect either a single set of annual
+mean fields (constant boundary forcing) or 12 sets of monthly mean
+fields in the input files.
+
+As in the OBC module there is an option to use initial conditions as
+boundary conditions. This is chosen by setting
+$\np{nb\_dta}=0$. However, since the model defines the boundary
+geometry by reading the boundary index arrays from the input files,
+it is still necessary to provide a set of input files in this
+case. They need only contain the boundary index arrays, $nbidta$,
+$nbjdta$, $nbrdta$.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_LBC_nc_header}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_LBC_nc_header.pdf}
+\caption {Example of header of netcdf input data file for BDY}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+%----------------------------------------------
+\subsection{Volume correction}
+\label{BDY_vol_corr}
+
+There is an option to force the total volume in the regional model to be constant, similar to the option in the OBC module. This is controlled  by the \np{volbdy} parameter in the namelist. A value of $\np{volbdy} = 0$ indicates that this option is not used. If  $\np{volbdy} = 1$ then a correction is applied to the normal velocities around the boundary at each timestep to ensure that the integrated volume flow through the boundary is zero. If $\np{volbdy} = 2$ then the calculation of the volume change on the timestep includes the change due to the freshwater flux across the surface and the correction velocity corrects for this as well.
+
+
+%----------------------------------------------
+\subsection{Tidal harmonic forcing}
+\label{BDY_tides}
+
+To be written....
+
+
+
+
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LDF.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_LDF.tex	(revision 2282)
@@ -7,5 +7,8 @@
 \minitoc
 
+
+\newpage
 $\ $\newline    % force a new ligne
+
 
 The lateral physics terms in the momentum and tracer equations have been 
@@ -18,9 +21,9 @@
 and for tracers only, eddy induced advection on tracers). These three aspects 
 of the lateral diffusion are set through namelist parameters and CPP keys 
-(see the nam\_traldf and nam\_dynldf below).
+(see the \textit{nam\_traldf} and \textit{nam\_dynldf} below).
 
 %-----------------------------------nam_traldf - nam_dynldf--------------------------------------------
-\namdisplay{nam_traldf} 
-\namdisplay{nam_dynldf} 
+\namdisplay{namtra_ldf} 
+\namdisplay{namdyn_ldf} 
 %--------------------------------------------------------------------------------------------------------------
 
@@ -29,5 +32,6 @@
 % Lateral Mixing Coefficients
 % ================================================================
-\section {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn)} }
+\section [Lateral Mixing Coefficient (\textit{ldftra}, \textit{ldfdyn})] 
+		  {Lateral Mixing Coefficient (\mdl{ldftra}, \mdl{ldfdyn}) }
 \label{LDF_coef}
 
@@ -38,8 +42,9 @@
 momentum. Six CPP keys control the space variation of eddy coefficients: 
 three for momentum and three for tracer. The three choices allow: 
-a space variation in the three space directions, in the horizontal plane, 
-or in the vertical only. The default option is a constant value over the whole 
-ocean on both momentum and tracers. 
-
+a space variation in the three space directions (\key{traldf\_c3d},  \key{dynldf\_c3d}), 
+in the horizontal plane (\key{traldf\_c2d},  \key{dynldf\_c2d}), 
+or in the vertical only (\key{traldf\_c1d},  \key{dynldf\_c1d}). 
+The default option is a constant value over the whole ocean on both momentum and tracers. 
+   
 The number of additional arrays that have to be defined and the gridpoint 
 position at which they are defined depend on both the space variation chosen 
@@ -60,5 +65,5 @@
 When none of the \textbf{key\_ldfdyn\_...} and \textbf{key\_ldftra\_...} keys are 
 defined, a constant value is used over the whole ocean for momentum and 
-tracers, which is specified through the \np{ahm0} and \np{aht0} namelist 
+tracers, which is specified through the \np{rn\_ahm0} and \np{rn\_aht0} namelist 
 parameters.
 
@@ -67,9 +72,9 @@
 Indeed in all the other types of vertical coordinate, the depth is a 3D function 
 of (\textbf{i},\textbf{j},\textbf{k}) and therefore, introducing depth-dependent 
-mixing coefficients will require 3D arrays, $i.e.$ \key{ldftra\_c3d} and \key{ldftra\_c3d}.  
-In the 1D option, a hyperbolic variation of the lateral mixing coefficient is introduced 
-in which the surface value is \np{aht0} (\np{ahm0}), the bottom value is 1/4 of 
-the surface value, and the transition takes place around z=300~m with a width 
-of 300~m ($i.e.$ both the depth and the width of the inflection point are set to 300~m). 
+mixing coefficients will require 3D arrays. In the 1D option, a hyperbolic variation 
+of the lateral mixing coefficient is introduced in which the surface value is 
+\np{rn\_aht0} (\np{rn\_ahm0}), the bottom value is 1/4 of the surface value, 
+and the transition takes place around z=300~m with a width of 300~m 
+($i.e.$ both the depth and the width of the inflection point are set to 300~m). 
 This profile is hard coded in file \hf{ldftra\_c1d}, but can be easily modified by users.
 
@@ -81,23 +86,23 @@
    \begin{aligned}
          & \frac{\max(e_1,e_2)}{e_{max}} A_o^l  			& \text{for laplacian operator } \\
-         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l  & \text{for bilaplacian operator } 
+         & \frac{\max(e_1,e_2)^{3}}{e_{max}^{3}} A_o^l          & \text{for bilaplacian operator } 
    \end{aligned}    \right.
-\quad \text{comments}
 \end{equation}
 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked 
-ocean domain, and $A_o^l$ is the \np{ahm0} (momentum) or \np{aht0} (tracer) 
+ocean domain, and $A_o^l$ is the \np{rn\_ahm0} (momentum) or \np{rn\_aht0} (tracer) 
 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}. 
-%%%
-\gmcomment { not only that! stability reasons: with non uniform grid size, it is common 
-to face a blow up of the model due to to large diffusive coefficient compare to the smallest 
-grid size... this is especially true for bilaplacian (to be added in the text!)  }
+the context of the DYNAMO modelling project \citep{Willebrand_al_PO01}. 
+Note that such a grid scale dependance of mixing coefficients significantly increase 
+the range of stability of model configurations presenting large changes in grid pacing 
+such as global ocean models. Indeed, in such a case, a constant mixing coefficient 
+can lead to a blow up of the model due to large coefficient compare to the smallest 
+grid size (see \S\ref{STP_forward_imp}), especially when using a bilaplacian operator.
 
 Other formulations can be introduced by the user for a given configuration. 
 For example, in the ORCA2 global ocean model (\key{orca\_r2}), the laplacian 
-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 
+viscosity operator uses \np{rn\_ahm0}~= 4.10$^4$ m$^2$/s poleward of 20$^{\circ}$ 
+north and south and decreases linearly to \np{rn\_aht0}~= 2.10$^3$ m$^2$/s 
+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 
@@ -121,4 +126,6 @@
 and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
 
+$\ $\newline    % force a new ligne
+
 A space variation in the eddy coefficient appeals several remarks:
 
@@ -135,5 +142,5 @@
 (3) for isopycnal diffusion on momentum or tracers, an additional purely 
 horizontal background diffusion with uniform coefficient can be added by 
-setting a non zero value of \np{ahmb0} or \np{ahtb0}, a background horizontal 
+setting a non zero value of \np{rn\_ahmb0} or \np{rn\_ahtb0}, a background horizontal 
 eddy viscosity or diffusivity coefficient (namelist parameters whose default 
 values are $0$). However, the technique used to compute the isopycnal 
@@ -150,6 +157,8 @@
 (6) it is possible to use both the laplacian and biharmonic operators concurrently.
 
-(7) for testing purposes it is possible to run without lateral diffusion on momentum.
-
+(7) it is possible to run without explicit lateral diffusion on momentum (\np{ln\_dynldf\_lap} = 
+\np{ln\_dynldf\_bilap} = false). This is recommended when using the UBS advection 
+scheme on momentum (\np{ln\_dynadv\_ubs} = true, see \ref{DYN_adv_ubs}) 
+and can be useful for testing purposes.
 
 % ================================================================
@@ -162,5 +171,5 @@
 %%%
 \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 
@@ -172,6 +181,6 @@
 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
@@ -190,15 +199,15 @@
 \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}] 
  \\
@@ -268,5 +277,5 @@
 there is no specific treatment for iso-neutral mixing in the $s$-coordinate. 
 In other words, iso-neutral mixing will only be accurately represented with a 
-linear equation of state (\np{neos}=1 or 2). In the case of a "true" equation 
+linear equation of state (\np{nn\_eos}=1 or 2). In the case of a "true" equation 
 of state, the evaluation of $i$ and $j$ derivatives in \eqref{Eq_ldfslp_iso} 
 will include a pressure dependent part, leading to the wrong evaluation of 
@@ -276,5 +285,5 @@
 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 
@@ -328,6 +337,6 @@
 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 
@@ -341,5 +350,5 @@
 
 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. 
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -359,10 +368,10 @@
 
 
-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 
@@ -402,6 +411,6 @@
 \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}\\
@@ -437,5 +446,5 @@
 \end{equation}
 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set 
-through \np{aeiv}, a \textit{nam\_traldf} namelist parameter. 
+through \np{rn\_aeiv}, a \textit{nam\_traldf} namelist parameter. 
 The three components of the eddy induced velocity are computed and add 
 to the eulerian velocity in \mdl{traadv\_eiv}. This has been preferred to a 
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_MISC.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_MISC.tex	(revision 2282)
@@ -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.
 
 % ================================================================
@@ -145,27 +174,27 @@
 %--------------------------------------------------------------------------------------------------------------
 
-% 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:
+$\rdt=rn\_rdt$ is the time step of dynamics while $\widetilde{\rdt}=rdttra$ 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}
 \frac{\partial \textbf{U}_h }{\partial t} 
-	&\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\Delta t} = \ldots \\ 
-\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\Delta t}} = \ldots \\ 
-\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\Delta t}} = \ldots \\ 
+	&\equiv \frac{\textbf{U}_h ^{t+1}-\textbf{U}_h^{t-1} }{2\rdt} = \ldots \\ 
+\frac{\partial T}{\partial t} &\equiv \frac{T^{t+1}-T^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
+\frac{\partial S}{\partial t} &\equiv \frac{S^{t+1} -S^{t-1}}{2 \widetilde{\rdt}} = \ldots \\ 
 \end{split}
 \end{equation}
@@ -181,9 +210,8 @@
 and it cannot be used in high-resolution models where baroclinically 
 unstable processes are important. Moreover, the vertical variation of 
-$\widetilde{ \Delta t}$ implies that the heat and salt contents are no longer 
+$\widetilde{ \rdt}$ 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 \rdt }^2 \; \frac{H_v (i,j) \; e_{1v} (i,j)}{e_{2v}(i,j)}              \\
+&c_{i,j}^{EW} &&= {2 \rdt }^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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_Model_Basics.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_Model_Basics.tex	(revision 2282)
@@ -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 $T_c > \rdt$. \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 
@@ -272,9 +275,9 @@
 Firstly, the damping of EGWs can be quantified through the magnitude of the additional term. 
 Secondly, the numerical scheme does not need any tuning. Numerical stability is ensured as 
-soon as $T_c > \Delta t$.
+soon as $T_c > \rdt$.
 
 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, 
@@ -309,108 +312,10 @@
 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}.
+
+\newpage
+$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -435,5 +340,5 @@
 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 +349,5 @@
 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 +587,6 @@
 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 +595,34 @@
 the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
 
-To sum up, the equations solved by the ocean model can be written in the following tensorial formalism:
+
+$\ $\newline    % force a new ligne
+
+To sum up, the curvilinear $z$-coordinate 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 +651,14 @@
 \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 +669,4 @@
 \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}.
 
@@ -809,136 +694,39 @@
 in Chapter~\ref{SBC}.
 
+
 \newpage 
+$\ $\newline    % force a new ligne
 % ================================================================
-% Curvilinear z*-coordinate System
+% Curvilinear generalised vertical coordinate System
 % ================================================================
-\section{Curvilinear \textit{z*}-coordinate System}
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!b] \label{Fig_z_zstar}  \begin{center}
-\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf}
-\caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear 
-free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate 
-\citep{Adcroft_Campin_OM04} ).}
-\end{center}   \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
-
-In that case, the free surface equation is nonlinear, and the variations of volume are fully 
-taken into account. These coordinates systems is presented in a report \citep{Levier2007} 
-available on the \NEMO web site. 
-
-\gmcomment{
-The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation 
-which allows one to deal with large amplitude free-surface
-variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
-the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
-undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
-but is equally distributed over the full water column. Thus vertical
-levels naturally follow sea-surface variations, with a linear attenuation with
-depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
-fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
-The definition and modified oceanic equations for the rescaled vertical coordinate
- \textit{z*}, including the treatment of fresh-water flux at the surface, are
-detailed in Adcroft and Campin (2004). The major points are summarized
-here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
-
-$H +  \textit{z*} = (H + z) / r$ and  $\delta \textit{z*} = \delta z / r$ with $r = \frac{H+\eta} {H}$
-
-Since the vertical displacement of the free surface is incorporated in the vertical
-coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position,  
-$\textit{z*} = 0$ and  $\textit{z*} = ?H$ respectively. Also the divergence of the flow field 
-is no longer zero as shown by the continuity equation:
-
-$\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
-		\left( r \; w\textit{*} \right) = 0 $
-
-}
-
-
-\newpage 
-% ================================================================
-% Curvilinear s-coordinate System
-% ================================================================
-\section{Curvilinear \textit{s}-coordinate System}
-\label{PE_sco}
-
-% -------------------------------------------------------------------------------------------------------------
-% Introduction
-% -------------------------------------------------------------------------------------------------------------
-\subsection{Introduction}
-
-Several important aspects of the ocean circulation are influenced by bottom topography. 
-Of course, the most important is that bottom topography determines deep ocean sub-basins, 
-barriers, sills and channels that strongly constrain the path of water masses, but more subtle 
-effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary 
-one along continental slopes. Topographic Rossby waves can be excited and can interact 
-with the mean current. In the $z-$coordinate system presented in the previous section 
-(\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is 
-discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom 
-and to large localized depth gradients associated with large localized vertical velocities. 
-The response to such a velocity field often leads to numerical dispersion effects. 
-One solution to strongly reduce this error is to use a partial step representation of bottom 
-topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. 
-Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 
-
-The $s$-coordinate avoids the discretisation error in the depth field since the layers of 
-computation are gradually adjusted with depth to the ocean bottom. Relatively small 
-topographic features as well as  gentle, large-scale slopes of the sea floor in the deep 
-ocean, which would be ignored in typical $z$-model applications with the largest grid 
-spacing at greatest depths, can easily be represented (with relatively low vertical resolution). 
-A terrain-following model (hereafter $s-$model) also facilitates the modelling of the 
-boundary layer flows over a large depth range, which in the framework of the $z$-model 
-would require high vertical resolution over the whole depth range. Moreover, with a 
-$s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface 
-as the only boundaries of the domain (nomore lateral boundary condition to specify). 
-Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a 
-homogeneous ocean, it has strong limitations as soon as stratification is introduced. 
-The main two problems come from the truncation error in the horizontal pressure 
-gradient and a possibly increased diapycnal diffusion. The horizontal pressure force 
-in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
-
-\begin{equation} \label{Eq_PE_p_sco}
-\left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 
-p}{\partial s}\left. {\nabla z} \right|_s 
-\end{equation}
-
-The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface 
-and introduces a truncation error that is not present in a $z$-model. In the special case 
-of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 
-\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude 
-of this truncation error. It depends on topographic slope, stratification, horizontal and 
-vertical resolution, the equation of state, and the finite difference scheme. This error 
-limits the possible topographic slopes that a model can handle at a given horizontal 
-and vertical resolution. This is a severe restriction for large-scale applications using 
-realistic bottom topography. The large-scale slopes require high horizontal resolution, 
-and the computational cost becomes prohibitive. This problem can be at least partially 
-overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,Madec1996}. However, the definition of the model 
-domain vertical coordinate becomes then a non-trivial thing for a realistic bottom 
-topography: a envelope topography is defined in $s$-coordinate on which a full or 
-partial step bottom topography is then applied in order to adjust the model depth to 
-the observed one (see \S\ref{DOM_zgr}.
-
-For numerical reasons a minimum of diffusion is required along the coordinate surfaces 
-of any finite difference model. It causes spurious diapycnal mixing when coordinate 
-surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as 
-well as for a $s$-model. However, density varies more strongly on $s-$surfaces than 
-on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal 
-diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a 
-$z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal 
-circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. 
-For example, imagine an isolated bump of topography in an ocean at rest with a horizontally 
-uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral 
-surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, 
-the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column 
-($i.e.$ the main thermocline) \citep{Madec1996}. An alternate solution consists of rotating 
-the lateral diffusive tensor to geopotential or to isoneutral surfaces (see \S\ref{PE_ldf}. 
-Unfortunately, the slope of isoneutral surfaces relative to the $s$-surfaces can very large, 
-strongly exceeding the stability limit of such a operator when it is discretized (see Chapter~\ref{LDF}). 
-
-The $s-$coordinates introduced here \citep{Lott1990,Madec1996} differ mainly in two 
-aspects from similar models:  it allows  a representation of bottom topography with mixed 
-full or partial step-like/terrain following topography ; It also offers a completely general 
-transformation, $s=s(i,j,z)$ for the vertical coordinate.
+\section{Curvilinear generalised vertical coordinate System}
+\label{PE_gco}
+
+%\gmcomment{
+The ocean domain presents a huge diversity of situation in the vertical. First the ocean surface is a time dependent surface (moving surface). Second the ocean floor depends on the geographical position, varying from more than 6,000 meters in abyssal trenches to zero at the coast. Last but not least, the ocean stratification exerts a strong barrier to vertical motions and mixing. 
+Therefore, in order to represent the ocean with respect to the first point a space and time dependent vertical coordinate that follows the variation of the sea surface height $e.g.$ an $z$*-coordinate; for the second point, a space variation to fit the change of bottom topography $e.g.$ a terrain-following or $\sigma$-coordinate; and for the third point, one will be tempted to use a space and time dependent coordinate that follows the isopycnal surfaces, $e.g.$ an isopycnic coordinate.
+
+In order to satisfy two or more constrains one can even be tempted to mixed these coordinate systems, as in HYCOM (mixture of $z$-coordinate at the surface, isopycnic coordinate in the ocean interior and $\sigma$ at the ocean bottom) \citep{Chassignet_al_JPO03}  or OPA (mixture of $z$-coordinate in vicinity the surface and steep topography areas and $\sigma$-coordinate elsewhere) \citep{Madec_al_JPO96} among others.
+
+In fact one is totally free to choose any space and time vertical coordinate by introducing an arbitrary vertical coordinate :
+\begin{equation} \label{Eq_s}
+s=s(i,j,k,t)
+\end{equation}
+with the restriction that the above equation gives a single-valued monotonic relationship between $s$ and $k$, when $i$, $j$ and $t$ are held fixed. \eqref{Eq_s} is a transformation from the $(i,j,k,t)$ coordinate system with independent variables into the $(i,j,s,t)$ generalised coordinate system with $s$ depending on the other three variables through \eqref{Eq_s}.
+This so-called \textit{generalised vertical coordinate} \citep{Kasahara_MWR74} is in fact an Arbitrary Lagrangian--Eulerian (ALE) coordinate. Indeed, choosing an expression for $s$ is an arbitrary choice that determines which part of the vertical velocity (defined from a fixed referential) will cross the levels (Eulerian part) and which part will be used to move them (Lagrangian part).
+The coordinate is also sometime referenced as an adaptive coordinate \citep{Hofmeister_al_OM09}, since the coordinate system is adapted in the course of the simulation. Its most often used implementation is via an ALE algorithm, in which a pure lagrangian step is followed by regridding and remapping steps, the later step implicitly embedding the vertical advection \citep{Hirt_al_JCP74, Chassignet_al_JPO03, White_al_JCP09}. Here we follow the \citep{Kasahara_MWR74} strategy : a regridding step (an update of the vertical coordinate) followed by an eulerian step with an explicit computation of vertical advection relative to the moving s-surfaces.
+
+A key point here is that the $s$-coordinate depends on $(i,j)$ ==> horizontal pressure gradient...
+
+the generalized vertical coordinates used in ocean modelling are not orthogonal, which contrasts with many other applications in mathematical physics. Hence, it is useful to keep in mind the following properties that may seem odd on initial encounter.
+
+the horizontal velocity in ocean models measures motions in the horizontal plane, perpendicular to the local gravitational field. That is, horizontal velocity is mathematically the same regardless the vertical coordinate, be it geopotential, isopycnal, pressure, or terrain following. The key motivation for maintaining the same horizontal velocity component is that the hydrostatic and geostrophic balances are dominant in the large-scale ocean. Use of an alternative quasi-horizontal velocity, for example one oriented parallel to the generalized surface, would lead to unacceptable numerical errors. Correspondingly, the vertical direction is anti-parallel to the gravitational force in all of the coordinate systems. We do not choose the alternative of a quasi-vertical direction oriented normal to the surface of a constant generalized vertical coordinate. 
+
+It is the method used to measure transport across the generalized vertical coordinate surfaces which differs between the vertical coordinate choices. That is, computation of the dia-surface velocity component represents the fundamental distinction between the various coordinates. In some models, such as geopotential, pressure, 
+and terrain following, this transport is typically diagnosed from volume or mass conservation. In other models, such as isopycnal layered models, this transport is prescribed based on assumptions about the physical processes producing a flux across the layer interfaces. 
+
+
+In this section we first establish the PE in the generalised vertical $s$-coordinate, then we discuss the particular cases available in \NEMO, namely $z$, $z$*, $s$, and $\tilde z$.  
+%}
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1023,4 +811,195 @@
 Add a few works on z and zps and s and underlies the differences between all of them
 \colorbox{yellow}{ $< = =$ end update}  }
+
+
+
+% -------------------------------------------------------------------------------------------------------------
+% Curvilinear z*-coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear \textit{z*}--coordinate System}
+\label{PE_zco_star}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!b] \label{Fig_z_zstar}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_z_zstar.pdf}
+\caption{(a) $z$-coordinate in linear free-surface case ; (b) $z-$coordinate in non-linear 
+free surface case (c) re-scaled height coordinate (become popular as the \textit{z*-}coordinate 
+\citep{Adcroft_Campin_OM04} ).}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+
+In that case, the free surface equation is nonlinear, and the variations of volume are fully 
+taken into account. These coordinates systems is presented in a report \citep{Levier2007} 
+available on the \NEMO web site. 
+
+%\gmcomment{
+The \textit{z*} coordinate approach is an unapproximated, non-linear free surface implementation 
+which allows one to deal with large amplitude free-surface
+variations relative to the vertical resolution \citep{Adcroft_Campin_OM04}. In
+the  \textit{z*} formulation, the variation of the column thickness due to sea-surface
+undulations is not concentrated in the surface level, as in the $z$-coordinate formulation,
+but is equally distributed over the full water column. Thus vertical
+levels naturally follow sea-surface variations, with a linear attenuation with
+depth, as illustrated by figure fig.1c . Note that with a flat bottom, such as in
+fig.1c, the bottom-following  $z$ coordinate and  \textit{z*} are equivalent.
+The definition and modified oceanic equations for the rescaled vertical coordinate
+ \textit{z*}, including the treatment of fresh-water flux at the surface, are
+detailed in Adcroft and Campin (2004). The major points are summarized
+here. The position ( \textit{z*}) and vertical discretization (\textit{z*}) are expressed as:
+\begin{equation} \label{Eq_z-star}
+H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
+\end{equation} 
+Since the vertical displacement of the free surface is incorporated in the vertical
+coordinate  \textit{z*}, the upper and lower boundaries are at fixed  \textit{z*} position,  
+$\textit{z*} = 0$ and  $\textit{z*} = -H$ respectively. Also the divergence of the flow field 
+is no longer zero as shown by the continuity equation:
+\begin{equation*} 
+\frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
+		\left( r \; w\textit{*} \right) = 0 
+\end{equation*} 
+%}
+
+
+% from MOM4p1 documentation
+
+To overcome problems with vanishing surface and/or bottom cells, we consider the 
+zstar coordinate 
+\begin{equation} \label{PE_}
+	z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\end{equation}
+
+This coordinate is closely related to the "eta" coordinate used in many atmospheric 
+models (see Black (1994) for a review of eta coordinate atmospheric models). It 
+was originally used in ocean models by Stacey et al. (1995) for studies of tides 
+next to shelves, and it has been recently promoted by Adcroft and Campin (2004) 
+for global climate modelling.
+
+The surfaces of constant $z^\star$ are quasi-horizontal. Indeed, the $z^\star$ coordinate reduces to $z$ when $\eta$ is zero. In general, when noting the large differences between 
+undulations of the bottom topography versus undulations in the surface height, it 
+is clear that surfaces constant $z^\star$ are very similar to the depth surfaces. These properties greatly reduce difficulties of computing the horizontal pressure gradient relative to terrain following sigma models discussed in \S\ref{PE_sco}. 
+Additionally, since $z^\star$ when $\eta = 0$, no flow is spontaneously generated in an 
+unforced ocean starting from rest, regardless the bottom topography. This behaviour is in contrast to the case with "s"-models, where pressure gradient errors in 
+the presence of nontrivial topographic variations can generate nontrivial spontaneous flow from a resting state, depending on the sophistication of the pressure 
+gradient solver. The quasi-horizontal nature of the coordinate surfaces also facilitates the implementation of neutral physics parameterizations in $z^\star$ models using 
+the same techniques as in $z$-models (see Chapters 13-16 of \cite{Griffies_Bk04}) for a 
+discussion of neutral physics in $z$-models, as well as Section \S\ref{LDF_slp} 
+in this document for treatment in \NEMO). 
+
+The range over which $z^\star$ varies is time independent $-H \leq z^\star \leq 0$. Hence, all 
+cells remain nonvanishing, so long as the surface height maintains $\eta > ?H$. This 
+is a minor constraint relative to that encountered on the surface height when using 
+$s = z$ or $s = z - \eta$. 
+
+Because $z^\star$ has a time independent range, all grid cells have static increments 
+ds, and the sum of the ver tical increments yields the time independent ocean 
+depth %·k ds = H. 
+The $z^\star$ coordinate is therefore invisible to undulations of the 
+free surface, since it moves along with the free surface. This proper ty means that 
+no spurious ver tical transpor t is induced across surfaces of constant $z^\star$ by the 
+motion of external gravity waves. Such spurious transpor t can be a problem in 
+z-models, especially those with tidal forcing. Quite generally, the time independent 
+range for the $z^\star$ coordinate is a very convenient proper ty that allows for a nearly 
+arbitrary ver tical resolution even in the presence of large amplitude fluctuations of 
+the surface height, again so long as $\eta > -H$. 
+
+%end MOM doc %%%
+
+
+
+\newpage 
+% -------------------------------------------------------------------------------------------------------------
+% Terrain following  coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear Terrain-following \textit{s}--coordinate}
+\label{PE_sco}
+
+% -------------------------------------------------------------------------------------------------------------
+% Introduction
+% -------------------------------------------------------------------------------------------------------------
+\subsubsection{Introduction}
+
+Several important aspects of the ocean circulation are influenced by bottom topography. 
+Of course, the most important is that bottom topography determines deep ocean sub-basins, 
+barriers, sills and channels that strongly constrain the path of water masses, but more subtle 
+effects exist. For example, the topographic $\beta$-effect is usually larger than the planetary 
+one along continental slopes. Topographic Rossby waves can be excited and can interact 
+with the mean current. In the $z-$coordinate system presented in the previous section 
+(\S\ref{PE_zco}), $z-$surfaces are geopotential surfaces. The bottom topography is 
+discretised by steps. This often leads to a misrepresentation of a gradually sloping bottom 
+and to large localized depth gradients associated with large localized vertical velocities. 
+The response to such a velocity field often leads to numerical dispersion effects. 
+One solution to strongly reduce this error is to use a partial step representation of bottom 
+topography instead of a full step one \cite{Pacanowski_Gnanadesikan_MWR98}. 
+Another solution is to introduce a terrain-following coordinate system (hereafter $s-$coordinate) 
+
+The $s$-coordinate avoids the discretisation error in the depth field since the layers of 
+computation are gradually adjusted with depth to the ocean bottom. Relatively small 
+topographic features as well as  gentle, large-scale slopes of the sea floor in the deep 
+ocean, which would be ignored in typical $z$-model applications with the largest grid 
+spacing at greatest depths, can easily be represented (with relatively low vertical resolution). 
+A terrain-following model (hereafter $s-$model) also facilitates the modelling of the 
+boundary layer flows over a large depth range, which in the framework of the $z$-model 
+would require high vertical resolution over the whole depth range. Moreover, with a 
+$s$-coordinate it is possible, at least in principle, to have the bottom and the sea surface 
+as the only boundaries of the domain (nomore lateral boundary condition to specify). 
+Nevertheless, a $s$-coordinate also has its drawbacks. Perfectly adapted to a 
+homogeneous ocean, it has strong limitations as soon as stratification is introduced. 
+The main two problems come from the truncation error in the horizontal pressure 
+gradient and a possibly increased diapycnal diffusion. The horizontal pressure force 
+in $s$-coordinate consists of two terms (see Appendix~\ref{Apdx_A}),
+
+\begin{equation} \label{Eq_PE_p_sco}
+\left. {\nabla p} \right|_z =\left. {\nabla p} \right|_s -\frac{\partial 
+p}{\partial s}\left. {\nabla z} \right|_s 
+\end{equation}
+
+The second term in \eqref{Eq_PE_p_sco} depends on the tilt of the coordinate surface 
+and introduces a truncation error that is not present in a $z$-model. In the special case 
+of a $\sigma-$coordinate (i.e. a depth-normalised coordinate system $\sigma = z/H$), 
+\citet{Haney1991} and \citet{Beckmann1993} have given estimates of the magnitude 
+of this truncation error. It depends on topographic slope, stratification, horizontal and 
+vertical resolution, the equation of state, and the finite difference scheme. This error 
+limits the possible topographic slopes that a model can handle at a given horizontal 
+and vertical resolution. This is a severe restriction for large-scale applications using 
+realistic bottom topography. The large-scale slopes require high horizontal resolution, 
+and the computational cost becomes prohibitive. This problem can be at least partially 
+overcome by mixing $s$-coordinate and step-like representation of bottom topography \citep{Gerdes1993a,Gerdes1993b,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 
+partial step bottom topography is then applied in order to adjust the model depth to 
+the observed one (see \S\ref{DOM_zgr}.
+
+For numerical reasons a minimum of diffusion is required along the coordinate surfaces 
+of any finite difference model. It causes spurious diapycnal mixing when coordinate 
+surfaces do not coincide with isoneutral surfaces. This is the case for a $z$-model as 
+well as for a $s$-model. However, density varies more strongly on $s-$surfaces than 
+on horizontal surfaces in regions of large topographic slopes, implying larger diapycnal 
+diffusion in a $s$-model than in a $z$-model. Whereas such a diapycnal diffusion in a 
+$z$-model tends to weaken horizontal density (pressure) gradients and thus the horizontal 
+circulation, it usually reinforces these gradients in a $s$-model, creating spurious circulation. 
+For example, imagine an isolated bump of topography in an ocean at rest with a horizontally 
+uniform stratification. Spurious diffusion along $s$-surfaces will induce a bump of isoneutral 
+surfaces over the topography, and thus will generate there a baroclinic eddy. In contrast, 
+the ocean will stay at rest in a $z$-model. As for the truncation error, the problem can be reduced by introducing the terrain-following coordinate below the strongly stratified portion of the water column 
+($i.e.$ the main thermocline) \citep{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{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 
+transformation, $s=s(i,j,z)$ for the vertical coordinate.
+
+
+\newpage 
+% -------------------------------------------------------------------------------------------------------------
+% Curvilinear z-tilde coordinate System
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Curvilinear $\tilde{z}$--coordinate}
+\label{PE_zco_tilde}
+
+
 
 \newpage 
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_SBC.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_SBC.tex	(revision 2282)
@@ -1,3 +1,3 @@
-% ================================================================
+ ================================================================
 % Chapter Ñ Surface Boundary Condition (SBC) 
 % ================================================================
@@ -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 +86,28 @@
 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 }$.
-
-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.
+For example, 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 $\textit{4 psu}). In this case, 
+$\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 +120,5 @@
 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 +186,5 @@
 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 +260,5 @@
 %-------------------------------------------------------------------------------------------------------------
 
-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 
@@ -361,4 +354,177 @@
 
 % ================================================================
+%        River runoffs
+% ================================================================
+\section   [river runoffs (\textit{sbcrnf})]
+			{river runoffs (\mdl{sbcrnf})}
+\label{SBC_rnf}
+%------------------------------------------namsbc_rnf----------------------------------------------------
+\namdisplay{namsbc_rnf} 
+%-------------------------------------------------------------------------------------------------------------
+
+%River runoff generally enters the ocean at a nonzero depth rather than through the surface. 
+%Many models, however, have traditionally inserted river runoff to the top model cell.
+%This was the case in \NEMO prior to the version 3.3. The switch toward a input of runoff 
+%throughout a nonzero depth has been motivated by the numerical and physical problems 
+%that arise when the top grid cells are of the order of one meter. This situation is common in 
+%coastal modelling and becomes more and more often open ocean and climate modelling 
+%\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
+%required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.
+
+
+%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the 
+%\mdl{tra\_sbc} module.  We decided to separate them throughout the code, so that the variable 
+%\textit{emp} represented solely evaporation minus precipitation fluxes, and a new 2d variable 
+%rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with 
+%emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use 
+%emp or emps and the changes made are below:
+
+
+%Rachel:
+River runoff generally enters the ocean at a nonzero depth rather than through the surface.
+Many models, however, have traditionally inserted river runoff to the top model cell.
+This was the case in \NEMO prior to the version 3.3, and was combined with an option to increase vertical mixing near the river mouth.
+
+However, with this method numerical and physical problems arise when the top grid cells are 
+of the order of one meter. This situation is common in coastal modelling and is becoming 
+more common in open ocean and climate modelling 
+\footnote{At least a top cells thickness of 1~meter and a 3 hours forcing frequency are
+required to properly represent the diurnal cycle \citep{Bernie_al_JC05}. see also \S\ref{SBC_dcy}.}.
+
+As such from VN3.3 onwards it is possible to add river runoff through a non-zero depth, and for the 
+temperature and salinity of the river to effect the surrounding ocean.
+The user is able to specify, in a NetCDF input file, the temperature and salinity of the river, along with the   
+depth (in metres) which the river should be added to.
+
+Namelist options, \np{ln\_rnf\_depth}, \np{ln\_rnf\_sal} and \np{ln\_rnf\_temp} control whether 
+the river attributes (depth, salinity and temperature) are read in and used.  If these are set 
+as false the river is added to the surface box only, assumed to be fresh (0~psu), and/or 
+taken as surface temperature respectively.
+
+The runoff value and attributes are read in in sbcrnf.  
+For temperature -999 is taken as missing data and the river temperature is taken to be the 
+surface temperatue at the river point.
+For the depth parameter a value of -1 means the river is added to the surface box only, 
+and a value of -999 means the river is added through the entire water column. 
+After being read in the temperature and salinity variables are multiplied by the amount of runoff (converted into m/s) 
+to give the heat and salt content of the river runoff.
+After the user specified depth is read ini, the number of grid boxes this corresponds to is 
+calculated and stored in the variable \np{nz\_rnf}.
+The variable \np{h\_dep} is then calculated to be the depth (in metres) of the bottom of the 
+lowest box the river water is being added to (i.e. the total depth that river water is being added to in the model).
+
+The mass/volume addition due to the river runoff is, at each relevant depth level, added to the horizontal divergence (\np{hdivn}) 
+in the subroutine \np{sbc\_rnf\_div} (called from \np{divcur}).
+This increases the diffusion term in the vicinity of the river, thereby simulating a momentum flux.
+The sea surface height is calculated using the sum of the horizontal divergence terms, and so the 
+river runoff indirectly forces an increase in sea surface height. 
+
+The \np{hdivn} terms are used in the tracer advection modules to force vertical velocities.
+This causes a mass of water, equal to the amount of runoff, to be moved into the box above. 
+The heat and salt content of the river runoff is not included in this step, and so the tracer 
+concentrations are diluted as water of ocean temperature and salinity is moved upward out of the box 
+and replaced by the same volume of river water with no corresponding heat and salt addition.
+
+For the linear free surface case, at the surface box the tracer advection causes a flux of water 
+(of equal volume to the runoff) through the sea surface out of the domain, which causes a salt and heat flux out of the model.
+As such the volume of water does not change, but the water is diluted.
+
+For the non-linear free surface case (vvl), no flux is allowed through the surface.
+Instead in the surface box (as well as water moving up from the boxes below) a volume of runoff water 
+is added with no corresponding heat and salt addition and so as happens in the lower boxes there is a dilution effect.
+(The runoff addition to the top box along with the water being moved up through boxes below means the surface box has a large 
+increase in volume, whilst all other boxes remain the same size)
+
+In trasbc the addition of heat and salt due to the river runoff is added.
+This is done in the same way for both vvl and non-vvl.
+The temperature and salinity are increased through the specified depth according to the heat and salt content of the river. 
+
+In the non-linear free surface case (vvl), near the end of the time step the change in sea surface height is redistrubuted 
+through the grid boxes, so that the original ratios of grid box heights are restored.
+In doing this water is moved into boxes below, throughout the water column, so the large volume addition to the surface box is spread between all the grid boxes.
+
+It is also possible for runnoff to be specified as a negative value for modelling flow through straits, i.e. modelling the Baltic flow in and out of the North Sea.
+When the flow is out of the domain there is no change in temperature and salinity, regardless of the namelist options used, as the ocean water leaving the domain removes heat and salt (at the same concentration) with it. 
+
+
+%\colorbox{yellow}{Nevertheless, Pb of vertical resolution and 3D input : increase vertical mixing near river mouths to mimic a 3D river 
+
+%All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.}
+
+%\colorbox{yellow}{river mouths{\ldots}}
+
+%IF( ln_rnf ) THEN                                     ! increase diffusivity at rivers mouths
+%        DO jk = 2, nkrnf   ;   avt(:,:,jk) = avt(:,:,jk) + rn_avt_rnf * rnfmsk(:,:)   ;   END DO
+%ENDIF
+
+%\gmcomment{  word doc of runoffs:
+%
+%In the current \NEMO setup river runoff is added to emp fluxes, these are then applied at just the sea surface as a volume change (in the variable volume case this is a literal volume change, and in the linear free surface case the free surface is moved) and a salt flux due to the concentration/dilution effect.  There is also an option to increase vertical mixing near river mouths; this gives the effect of having a 3d river.  All river runoff and emp fluxes are assumed to be fresh water (zero salinity) and at the same temperature as the sea surface.
+%Our aim was to code the option to specify the temperature and salinity of river runoff, (as well as the amount), along with the depth that the river water will affect.  This would make it possible to model low salinity outflow, such as the Baltic, and would allow the ocean temperature to be affected by river runoff.  
+
+%The depth option makes it possible to have the river water affecting just the surface layer, throughout depth, or some specified point in between.
+
+%To do this we need to treat evaporation/precipitation fluxes and river runoff differently in the tra_sbc module.  We decided to separate them throughout the code, so that the variable emp represented solely evaporation minus precipitation fluxes, and a new 2d variable rnf was added which represents the volume flux of river runoff (in kg/m2s to remain consistent with emp).  This meant many uses of emp and emps needed to be changed, a list of all modules which use emp or emps and the changes made are below:
+
+}
+
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_SBC_diurnal}  \begin{center}
+\includegraphics[width=0.8\textwidth]{./TexFiles/Figures/Fig_SBC_diurnal.pdf}
+\caption{Example of recontruction of the diurnal cycle variation of short wave flux  
+from daily mean values. The reconstructed diurnal cycle (black line) is chosen 
+as the mean value of the analytical cycle (blue line) over a time step, not 
+as the mid time step value of the analytically cycle (red square). From \citet{Bernie_al_CD07}.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% ================================================================
+%        Diurnal cycle
+% ================================================================
+\section   [Diurnal  cycle (\textit{sbcdcy})]
+			{Diurnal cycle (\mdl{sbcdcy})}
+\label{SBC_dcy}
+%------------------------------------------namsbc_rnf----------------------------------------------------
+%\namdisplay{namsbc} 
+%-------------------------------------------------------------------------------------------------------------
+
+\cite{Bernie_al_JC05} have shown that to capture 90$\%$ of the diurnal variability of 
+SST requires a vertical resolution in upper ocean of 1~m or better and a temporal resolution 
+of the surface fluxes of 3~h or less. Unfortunately high frequency forcing fields are rare, 
+not to say inexistent. Nevertheless, it is possible to obtain a reasonable diurnal cycle 
+of the SST knowning only short wave flux (SWF) at high frequency \citep{Bernie_al_CD07}.
+Furthermore, only the knowledge of daily mean value of SWF is needed, 
+as higher frequency variations can be reconstructed from them, assuming that 
+the diurnal cycle of SWF is a scaling of the top of the atmosphere diurnal cycle 
+of incident SWF. The \cite{Bernie_al_CD07} reconstruction algorithm is available
+in \NEMO by setting \np{ln\_dm2dc}=true (a \textit{namsbc} namelist parameter) when using 
+CORE bulk formulea (\np{ln\_blk\_core}=true) or the flux formulation (\np{ln\_flx}=true). 
+The reconstruction is performed in the \mdl{sbcdcy} module. The detail of the algoritm used 
+can be found in the appendix~A of \cite{Bernie_al_CD07}. The algorithm preserve the daily 
+mean incomming SWF as the reconstructed SWF at a given time step is the mean value 
+of the analytical cycle over this time step (Fig.\ref{Fig_SBC_diurnal}). 
+The use of diurnal cycle reconstruction requires the input SWF to be daily 
+($i.e.$ a frequency of 24 and a time interpolation set to true in \np{sn\_qsr} namelist parameter).
+Furthermore, it is recommended to have a least 8 surface module time step per day,
+that is  $\rdt \ \np{nn\_fsbc} < 10,800~s = 3~h$. An example of recontructed SWF 
+is given in Fig.\ref{Fig_SBC_dcy} for a 12 reconstructed diurnal cycle, one every 2~hours 
+(from 1am to 11pm).
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_SBC_dcy}  \begin{center}
+\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_SBC_dcy.pdf}
+\caption{Example of recontruction of the diurnal cycle variation of short wave flux  
+from daily mean values on an ORCA2 grid with a time sampling of 2~hours (from 1am to 11pm). 
+The display is on (i,j) plane. }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+Note also that the setting a diurnal cycle in SWF is highly recommended  when 
+the top layer thickness approach 1~m or less, otherwise large error in SST can 
+appear due to an inconsistency between the scale of the vertical resolution 
+and the forcing acting on that scale.
+
+% ================================================================
 % Interpolation on the Fly
 % ================================================================
@@ -408,13 +574,11 @@
 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 +679,9 @@
 
 \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}), 
@@ -562,32 +726,4 @@
 
 % -------------------------------------------------------------------------------------------------------------
-%        Addition of river runoffs
-% -------------------------------------------------------------------------------------------------------------
-\subsection   [Addition of river runoffs (\textit{sbcrnf})]
-			{Addition of river runoffs (\mdl{sbcrnf})}
-\label{SBC_rnf}
-%------------------------------------------namsbc_rnf----------------------------------------------------
-\namdisplay{namsbc_rnf} 
-%-------------------------------------------------------------------------------------------------------------
-
-The river runoffs 
-
-It is convenient to introduce the river runoff in the model as a surface 
-fresh water flux. 
-
-
-%Griffies:  River runoff generally enters the ocean at a nonzero depth rather than through the surface. Many global models, however, have traditionally inserted river runoff to the top model cell. Such can become problematic numerically and physically when the top grid cells are reÞned to levels common in coastal modelling. Hence, more applications are now considering the input of runoff throughout a nonzero depth. Likewise, sea ice can melt at depth, thus necessitating a mass transport to occur within the ocean between the liquid and solid water masses.
-
-\colorbox{yellow}{Nevertheless, Pb of vertical resolution and increase of Kz in vicinity of }
-
-\colorbox{yellow}{river mouths{\ldots}}
-
-%IF( ln_rnf ) THEN                                     ! increase diffusivity at rivers mouths
-%        DO jk = 2, nkrnf   ;   avt(:,:,jk) = avt(:,:,jk) + rn_avt_rnf * rnfmsk(:,:)   ;   END DO
-%ENDIF
-
-
-
-% -------------------------------------------------------------------------------------------------------------
 %        Freshwater budget control 
 % -------------------------------------------------------------------------------------------------------------
@@ -602,5 +738,5 @@
 \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/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_STP.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_STP.tex	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_STP.tex	(revision 2282)
@@ -0,0 +1,422 @@
+
+% ================================================================
+% Chapter 2 Ñ Time Domain (step.F90)
+% ================================================================
+\chapter{Time Domain (STP) }
+\label{STP}
+\minitoc
+
+% Missing things:
+%	- daymod: definition of the time domain (nit000, nitend andd the calendar)
+
+
+\gmcomment{STEVEN :maybe a picture of the directory structure in the introduction 
+which could be referred to here, would help  ==> to be added}
+%%%%
+
+
+\newpage
+$\ $\newline    % force a new ligne
+
+
+Having defined the continuous equations in Chap.~\ref{PE}, we need now to choose 
+a time discretization. In the present chapter, we provide a general description of the \NEMO 
+time stepping strategy and the consequences for the order in which the equations are
+solved.
+
+$\ $\newline    % force a new ligne
+% ================================================================
+% Time Discretisation
+% ================================================================
+\section{Time stepping environment}
+\label{STP_environment}
+
+The time stepping used in \NEMO is a three level scheme that can be 
+represented as follows:
+\begin{equation} \label{Eq_STP}
+   x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t-\rdt,\,t,\,t+\rdt}
+\end{equation} 
+where $x$ stands for $u$, $v$, $T$ or $S$; RHS is the Right-Hand-Side of the 
+corresponding time evolution equation; $\rdt$ is the time step; and the 
+superscripts indicate the time at which a quantity is evaluated. Each term of the 
+RHS is evaluated at a specific time step depending on the physics with which 
+it is associated. 
+
+The choice of the time step used for this evaluation is discussed below as 
+well as the implications for starting or restarting a model 
+simulation. Note that the time stepping calculation is generally performed in a single 
+operation. With such a complex and nonlinear system of equations it would be 
+dangerous to let a prognostic variable evolve in time for each term separately.
+
+The three level scheme requires three arrays for each prognostic variable. 
+For each variable $x$ there is $x_b$ (before), $x_n$ (now) and $x_a$. The third array, 
+although referred to as $x_a$ (after) in the code, is usually not the variable at 
+the after time step; but rather it is used to store the time derivative (RHS in 
+\eqref{Eq_STP}) prior to time-stepping the equation. Generally, the time 
+stepping is performed once at each time step in the \mdl{tranxt} and \mdl{dynnxt} 
+modules, except when using implicit vertical diffusion or calculating sea surface height 
+in which case time-splitting options are used.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Non-Diffusive Part---Leapfrog Scheme
+% -------------------------------------------------------------------------------------------------------------
+\section{Non-Diffusive Part --- Leapfrog Scheme}
+\label{STP_leap_frog}
+
+The time stepping used for processes other than diffusion is the well-known leapfrog
+scheme \citep{Mesinger_Arakawa_Bk76}.  This scheme is widely used for advection 
+processes in low-viscosity fluids. It is a time centred scheme, $i.e.$ 
+the RHS in \eqref{Eq_STP} is evaluated at time step $t$, the now time step. 
+It may be used for momentum and tracer advection, 
+pressure gradient, and Coriolis terms, but not for diffusion terms.
+It is an efficient method that achieves 
+second-order accuracy with just one right hand side evaluation per time step. 
+Moreover, it does not artificially damp linear oscillatory motion nor does it produce 
+instability by amplifying the oscillations. These advantages are somewhat diminished 
+by the large phase-speed error of the leapfrog scheme, and the unsuitability 
+of leapfrog differencing for the representation of diffusion and Rayleigh 
+damping processes. However, the scheme allows the coexistence of a numerical 
+and a physical mode due to its leading third order dispersive error. In other words a 
+divergence of odd and even time steps may occur. To prevent it, the leapfrog scheme 
+is often used in association with a Robert-Asselin time filter (hereafter the LF-RA scheme). 
+This filter, first designed by \citet{Robert_JMSJ66} and more comprehensively studied 
+by \citet{Asselin_MWR72}, is a kind of laplacian diffusion in time that mixes odd and 
+even time steps:
+\begin{equation} \label{Eq_STP_asselin}
+x_F^t  = x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right]
+\end{equation} 
+where the subscript $F$ denotes filtered values and $\gamma$ is the Asselin 
+coefficient. $\gamma$ is initialized as \np{rn\_atfp} (namelist parameter). 
+Its default value is \np{rn\_atfp}=$10^{-3}$ (see \S~\ref{STP_mLF}), 
+causing only a weak dissipation of high frequency motions (\citep{Farge1987}). 
+The addition of a time filter degrades the accuracy of the 
+calculation from second to first order. However, the second order truncation 
+error is proportional to $\gamma$, which is small compared to 1. Therefore, 
+the LF-RA is a quasi second order accurate scheme. The LF-RA scheme 
+is preferred to other time differencing schemes such as 
+predictor corrector or trapezoidal schemes, because the user has an explicit 
+and simple control of the magnitude of the time diffusion of the scheme. 
+When used with the 2nd order space centred discretisation of the 
+advection terms in the momentum and tracer equations, LF-RA avoids implicit 
+numerical diffusion: diffusion is set explicitly by the user through the Robert-Asselin 
+filter parameter and the viscosity and diffusion coefficients.
+
+% -------------------------------------------------------------------------------------------------------------
+%        Diffusive Part---Forward or Backward Scheme
+% -------------------------------------------------------------------------------------------------------------
+\section{Diffusive Part --- Forward or Backward Scheme}
+\label{STP_forward_imp}
+
+The leapfrog differencing scheme is unsuitable for the representation of 
+diffusion and damping processes. For a tendancy $D_x$, representing a 
+diffusion term or a restoring term to a tracer climatology 
+(when present, see \S~\ref{TRA_dmp}), a forward time differencing scheme
+ is used :
+\begin{equation} \label{Eq_STP_euler}
+   x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt}
+\end{equation} 
+
+This is diffusive in time and conditionally stable. The 
+conditions for stability of second and fourth order horizontal diffusion schemes are \citep{Griffies_Bk04}:
+\begin{equation} \label{Eq_STP_euler_stability}
+A^h < \left\{
+\begin{aligned}
+                    &\frac{e^2}{  8 \; \rdt } 	&&\quad \text{laplacian diffusion} 	\\
+                    &\frac{e^4}{64 \; \rdt } 	&&\quad \text{bilaplacian diffusion} 
+            \end{aligned}
+\right.
+\end{equation}
+where $e$ is the smallest grid size in the two horizontal directions and $A^h$ is 
+the mixing coefficient. The linear constraint \eqref{Eq_STP_euler_stability} 
+is a necessary condition, but not sufficient. If it is not satisfied, even mildly, 
+then the model soon becomes wildly unstable. The instability can be removed 
+by either reducing the length of the time steps or reducing the mixing coefficient.
+
+For the vertical diffusion terms, a forward time differencing scheme can be 
+used, but usually the numerical stability condition imposes a strong 
+constraint on the time step. Two solutions are available in \NEMO to overcome 
+the stability constraint: $(a)$ a forward time differencing scheme using a 
+time splitting technique (\np{ln\_zdfexp} = true) or $(b)$ a backward (or implicit) 
+time differencing scheme (\np{ln\_zdfexp} = false). In $(a)$, the master 
+time step $\Delta $t is cut into $N$ fractional time steps so that the 
+stability criterion is reduced by a factor of $N$. The computation is performed as 
+follows:
+\begin{equation} \label{Eq_STP_ts}
+\begin{split}
+& x_\ast ^{t-\rdt} = x^{t-\rdt}   \\
+& x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1} 
+\right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}}
+        \quad \text{for $L=1$ to $N$}      \\
+& x^{t+\rdt} = x_\ast^{t+\rdt}
+\end{split}
+\end{equation}
+with DF a vertical diffusion term. The number of fractional time steps, $N$, is given 
+by setting \np{nn\_zdfexp}, (namelist parameter). The scheme $(b)$ is unconditionally 
+stable but diffusive. It can be written as follows:
+\begin{equation} \label{Eq_STP_imp}
+   x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t+\rdt}
+\end{equation} 
+
+This scheme is rather time consuming since it requires a matrix inversion, 
+but it becomes attractive since a value of 3 or more is needed for N in
+the forward time differencing scheme. For example, the finite difference 
+approximation of the temperature equation is:
+\begin{equation} \label{Eq_STP_imp_zdf}
+\frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\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]
+\end{equation}
+where RHS is the right hand side of the equation except for the vertical diffusion term. 
+We rewrite \eqref{Eq_STP_imp} as:
+\begin{equation} \label{Eq_STP_imp_mat}
+-c(k+1)\;T^{t+1}(k+1) + d(k)\;T^{t+1}(k) - \;c(k)\;T^{t+1}(k-1) \equiv b(k)
+\end{equation}
+where 
+\begin{align*} 
+ c(k) &= A_w^{vT} (k) \, / \, e_{3w} (k)     \\
+ d(k) &= e_{3t} (k)       \, / \, (2\rdt) + c_k + c_{k+1}    \\
+ b(k) &= e_{3t} (k) \; \left( T^{t-1}(k) \, / \, (2\rdt) + \text{RHS} \right)    
+\end{align*}
+
+\eqref{Eq_STP_imp_mat} is a linear system of equations with an associated 
+matrix which is tridiagonal. Moreover, $c(k)$ and $d(k)$ are positive and the diagonal 
+term is greater than the sum of the two extra-diagonal terms, therefore a special 
+adaptation of the Gauss elimination procedure is used to find the solution 
+(see for example \citet{Richtmyer1967}).
+
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        Hydrostatic Pressure gradient
+% -------------------------------------------------------------------------------------------------------------
+\section{Hydrostatic Pressure Gradient --- semi-implicit scheme}
+\label{STP_hpg_imp}
+
+%\gmcomment{ 
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_TimeStep_flowchart}  \begin{center}
+\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_TimeStepping_flowchart.pdf}
+\caption{Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}. 
+The use of a semi-implicit computation of the hydrostatic pressure gradient requires
+the tracer equation to be stepped forward prior to the momentum equation. 
+The need for knowledge of the vertical scale factor (here denoted as $h$)
+requires the sea surface height and the continuity equation to be stepped forward
+prior to the computation of the tracer equation.
+Note that the method for the evaluation of the surface pressure gradient $\nabla p_s$ is not presented here 
+(see \S~\ref{DYN_spg}). }
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+%}
+
+The range of stability of the Leap-Frog scheme can be extended by a factor of two
+by introducing a semi-implicit computation of the hydrostatic pressure gradient term
+\citep{Brown_Campana_MWR78}. Instead of evaluating the pressure at $t$, a linear 
+combination of values at $t-\rdt$, $t$ and $t+\rdt$ is used (see \S~\ref{DYN_hpg_imp}).  
+This technique, controlled by the \np{nn\_dynhpg\_rst} namelist parameter, does not 
+introduce a significant additional computational cost when tracers and thus density 
+is time stepped before the dynamics. This time step ordering is used in \NEMO 
+(Fig.\ref{Fig_TimeStep_flowchart}).
+
+
+This technique, used in several GCMs (\NEMO, POP or MOM for instance), 
+makes the Leap-Frog scheme as efficient 
+\footnote{The efficiency is defined as the maximum allowed Courant number of the time 
+stepping scheme divided by the number of computations of the right-hand side per time step.} 
+as the Forward-Backward scheme used in MOM \citep{Griffies_al_OS05} and more 
+efficient than the LF-AM3 scheme (leapfrog time stepping combined with a third order
+Adams-Moulthon interpolation for the predictor phase) used in ROMS 
+\citep{Shchepetkin_McWilliams_OM05}. 
+
+In fact, this technique is efficient when the physical phenomenon that 
+limits the time-step is internal gravity waves (IGWs). Indeed, it is 
+equivalent to applying a time filter to the pressure gradient to eliminate high 
+frequency IGWs. Obviously, the doubling of the time-step is achievable only 
+if no other factors control the time-step, such as the stability limits associated 
+with advection, diffusion or Coriolis terms. For example, it is inefficient in low resolution
+global ocean configurations, since inertial oscillations in the vicinity of the North Pole 
+are the limiting factor for the time step. It is also often inefficient in very high 
+resolution configurations where strong currents and small grid cells exert 
+the strongest constraint on the time step.
+
+% -------------------------------------------------------------------------------------------------------------
+%        The Modified Leapfrog -- Asselin Filter scheme
+% -------------------------------------------------------------------------------------------------------------
+\section{The Modified Leapfrog -- Asselin Filter scheme}
+\label{STP_mLF}
+
+Significant changes have been introduced by \cite{Leclair_Madec_OM09} in the 
+LF-RA scheme in order to ensure tracer conservation and to allow the use of 
+a much smaller value of the Asselin filter parameter. The modifications affect 
+both the forcing and filtering treatments in the LF-RA scheme.
+
+In a classical LF-RA environment, the forcing term is centred in time, $i.e.$ 
+it is time-stepped over a $2\rdt$ period:  $x^t  = x^t + 2\rdt Q^t $ where $Q$ 
+is the forcing applied to $x$, and the time filter is given by \eqref{Eq_STP_asselin} 
+so that $Q$ is redistributed over several time step. 
+In the modified LF-RA environment, these two formulations have been replaced by:
+\begin{align} 
+x^{t+\rdt}  &= x^{t-\rdt} + \rdt \left( Q^{t-\rdt/2} + Q^{t+\rdt/2} \right)                   \label{Eq_STP_forcing} \\
+%
+x_F^t  &= x^t + \gamma \, \left[ x_F^{t-\rdt} - 2 x^t + x^{t+\rdt} \right] 
+           - \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]                          \label{Eq_STP_RA}
+\end{align}
+The change in the forcing formulation given by \eqref{Eq_STP_forcing} 
+(see Fig.\ref{Fig_MLF_forcing}) has a significant effect: the forcing term no 
+longer excites the divergence of odd and even time steps \citep{Leclair_Madec_OM09}. 
+% forcing seen by the model....
+This property improves the LF-RA scheme in two respects. 
+First, the LF-RA can now ensure the local and global conservation of tracers.
+Indeed, time filtering is no longer required on the forcing part. The influence of 
+the Asselin filter on the forcing is be removed by adding a new term in the filter
+(last term in \eqref{Eq_STP_RA} compared to \eqref{Eq_STP_asselin}). Since 
+the filtering of the forcing was the source of non-conservation in the classical 
+LF-RA scheme, the modified formulation becomes conservative  \citep{Leclair_Madec_OM09}.
+Second, the LF-RA becomes a truly quasi-second order scheme. Indeed, 
+\eqref{Eq_STP_forcing} used in combination with a careful treatment of static 
+instability (\S\ref{ZDF_evd}) and of the TKE physics (\S\ref{ZDF_tke_ene}),
+the two other main sources of time step divergence, allows a reduction by 
+two orders of magnitude of the Asselin filter parameter. 
+
+Note that the forcing is now provided at the middle of a time step: $Q^{t+\rdt/2}$ 
+is the forcing applied over the $[t,t+\rdt]$ time interval. This and the change 
+in the time filter, \eqref{Eq_STP_RA}, allows an exact evaluation of the 
+contribution due to the forcing term between any two time steps, 
+even if separated by only $\rdt$ since the time filter is no longer applied to the
+forcing term.
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_MLF_forcing}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_MLF_forcing.pdf}
+\caption{Illustration of forcing integration methods. 
+(top) ''Traditional'' formulation : the forcing is defined at the same time as the variable 
+to which it is applied (integer value of the time step index) and it is applied over a $2\rdt$ period. 
+(bottom)  modified formulation : the forcing is defined in the middle of the time (integer and a half 
+value of the time step index) and the mean of two successive forcing values ($n-1/2$, $n+1/2$).
+is applied over a $2\rdt$ period.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+
+% -------------------------------------------------------------------------------------------------------------
+%        Start/Restart strategy
+% -------------------------------------------------------------------------------------------------------------
+\section{Start/Restart strategy}
+\label{STP_rst}
+%--------------------------------------------namrun-------------------------------------------
+\namdisplay{namrun}         
+%--------------------------------------------------------------------------------------------------------------
+
+The first time step of this three level scheme when starting from initial conditions 
+is a forward step (Euler time integration): 
+\begin{equation} \label{Eq_DOM_euler}
+	x^1 = x^0 + \rdt \ \text{RHS}^0
+\end{equation}
+This is done simply by keeping the leapfrog environment ($i.e.$ the \eqref{Eq_STP} 
+three level time stepping) but setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields 
+equal at the first time step and using half the value of $\rdt$.
+
+It is also possible to restart from a previous computation, by using a 
+restart file. The restart strategy is designed to ensure perfect 
+restartability of the code: the user should obtain the same results to 
+machine precision either by running the model for $2N$ time steps in one go, 
+or by performing two consecutive experiments of $N$ steps with a restart. 
+This requires saving two time levels and many auxiliary data in the restart 
+files in machine precision. 
+
+Note that when a semi-implicit scheme is used to evaluate the hydrostatic pressure 
+gradient (see \S\ref{DYN_hpg_imp}), an extra three-dimensional field has to be 
+added to the restart file to ensure an exact restartability. This is done optionally 
+via the  \np{nn\_dynhpg\_rst} namelist parameter, so that the size of the
+restart file can be reduced when restartability is not a key issue (operational 
+oceanography or in ensemble simulations for seasonal forecasting).
+
+Note the size of the time step used, $\rdt$, is also saved in the restart file. 
+When restarting, if the the time step has been changed, a restart using an Euler time 
+stepping scheme is imposed. 
+%%%
+\gmcomment{
+add here how to force the restart to contain only one time step for operational purposes
+
+add also the idea of writing several restart for seasonal forecast : how is it done ?
+
+verify that all namelist pararmeters are truly described 
+
+a word on the check of restart  .....
+}
+%%%
+
+\gmcomment{       % add a subsection here  
+
+%-------------------------------------------------------------------------------------------------------------
+%        Time Domain
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Time domain}
+\label{STP_time}
+%--------------------------------------------namrun-------------------------------------------
+\namdisplay{namdom}         
+%--------------------------------------------------------------------------------------------------------------
+
+
+ \colorbox{yellow}{add here a few word on nit000 and nitend}
+
+ \colorbox{yellow}{Write documentation on the calendar and the key variable adatrj}
+
+add a description of daymod, and the model calandar (leap-year and co)
+
+}     	%% end add
+
+
+
+%%
+\gmcomment{       % add implicit in vvl case  and Crant-Nicholson scheme   
+
+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\rdt}
+\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\rdt} \ \text{RHS}+ {2\rdt} \ \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\rdt \ \text{RHS}
++ 2\rdt \ \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\rdt} \           \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}                  T_{k+1}^{t+1} 
++ {2\rdt} \ \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\rdt} \           \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\rdt} \ \text{RHS}    \\
+%
+\end{flalign*}
+
+\begin{flalign*}
+\allowdisplaybreaks
+\intertext{ Tracer case }
+%
+&  \qquad \qquad  \quad   -  {2\rdt}                  \ \left[ \frac{A_w^{vt}}{e_{3w}^{t+1}} \right]_{k+1/2}   
+                                                                                                      \qquad \qquad \qquad  \qquad  T_{k+1}^{t+1}   \\
+&+ {2\rdt} \ \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\rdt} \                          \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\rdt} \ \text{RHS}  \\
+%
+\end{flalign*}
+\begin{flalign*}
+\allowdisplaybreaks
+\intertext{ Tracer content case }
+%
+& -  {2\rdt} \              & \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\rdt} \ \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\rdt} \               & \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\rdt} \ \text{RHS}  &
+\end{flalign*}
+
+%%
+}
+%%
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_TRA.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_TRA.tex	(revision 2282)
@@ -8,7 +8,4 @@
 % missing/update 
 % traqsr: need to coordinate with SBC module
-% trabbl : advective case to be discussed
-%			diffusive case : add : only the bottom ocean cell is concerned
-%			==> addfigure on bbl
 
 %STEVEN :  is the use of the word "positive" to describe a scheme enough, or should it be "positive definite"? I added a comment to this effect on some instances of this below
@@ -51,11 +48,10 @@
 In the present chapter we also describe the diagnostic equations used to compute 
 the sea-water properties (density, Brunt-Vais\"{a}l\"{a} frequency, specific heat and 
-freezing point) although the associated modules ($i.e.$ \mdl{eosbn2}, \mdl{ocfzpt} 
-and \mdl{phycst}) are (temporarily) located in the NEMO/OPA directory.
-
-The different options available to the user are managed by namelist logical or 
+freezing point with associated modules \mdl{eosbn2}, \mdl{ocfzpt} and \mdl{phycst}).
+
+The different options available to the user are managed by namelist logicals or 
 CPP keys. For each equation term \textit{ttt}, the namelist logicals are \textit{ln\_trattt\_xxx}, 
-where \textit{xxx} is a 3 or 4 letter acronym accounting for each optional scheme. 
-The CPP key (when it exists) is \textbf{key\_trattt}. The corresponding code can be 
+where \textit{xxx} is a 3 or 4 letter acronym corresponding to each optional scheme. 
+The CPP key (when it exists) is \textbf{key\_trattt}. The equivalent code can be 
 found in the \textit{trattt} or \textit{trattt\_xxx} module, in the NEMO/OPA/TRA directory.
 
@@ -70,6 +66,6 @@
 		{Tracer Advection (\mdl{traadv})}
 \label{TRA_adv}
-%------------------------------------------nam_traadv-----------------------------------------------------
-\namdisplay{nam_traadv}
+%------------------------------------------namtra_adv-----------------------------------------------------
+\namdisplay{namtra_adv}
 %-------------------------------------------------------------------------------------------------------------
 
@@ -77,28 +73,27 @@
 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} $. The flux form in \eqref{Eq_tra_adv} 
-requires implicitly the use of the continuity equation. Indeed, it is obtained
+$e_{3u} =e_{3v} =e_{3t} $. The flux form in \eqref{Eq_tra_adv} 
+implicitly requires the use of the continuity equation. Indeed, it is obtained
 by using the following equality : $\nabla \cdot \left( \vect{U}\,T \right)=\vect{U} \cdot \nabla T$ 
 which results from the use of the continuity equation, $\nabla \cdot \vect{U}=0$ or 
-$\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant (default option) 
-or variable (\key{vvl} defined) volume case, respectively. 
+$\partial _t e_3 + e_3\;\nabla \cdot \vect{U}=0$ in constant volume (default option) 
+or variable volume (\key{vvl} defined) case, respectively. 
 Therefore it is of paramount importance to design the discrete analogue of the 
 advection tendency so that it is consistent with the continuity equation in order to 
 enforce the conservation properties of the continuous equations. In other words, 
-by substituting $\tau$ by 1 in (\ref{Eq_tra_adv}) we recover the discrete form of 
+by replacing $\tau$ by the number 1 in (\ref{Eq_tra_adv}) we recover the discrete form of 
 the continuity equation which is used to calculate the vertical velocity.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -117,20 +112,18 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-The key difference between the advection schemes used in \NEMO is the choice 
+The key difference between the advection schemes available in \NEMO is the choice 
 made in space and time interpolation to define the value of the tracer at the 
 velocity points (Fig.~\ref{Fig_adv_scheme}). 
 
-Along solid lateral and bottom boundaries a zero tracer flux is naturally 
+Along solid lateral and bottom boundaries a zero tracer flux is automatically 
 specified, since the normal velocity is zero there. At the sea surface the 
 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 
@@ -144,13 +137,12 @@
 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}) 
 is the centred (\textit{now}) \textit{eulerian} ocean velocity (see Chap.~\ref{DYN}). 
-When advective bottom boundary layer (\textit{bbl}) and/or eddy induced velocity 
-(\textit{eiv}) parameterisations are used it is the \textit{now} \textit{effective} 
-velocity ($i.e.$ the sum of the eulerian, the bbl and/or the eiv velocities) which is used.
-
-The choice of an advection scheme is made in the \np{nam\_traadv} namelist, by 
+When eddy induced velocity (\textit{eiv}) parameterisation is used it is the \textit{now} 
+\textit{effective} velocity ($i.e.$ the sum of the eulerian and eiv velocities) which is used.
+
+The choice of an advection scheme is made in the \textit{nam\_traadv} namelist, by 
 setting to \textit{true} one and only one of the logicals \textit{ln\_traadv\_xxx}. The 
 corresponding code can be found in the \textit{traadv\_xxx.F90} module, where 
@@ -174,9 +166,10 @@
 that are pure numerical artefacts. 
 
+\sgacomment{not sure whether 3 is still relevant after TRA-TRC branch is merged in?}
 % -------------------------------------------------------------------------------------------------------------
 %        2nd order centred scheme  
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2})]
-			{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=.true.)}
+			{$2^{nd}$ order centred scheme (cen2) (\np{ln\_traadv\_cen2}=true)}
 \label{TRA_adv_cen2}
 
@@ -195,13 +188,14 @@
 (\ref{Eq_tra_adv_cen2}) is the \textit{now} tracer value. The centered second 
 order advection is computed in the \mdl{traadv\_cen2} module. In this module,
-it is also proposed to combine the \textit{cen2} scheme with an upstream scheme
-in specific areas which requires a strong diffusion in order to avoid the generation 
+it is advantageous to combine the \textit{cen2} scheme with an upstream scheme
+in specific areas which require a strong diffusion in order to avoid the generation 
 of false extrema. These areas are the vicinity of large river mouths, some straits 
 with coarse resolution, and the vicinity of ice cover area ($i.e.$ when the ocean 
 temperature is close to the freezing point).
+This combined scheme has been included for specific grid points in the ORCA2 and ORCA4 configurations only.
 
 Note that using the cen2 scheme, the overall tracer advection is of second 
 order accuracy since both (\ref{Eq_tra_adv}) and (\ref{Eq_tra_adv_cen2}) 
-have this order of accuracy. Note also that 
+have this order of accuracy. \gmcomment{Note also that ... blah, blah}
 
 % -------------------------------------------------------------------------------------------------------------
@@ -209,9 +203,9 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4})]
-		     {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=.true.)}
+		     {$4^{nd}$ order centred scheme (cen4) (\np{ln\_traadv\_cen4}=true)}
 \label{TRA_adv_cen4}
 
 In the $4^{th}$ order formulation (to be implemented), tracer values are 
-evaluated at velocity points as a $4^{th}$ order interpolation, and thus uses 
+evaluated at velocity points as a $4^{th}$ order interpolation, and thus depend on 
 the four neighbouring $T$-points. For example, in the $i$-direction:
 \begin{equation} \label{Eq_tra_adv_cen4}
@@ -247,5 +241,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd})]
-			{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=.true.)}
+			{Total Variance Dissipation scheme (TVD) (\np{ln\_traadv\_tvd}=true)}
 \label{TRA_adv_tvd}
 
@@ -264,15 +258,15 @@
 \end{equation}
 where $c_u$ is a flux limiter function taking values between 0 and 1. 
-There exist many ways to define $c_u$, each correcponding to a different 
+There exist many ways to define $c_u$, each corresponding 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 
-transport) rather than TVD. The TVD scheme is computed in the \mdl{traadv\_tvd} module.
-
-For stability reasons (see \S\ref{DOM_nxt}), in (\ref{Eq_tra_adv_tvd}) 
-$\tau _u^{cen2}$ is evaluated using the \textit{now} tracer while $\tau _u^{ups}$ 
+\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 implemented in the \mdl{traadv\_tvd} module.
+
+For stability reasons (see \S\ref{DOM_nxt}),
+$\tau _u^{cen2}$ is evaluated  in (\ref{Eq_tra_adv_tvd}) using the \textit{now} tracer while $\tau _u^{ups}$ 
 is evaluated using the \textit{before} tracer. In other words, the advective part of 
 the scheme is time stepped with a leap-frog scheme while a forward scheme is 
@@ -287,12 +281,12 @@
 
 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 :
 \begin{equation} \label{Eq_tra_adv_muscl}
    \tau _u^{mus} = \left\{      \begin{aligned}
-         &\tau _i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\Delta t}{e_{1u}} \right)
+         &\tau _i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right)
          &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\
-         &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\Delta t}{e_{1u} } \right)
+         &\tau _{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right)
          &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
    \end{aligned}    \right.
@@ -306,8 +300,8 @@
 For an ocean grid point adjacent to land and where the ocean velocity is 
 directed toward land, two choices are available: an upstream flux 
-(\np{ln\_traadv\_muscl}=.true.) or a second order flux 
-(\np{ln\_traadv\_muscl2}=.true.). Note that the latter choice does not ensure 
+(\np{ln\_traadv\_muscl}=true) or a second order flux 
+(\np{ln\_traadv\_muscl2}=true). Note that the latter choice does not ensure 
 the \textit{positive} character of the scheme. Only the former can be used 
-on both active and passive tracers. The two MUSCL schemes are computed 
+on both active and passive tracers. The two MUSCL schemes are implemented 
 in the \mdl{traadv\_tvd} and \mdl{traadv\_tvd2} modules.
 
@@ -316,5 +310,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs})]
-			{Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=.true.)}
+			{Upstream-Biased Scheme (UBS) (\np{ln\_traadv\_ubs}=true)}
 \label{TRA_adv_ubs}
 
@@ -333,5 +327,5 @@
 
 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation 
-error \citep{Sacha2005}. The overall performance of the advection 
+error \citep{Shchepetkin_McWilliams_OM05}. The overall performance of the advection 
 scheme is similar to that reported in \cite{Farrow1995}. 
 It is a relatively good compromise between accuracy and smoothness. 
@@ -344,18 +338,27 @@
 where the control of artificial diapycnal fluxes is of paramount importance. 
 Therefore the vertical flux is evaluated using the TVD scheme when 
-\np{ln\_traadv\_ubs}=.true..
-
-For stability reasons  (see \S\ref{DOM_nxt}), in \eqref{Eq_tra_adv_ubs}, 
-the first term (which corresponds to a second order centred scheme) 
+\np{ln\_traadv\_ubs}=true.
+
+For stability reasons  (see \S\ref{DOM_nxt}),
+the first term  in \eqref{Eq_tra_adv_ubs} (which corresponds to a second order centred scheme) 
 is evaluated using the \textit{now} tracer (centred in time) while the 
 second term (which is the diffusive part of the scheme), is 
 evaluated using the \textit{before} tracer (forward in time). 
-This 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 
 substitution in the \mdl{traadv\_ubs} module and obtain a QUICK scheme.
+
+Four different options are possible for the vertical 
+component used in the UBS scheme. $\tau _w^{ubs}$ can be evaluated 
+using either \textit{(a)} a centred $2^{nd}$ order scheme, or  \textit{(b)} 
+a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
+parabolic splines following the \citet{Shchepetkin_McWilliams_OM05} 
+implementation of UBS in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case 
+has dispersion properties similar to an eighth-order accurate conventional scheme.
+The current reference version uses method b)
 
 Note that :
@@ -368,13 +371,5 @@
 in the current reference version. 
 
-(2) In a forthcoming release four options will be available for the vertical 
-component used in the UBS scheme. $\tau _w^{ubs}$ will be evaluated 
-using either \textit{(a)} a centred $2^{nd}$ order scheme, or  \textit{(b)} 
-a TVD scheme, or  \textit{(c)} an interpolation based on conservative 
-parabolic splines following the \citet{Sacha2005} implementation of UBS 
-in ROMS, or  \textit{(d)} a UBS. The $3^{rd}$ case has dispersion properties 
-similar to an eighth-order accurate conventional scheme.
-
-(3) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
+(2) It is straightforward to rewrite \eqref{Eq_tra_adv_ubs} as follows:
 \begin{equation} \label{Eq_traadv_ubs2}
 \tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{	 
@@ -398,7 +393,6 @@
 Thirdly, the diffusion term is in fact a biharmonic operator with an eddy 
 coefficient which is simply proportional to the velocity:
- $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v2.3 still uses 
- \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}. This should be 
- changed in forthcoming release.
+ $A_u^{lm}= - \frac{1}{12}\,{e_{1u}}^3\,|u|$. Note that NEMO v3.3 still uses 
+ \eqref{Eq_tra_adv_ubs}, not \eqref{Eq_traadv_ubs2}.
  %%%
  \gmcomment{the change in UBS scheme has to be done}
@@ -409,5 +403,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [QUICKEST scheme (QCK) (\np{ln\_traadv\_qck})]
-			{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=.true.)}
+			{QUICKEST scheme (QCK) (\np{ln\_traadv\_qck}=true)}
 \label{TRA_adv_qck}
 
@@ -419,9 +413,9 @@
 The resulting scheme is quite expensive but \emph{positive}. 
 It can be used on both active and passive tracers. 
-Nevertheless, the intrinsic diffusion of QCK makes its use risky in the vertical 
+However, the intrinsic diffusion of QCK makes its use risky in the vertical 
 direction where the control of artificial diapycnal fluxes is of paramount importance. 
 Therefore the vertical flux is evaluated using the CEN2 scheme. 
-This no more ensure the positivity of the scheme. The use of TVD in the vertical 
-direction as for the UBS case should be implemented to maintain the property.
+This no longer guarantees the positivity of the scheme. The use of TVD in the vertical 
+direction (as for the UBS case) should be implemented to restore this property.
 
 
@@ -430,12 +424,13 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm})]
-			{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=.true.)}
+			{Piecewise Parabolic Method (PPM) (\np{ln\_traadv\_ppm}=true)}
 \label{TRA_adv_ppm}
 
 The Piecewise Parabolic Method (PPM) proposed by Colella and Woodward (1984) 
-is based on a quadradic piecewise rebuilding. Like the QCK scheme, it is associated 
+\sgacomment{reference?}
+is based on a quadradic piecewise construction. Like the QCK scheme, it is associated 
 with the ULTIMATE QUICKEST limiter \citep{Leonard1991}. It has been implemented 
 in \NEMO by G. Reffray (MERCATOR-ocean) but is not yet offered in the reference 
-version 3.0.
+version 3.3.
 
 % ================================================================
@@ -446,5 +441,5 @@
 \label{TRA_ldf}
 %-----------------------------------------nam_traldf------------------------------------------------------
-\namdisplay{nam_traldf}
+\namdisplay{namtra_ldf}
 %-------------------------------------------------------------------------------------------------------------
  
@@ -465,5 +460,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap})]
-			{Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=.true.) }
+			{Iso-level laplacian operator (lap) (\np{ln\_traldf\_lap}=true) }
 \label{TRA_ldf_lap}
 
@@ -471,23 +466,23 @@
 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. 
-It can be found in the \mdl{traadv\_lap} module.
+where  $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells. 
+It is implemented in the \mdl{traadv\_lap} module.
 
 This lateral operator is computed in \mdl{traldf\_lap}. It is a \emph{horizontal} 
 operator ($i.e.$ acting along geopotential surfaces) in the $z$-coordinate with 
-or without partial step, but is simply an iso-level operator in the $s$-coordinate. 
-It is thus used when, in addition to \np{ln\_traldf\_lap}=.true., we have 
-\np{ln\_traldf\_level}=.true., or \np{ln\_traldf\_hor}=\np{ln\_zco}=.true.. 
+or without partial steps, but is simply an iso-level operator in the $s$-coordinate. 
+It is thus used when, in addition to \np{ln\_traldf\_lap}=true, we have 
+\np{ln\_traldf\_level}=true or \np{ln\_traldf\_hor}=\np{ln\_zco}=true. 
 In both cases, it significantly contributes to diapycnal mixing. 
 It is therefore not recommended.
 
 Note that 
-(a) In pure $z$-coordinate (\key{zco} is defined), $e_{3u}=e_{3v}=e_{3T}$, 
+(a) In the 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 
+(b) In the partial step $z$-coordinate (\np{ln\_zps}=true), tracers in horizontally 
 adjacent cells are located at different depths in the vicinity of the bottom. 
 In this case, horizontal derivatives in (\ref{Eq_tra_ldf_lap}) at the bottom level 
@@ -499,5 +494,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Rotated laplacian operator (iso) (\np{ln\_traldf\_lap})]
-			{Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=.true.)}
+			{Rotated laplacian operator (iso) (\np{ln\_traldf\_lap}=true)}
 \label{TRA_ldf_iso}
 
@@ -507,6 +502,6 @@
 \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.    \\ 
@@ -525,10 +520,10 @@
  \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 
 acts ($i.e.$ horizontal or iso-neutral surfaces).  It is thus used when, 
-in addition to \np{ln\_traldf\_lap}= .true., we have \np{ln\_traldf\_iso}=.true., 
-or both \np{ln\_traldf\_hor}=.true. and \np{ln\_zco}=.true.. The way these 
+in addition to \np{ln\_traldf\_lap}= true, we have \np{ln\_traldf\_iso}=true, 
+or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. The way these 
 slopes are evaluated is given in \S\ref{LDF_slp}. At the surface, bottom 
 and lateral boundaries, the turbulent fluxes of heat and salt are set to zero 
@@ -546,10 +541,10 @@
 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 
-is currently been tested in \NEMO. It should be available in a forthcoming
-release.
-
-Note that in the partial step $z$-coordinate (\np{ln\_zps}=.true.), the horizontal 
+background horizontal diffusion \citep{Guilyardi_al_CD01}. An alternative scheme 
+developed by \cite{Griffies_al_JPO98} which preserves both tracer and its variance 
+is also available in \NEMO (\np{ln\_traldf\_grif}=true). A complete description of 
+the algorithm is given in App.\ref{Apdx_Griffies}.
+
+Note that in the partial step $z$-coordinate (\np{ln\_zps}=true), the horizontal 
 derivatives at the bottom level in \eqref{Eq_tra_ldf_iso} require a specific 
 treatment. They are calculated in module zpshde, described in \S\ref{TRA_zpshde}.
@@ -559,13 +554,13 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap})]
-			{Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=.true.)}
+			{Iso-level bilaplacian operator (bilap) (\np{ln\_traldf\_bilap}=true)}
 \label{TRA_ldf_bilap}
 
 The lateral fourth order bilaplacian operator on tracers is obtained by 
-applying (\ref{Eq_tra_ldf_lap}) twice. It requires an additional assumption 
-on boundary conditions: the first and third derivative terms normal to the 
-coast are set to zero. It is used when, in addition to \np{ln\_traldf\_bilap}=.true., 
-we have \np{ln\_traldf\_level}=.true., or both \np{ln\_traldf\_hor}=.true. and 
-\np{ln\_zco}=.false.. In both cases, it can contribute diapycnal mixing, 
+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 
+\np{ln\_zco}=false. In both cases, it can contribute diapycnal mixing, 
 although less than in the laplacian case. It is therefore not recommended.
 
@@ -579,5 +574,5 @@
 where $a=\sqrt{|A|}$ and $A<0$. This was a mistake: both formulations 
 ensure the total variance decrease, but the former requires a larger 
-number of code-lines. It will be corrected in a forthcoming release.
+number of code-lines.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -585,5 +580,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap})]
-			{Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=.true.)}
+			{Rotated bilaplacian operator (bilapg) (\np{ln\_traldf\_bilap}=true)}
 \label{TRA_ldf_bilapg}
 
@@ -591,13 +586,13 @@
 applying (\ref{Eq_tra_ldf_iso}) twice. It requires an additional assumption 
 on boundary conditions: first and third derivative terms normal to the 
-coast, the bottom and the surface are set to zero. It can be found in the
+coast, normal to the bottom and normal to the surface are set to zero. It can be found in the
 \mdl{traldf\_bilapg}.
 
-It is used when, in addition to \np{ln\_traldf\_bilap}=.true., we have 
-\np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=.true. and \np{ln\_zco}=.true.. 
-Nevertheless, this rotated bilaplacian operator has never been seriously 
-tested. No warranties that it is neither free of bugs or correctly formulated. 
+It is used when, in addition to \np{ln\_traldf\_bilap}=true, we have 
+\np{ln\_traldf\_iso}= .true, or both \np{ln\_traldf\_hor}=true and \np{ln\_zco}=true. 
+This rotated bilaplacian operator has never been seriously 
+tested. There are no guarantees that it is either free of bugs or correctly formulated. 
 Moreover, the stability range of such an operator will be probably quite 
-narrow, requiring a significantly smaller time-step than the one used on 
+narrow, requiring a significantly smaller time-step than the one used with an
 unrotated operator.
 
@@ -618,7 +613,7 @@
 \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}
@@ -641,10 +636,10 @@
 The large eddy coefficient found in the mixed layer together with high 
 vertical resolution implies that in the case of explicit time stepping 
-(\np{ln\_zdfexp}=.true.) there would be too restrictive a constraint on 
+(\np{ln\_zdfexp}=true) there would be too restrictive a constraint on 
 the time step. Therefore, the default implicit time stepping is preferred 
 for the vertical diffusion since it overcomes the stability constraint. 
-A forward time differencing scheme (\np{ln\_zdfexp}=.true.) using a time 
-splitting technique (\np{n\_zdfexp} $> 1$) is provided as an alternative. 
-Namelist variables \np{ln\_zdfexp} and \np{n\_zdfexp} apply to both 
+A forward time differencing scheme (\np{ln\_zdfexp}=true) using a time 
+splitting technique (\np{nn\_zdfexp} $> 1$) is provided as an alternative. 
+Namelist variables \np{ln\_zdfexp} and \np{nn\_zdfexp} apply to both 
 tracers and dynamics. 
 
@@ -667,82 +662,92 @@
 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 
-forcing on an ocean tracer, $c$, can be split into two parts: $F_{ext}$, the 
-flux of tracer crossing the sea surface and not linked with the water 
-exchange with the atmosphere, $F_{wf}^d$, and $F_{wf}^i$ the forcing 
-on the tracer concentration associated with this water exchange. 
-The latter forcing has two components: a direct effect of change 
-in concentration associated with the tracer carried by the water flux, 
-and an indirect concentration/dilution effect :
-\begin{equation*}
-\begin{split}
- F^C &= F_{ext} + F_{wf}^d                                          +F_{wf}^i  	\\
-        &= F_{ext} - \left( c_E \, E - c_p \,P - c_R \,R \right) +c\left( E-P-R \right)
-\end{split}
-\end{equation*} 
-
-\gmcomment{add here a description of the variable names used in the above equation}
-
-Two cases must be distinguished, the nonlinear free surface case 
-(\key{vvl} is defined) and the linear free surface case. The first case 
-is simpler, because the indirect concentration/dilution effect is naturally 
-taken into account by letting the vertical scale factors vary in time. 
-The salinity of water exchanged at the surface is assumed to be zero, 
-so there is no salt flux at the free surface, except in the presence of sea ice. 
-The heat flux at the free surface is the sum of $F_{ext}$, the direct 
-heating/cooling (by the total non-penetrative heat flux) and $F_{wf}^e$ 
-the heat carried by the water exchanged through the surface (evaporation, 
-precipitation, runoff). The temperature of precipitation is not well known. 
-In the model we assume that this water has the same temperature as 
-the sea surface temperature. The resulting forcing terms for temperature 
-T and salinity S are: 
-\begin{equation} \label{Eq_tra_forcing}
+the thickness of the top model layer. 
+
+Due to interactions and mass exchange of water ($F_{mass}$) with other Earth system components ($i.e.$ atmosphere, sea-ice, land),
+the change in the heat and salt content of the surface layer of the ocean is due both 
+to the heat and salt fluxes crossing the sea surface (not linked with $F_{mass}$)
+ and to the heat and salt content of the mass exchange.
+\sgacomment{ the following does not apply to the release to which this documentation is 
+attached and so should not be included ....
+In a forthcoming release, these two parts, computed in the surface module (SBC), will be included directly
+in $Q_{ns}$, the surface heat flux and $F_{salt}$, the surface salt flux.
+The specification of these fluxes is further detailed in the SBC chapter (see \S\ref{SBC}). 
+This change will provide a forcing formulation which is the same for any tracer (including temperature and salinity).
+ 
+In the current version, the situation is a little bit more complicated. }
+The surface module (\mdl{sbcmod}, see \S\ref{SBC}) provides the following 
+forcing fields (used on tracers):
+
+$\bullet$ $Q_{ns}$, the non-solar part of the net surface heat flux that crosses the sea surface 
+(i.e. the difference between the total surface heat flux and the fraction of the short wave flux that 
+penetrates into the water column, see \S\ref{TRA_qsr})
+
+$\bullet$ \textit{emp}, the mass flux exchanged with the atmosphere (evaporation minus precipitation)
+
+$\bullet$ $\textit{emp}_S$, an equivalent mass flux taking into account the effect of ice-ocean mass exchange
+
+$\bullet$ \textit{rnf}, the mass flux associated with runoff (see \S\ref{SBC_rnf} for further detail of how it acts on temperature and salinity tendencies)
+
+The $\textit{emp}_S$ field is not simply the budget of evaporation-precipitation+freezing-melting because 
+the sea-ice is not currently embedded in the ocean but levitates above it. There is no mass
+exchanged between the sea-ice and the ocean. Instead we only take into account the salt
+flux associated with the non-zero salinity of sea-ice, and the concentration/dilution effect
+due to the freezing/melting (F/M) process. These two parts of the forcing are then converted into 
+an equivalent mass flux given by $\textit{emp}_S - \textit{emp}$. As a result of this mess, 
+the surface boundary condition on temperature and salinity is applied as follows:
+
+In the nonlinear free surface case (\key{vvl} is defined):
+\begin{equation} \label{Eq_tra_sbc}
 \begin{aligned}
- &F^T =\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }-\frac{\text{EMP}\;\left. T 
-\right|_{k=1} }{e_{3T} }  & \\ 
-\\
-& F^S =\frac{\text{EMP}_S\;\left. S \right|_{k=1} }{e_{3T} }   &
+ &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }   
+           &\overline{ \left( Q_{ns} - \textit{emp}\;C_p\,\left. T \right|_{k=1} \right) }^t  & \\ 
+%
+& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
+           &\overline{ \left( (\textit{emp}_S - \textit{emp})\;\left. S \right|_{k=1}  \right) }^t   & \\   
  \end{aligned}
 \end{equation} 
-where EMP is the freshwater budget (evaporation minus precipitation 
-minus river runoff) which forces the ocean volume, $Q_{ns}$ is the 
-non-penetrative part of the net surface heat flux (difference between 
-the total surface heat flux and the fraction of the short wave flux that 
-penetrates into the water column), the product EMP$_S\;.\left. S \right|_{k=1}$ 
-is  the ice-ocean salt flux, and $\left. S\right|_{k=1}$ is the sea surface 
-salinity (\textit{SSS}). The total salt content is conserved in this formulation 
-(except for the effect of the Asselin filter).
-
-%AMT note: the ice-ocean flux had been forgotten in the first release of the key_vvl option, has this been corrected in the code?     ===> gm :  NO to be added at NOCS 
-
-In the second case (linear free surface), the vertical scale factors are 
-fixed in time so that the concentration/dilution effect must be added in 
-the \mdl{trasbc} module. Because of the hypothesis made for the 
-temperature of precipitation and runoffs, $F_{wf}^e +F_{wf}^i =0$ 
-for temperature. The resulting forcing term for temperature is: 
-\begin{equation} \label{Eq_tra_forcing_q}
-F^T=\frac{Q_{ns} }{\rho _o \;C_p \,e_{3T} }
+
+In the linear free surface case (\key{vvl} not defined):
+\begin{equation} \label{Eq_tra_sbc_lin}
+\begin{aligned}
+ &F^T = \frac{ 1 }{\rho _o \;C_p \,\left. e_{3t} \right|_{k=1} }  &\overline{ Q_{ns} }^t  & \\ 
+%
+& F^S =\frac{ 1 }{\rho _o \,\left. e_{3t} \right|_{k=1} } 
+           &\overline{ \left( \textit{emp}_S\;\left. S \right|_{k=1}  \right) }^t   & \\   
+ \end{aligned}
 \end{equation} 
-
-The salinity forcing is still given by \eqref{Eq_tra_forcing} but the 
-definition of EMP$_S$ is different: it is the total surface freshwater 
-budget (evaporation minus precipitation minus river runoff plus 
-the rate of change of the sea ice thickness). The total salt content 
-is not exactly conserved (\citet{Roullet2000}. 
-See also \S\ref{PE_free_surface}).
-
-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 
-through the top level: this is because in the rigid lid case \textit{w(k=1) = 0} 
-(in contrast to the linear free surface case). As a result, even if the budget 
-of \textit{EMP} is zero on average over the whole ocean domain, the 
-associated salt flux is not, since sea-surface salinity and \textit{EMP} are 
-intrinsically correlated (high \textit{SSS} are found where evaporation is 
-strong whilst low \textit{SSS} is usually associated with high precipitation 
-or river runoff).
-
-The $Q_{ns} $ and \textit{EMP} fields are defined and updated in the 
-\mdl{sbcmod} module (see \S\ref{SBC}).
+where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps 
+($t-\rdt/2$ and $t+\rdt/2$). Such time averaging prevents the 
+divergence of odd and even time step (see \S\ref{STP}).
+
+The two set of equations, \eqref{Eq_tra_sbc} and \eqref{Eq_tra_sbc_lin}, are obtained 
+by assuming that the temperature of precipitation and evaporation are equal to
+the ocean surface temperature and that their salinity is zero. Therefore, the heat content
+of the \textit{emp} budget must be added to the temperature equation in the variable volume case, 
+while it does not appear in the constant volume case. Similarly, the \textit{emp} budget affects 
+the ocean surface salinity in the constant volume case (through the concentration dilution effect)
+while it does not appears explicitly in the variable volume case since salinity change will be
+induced by volume change. In both constant and variable volume cases, surface salinity 
+will change with ice-ocean salt flux and F/M flux (both contained in $\textit{emp}_S - \textit{emp}$) without mass exchanges.
+
+Note that the concentration/dilution effect due to F/M is computed using
+a constant ice salinity as well as a constant ocean salinity. 
+This approximation suppresses the correlation between \textit{SSS} 
+and F/M flux, allowing the ice-ocean salt exchanges to be conservative.
+Indeed, if this approximation is not made, even if the F/M budget is zero 
+on average over the whole ocean domain and over the seasonal cycle, 
+the associated salt flux is not zero, since sea-surface salinity and F/M flux are 
+intrinsically correlated (high \textit{SSS} are found where freezing is 
+strong whilst low \textit{SSS} is usually associated with high melting areas).
+
+Even using this approximation, an exact conservation of heat and salt content 
+is only achieved in the variable volume case. In the constant volume case, 
+there is a small imbalance associated with the product $(\partial_t\eta - \textit{emp}) * \textit{SSS}$.
+Nevertheless, the salt content variation is quite small and will not induce
+a long term drift as there is no physical reason for $(\partial_t\eta - \textit{emp})$ 
+and \textit{SSS} to be correlated \citep{Roullet_Madec_JGR00}. 
+Note that, while quite small, the imbalance in the constant volume case is larger 
+than the imbalance associated with the Asselin time filter \citep{Leclair_Madec_OM09}. 
+This is the reason why the modified filter is not applied in the constant volume case.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -753,12 +758,12 @@
 \label{TRA_qsr}
 %--------------------------------------------namqsr--------------------------------------------------------
-\namdisplay{namqsr}
+\namdisplay{namtra_qsr}
 %--------------------------------------------------------------------------------------------------------------
 
-When the penetrative solar radiation option is used (\np{ln\_flxqsr}=.true.), 
-the solar radiation penetrates the top few meters of the ocean, otherwise 
-all the heat flux is absorbed in the first ocean level (\np{ln\_flxqsr}=.false.). 
+When the penetrative solar radiation option is used (\np{ln\_flxqsr}=true), 
+the solar radiation penetrates the top few tens of meters of the ocean. If it is not used 
+(\np{ln\_flxqsr}=false) all the heat flux is absorbed in the first ocean level. 
 Thus, in the former case a term is added to the time evolution equation of 
-temperature \eqref{Eq_PE_tra_T} whilst the surface boundary condition is 
+temperature \eqref{Eq_PE_tra_T} and the surface boundary condition is 
 modified to take into account only the non-penetrative part of the surface 
 heat flux:
@@ -769,51 +774,81 @@
 \end{split}
 \end{equation}
-
-where $I$ is the downward irradiance. The additional term in \eqref{Eq_PE_qsr} 
-is discretized as follows:
+where $Q_{sr}$ is the penetrative part of the surface heat flux ($i.e.$ the shortwave radiation) 
+and $I$ is the downward irradiance ($\left. I \right|_{z=\eta}=Q_{sr}$). 
+The additional term in \eqref{Eq_PE_qsr} 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]
-\end{equation}
-
-A formulation involving two extinction coefficients is assumed for the 
-downward irradiance $I$ \citep{Paulson1977}:
+\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}
+
+The shortwave radiation,  $Q_{sr}$, consists of energy distributed across a wide spectral range. 
+The ocean is strongly absorbing for wavelengths longer than 700~nm and these 
+wavelengths contribute to heating the upper few tens of centimetres. The fraction of $Q_{sr}$ 
+that resides in these almost non-penetrative wavebands, $R$, is $\sim 58\%$ (specified 
+through namelist parameter \np{rn\_abs}).  It is assumed to penetrate the ocean 
+with a decreasing exponential profile, with an e-folding depth scale, $\xi_0$, 
+of a few tens of centimetres (typically $\xi_0=0.35~m$ set as \np{rn\_si0} in the namtra\_qsr namelist).
+For shorter wavelengths (400-700~nm), the ocean is more transparent, and solar energy 
+propagates to larger depths where it contributes to 
+local heating. 
+The way this second part of the solar energy penetrates into the ocean depends on 
+which formulation is chosen. In the simple 2-waveband light penetration scheme  (\np{ln\_qsr\_2bd}=true) 
+a chlorophyll-independent monochromatic formulation is chosen for the shorter wavelengths, 
+leading to the following expression  \citep{Paulson1977}:
 \begin{equation} \label{Eq_traqsr_iradiance}
-I(z) = Q_{sr} \left[Re^{-z / \xi_1} + \left( 1-R\right) e^{-z / \xi_2} \right]
-\end{equation}
-where $Q_{sr}$ is the penetrative part of the surface heat flux, 
-$\xi_1$ and $\xi_2$ are two extinction length scales and $R$ 
-determines the relative contribution of the two terms. 
-The default values used correspond to a Type I water in Jerlov's [1968] 
-%
-\gmcomment : Jerlov reference to be added
-%
-classification: $\xi_1 = 0.35~m$, $\xi_2 = 23~m$ and $R = 0.58$ 
-(corresponding to \np{xsi1}, \np{xsi2} and \np{rabs} namelist parameters, 
-respectively). $I$ is masked (no flux through the ocean bottom), 
-so all the solar radiation that reaches the last ocean level is absorbed 
-in that level. The trend in \eqref{Eq_tra_qsr} associated with the 
-penetration of the solar radiation is added to the temperature trend, 
-and the surface heat flux is modified in routine \mdl{traqsr}. 
-Note that in the $z$-coordinate, the depth of $T-$levels depends 
-on the single variable $k$. A one dimensional array of the coefficients 
-$gdsr(k) = Re^{-z_w (k)/\xi_1} + (1-R)e^{-z_w (k)/\xi_2}$ can then 
-be computed once and saved in memory. Moreover \textit{nksr}, 
-the level at which $gdrs$ becomes negligible (less than the 
-computer precision) is computed once, and the trend associated 
-with the penetration of the solar radiation is only added until that level. 
-Finally, note that when the ocean is shallow (< 200~m), part of the 
+I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
+\end{equation}
+where $\xi_1$ is the second extinction length scale associated with the shorter wavelengths.  
+It is usually chosen to be 23~m by setting the \np{rn\_si0} namelist parameter. 
+The set of default values ($\xi_0$, $\xi_1$, $R$) corresponds to a Type I water in 
+Jerlov's (1968) classification (oligotrophic waters).
+
+Such assumptions have been shown to provide a very crude and simplistic 
+representation of observed light penetration profiles (\cite{Morel_JGR88}, see also 
+Fig.\ref{Fig_traqsr_irradiance}). Light absorption in the ocean depends on 
+particle concentration and is spectrally selective. \cite{Morel_JGR88} has shown 
+that an accurate representation of light penetration can be provided by a 61 waveband 
+formulation. Unfortunately, such a model is very computationally expensive. 
+Thus, \cite{Lengaigne_al_CD07} have constructed a simplified version of this 
+formulation in which visible light is split into three wavebands: blue (400-500 nm), 
+green (500-600 nm) and red (600-700nm). For each wave-band, the chlorophyll-dependent 
+attenuation coefficient is fitted to the coefficients computed from the full spectral model 
+of \cite{Morel_JGR88} (as modified by \cite{Morel_Maritorena_JGR01}), assuming 
+the same power-law relationship. As shown in Fig.\ref{Fig_traqsr_irradiance}, 
+this formulation, called RGB (Red-Green-Blue), reproduces quite closely 
+the light penetration profiles predicted by the full spectal model, but with much greater 
+computational efficiency. The 2-bands formulation does not reproduce the full model very well. 
+
+The RGB formulation is used when \np{ln\_qsr\_rgb}=true. The RGB attenuation coefficients
+($i.e.$ the inverses of the extinction length scales) are tabulated over 61 nonuniform 
+chlorophyll classes ranging from 0.01 to 10 g.Chl/L (see the routine \rou{trc\_oce\_rgb} 
+in \mdl{trc\_oce} module). Three types of chlorophyll can be chosen in the RGB formulation:
+(1) a constant 0.05 g.Chl/L value everywhere (\np{nn\_chdta}=0) ; (2) an observed 
+time varying chlorophyll (\np{nn\_chdta}=1) ; (3) simulated time varying chlorophyll
+by TOP biogeochemical model (\np{ln\_qsr\_bio}=true). In the latter case, the RGB 
+formulation is used to calculate both the phytoplankton light limitation in PISCES 
+or LOBSTER and the oceanic heating rate. 
+
+The trend in \eqref{Eq_tra_qsr} associated with the penetration of the solar radiation 
+is added to the temperature trend, and the surface heat flux is modified in routine \mdl{traqsr}. 
+
+When the $z$-coordinate is preferred to the $s$-coordinate, the depth of $w-$levels does 
+not significantly vary with location. The level at which the light has been totally 
+absorbed ($i.e.$ it is less than the computer precision) is computed once, 
+and the trend associated with the penetration of the solar radiation is only added down to that level. 
+Finally, note that when the ocean is shallow ($<$ 200~m), part of the 
 solar radiation can reach the ocean floor. In this case, we have 
 chosen that all remaining radiation is absorbed in the last ocean 
-level ($i.e.$ $I_w$ is masked). 
-
-When coupling with a biological model (for example PISCES or LOBSTER), 
-it is possible to calculate the light attenuation using information from 
-the biology model. Without biological model, it is still possible to introduce 
-a horizontal variation of the light attenuation by using the observed ocean 
-surface color. At the time of writing, the latter has not been implemented
- in the reference version.
-%
-\gmcomment{  {yellow}{case 4 bands and bio-coupling to add !!!}  }
-%
+level ($i.e.$ $I$ is masked). 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_traqsr_irradiance}  \begin{center}
+\includegraphics[width=1.0\textwidth]{./TexFiles/Figures/Fig_TRA_Irradiance.pdf}
+\caption{Penetration profile of the downward solar irradiance 
+calculated by four models. Two waveband chlorophyll-independent formulation (blue), 
+a chlorophyll-dependent monochromatic formulation (green), 4 waveband RGB formulation (red), 
+61 waveband Morel (1988) formulation (black) for a chlorophyll concentration of 
+(a) Chl=0.05 mg/m$^3$ and (b) Chl=0.5 mg/m$^3$. From \citet{Lengaigne_al_CD07}.}
+\end{center}   \end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 % -------------------------------------------------------------------------------------------------------------
@@ -824,11 +859,11 @@
 \label{TRA_bbc}
 %--------------------------------------------nambbc--------------------------------------------------------
-\namdisplay{nambbc}
+\namdisplay{namtra_bbc}
 %--------------------------------------------------------------------------------------------------------------
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \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}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -840,22 +875,22 @@
 non-zero heat flux across the seafloor that is associated with solid 
 earth cooling. This flux is weak compared to surface fluxes (a mean 
-global value of $\sim0.1\;W/m^2$ \citep{Stein1992}), but it is 
-systematically positive and acts on the densest water masses. 
+global value of $\sim0.1\;W/m^2$ \citep{Stein_Stein_Nat92}), but it is 
+systematically positive \sgacomment{positive definite?} 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}. 
-
-The presence or not of geothermal heating is controlled by the namelist 
-parameter  \np{ngeo\_flux}. If this parameter is set to 1, a constant 
+Bottom Water) by a few Sverdrups  \citep{Emile-Geay_Madec_OS09}. 
+
+The presence of geothermal heating is controlled by the namelist 
+parameter  \np{nn\_geoflx}. When this parameter is set to 1, a constant 
 geothermal heating is introduced whose value is given by the 
-\np{ngeo\_flux\_const}, which is also a namelist parameter. If it is set to 2, 
+\np{nn\_geoflx\_cst}, which is also a namelist parameter. When it is set to 2, 
 a spatially varying geothermal heat flux is introduced which is provided 
-in the geothermal\_heating.nc NetCDF file (Fig.\ref{Fig_geothermal}).
+in the \ifile{geothermal\_heating} NetCDF file (Fig.\ref{Fig_geothermal}).
 
 % ================================================================
 % Bottom Boundary Layer
 % ================================================================
-\section  [Bottom Boundary Layer (\textit{trabbl}, \textit{trabbl\_adv} )]
-		{Bottom Boundary Layer (\mdl{trabbl}, \mdl{trabbl\_adv})}
+\section  [Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})]
+		{Bottom Boundary Layer (\mdl{trabbl} - \key{trabbl})}
 \label{TRA_bbl}
 %--------------------------------------------nambbl---------------------------------------------------------
@@ -865,40 +900,41 @@
 In a $z$-coordinate configuration, the bottom topography is represented by a 
 series of discrete steps. This is not adequate to represent gravity driven 
-downslope flows. Such flows arise downstream of sills such as the Strait of 
-Gibraltar, Bab El Mandeb, or Denmark Strait, where dense water formed in 
-marginal seas flows into a basin filled with less dense water. The amount of 
-entrainment that occurs in these gravity plumes is critical in determining the 
-density and volume flux of the densest waters of the ocean, such as 
-Antarctic Bottom Water, or North Atlantic Deep Water. $z$-coordinate 
-models tend to overestimate the entrainment, because the gravity flow is 
-mixed down vertically by convection as it goes ``downstairs'' following the 
-step topography, sometimes over a thickness much larger than the thickness 
-of the observed gravity plume. A similar problem occurs in the $s$-coordinate when 
-the thickness of the bottom level varies in large proportions downstream of 
-a sill \citep{Willebrand2001}, and the thickness of the plume is not resolved. 
-
-The idea of the bottom boundary layer (BBL) parameterisation first introduced by 
-\citet{BeckDos1998} is to allow a direct communication between 
+downslope flows. Such flows arise either downstream of sills such as the Strait of 
+Gibraltar or Denmark Strait, where dense water formed in marginal seas flows 
+into a basin filled with less dense water, or along the continental slope when dense 
+water masses are formed on a continental shelf. The amount of entrainment 
+that occurs in these gravity plumes is critical in determining the density 
+and volume flux of the densest waters of the ocean, such as Antarctic Bottom Water, 
+or North Atlantic Deep Water. $z$-coordinate models tend to overestimate the 
+entrainment, because the gravity flow is mixed vertically by convection 
+as it goes ''downstairs'' following the step topography, sometimes over a thickness 
+much larger than the thickness of the observed gravity plume. A similar problem 
+occurs in the $s$-coordinate when the thickness of the bottom level varies rapidly 
+downstream of 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 
+\citet{Beckmann_Doscher1997}, is to allow a direct communication between 
 two adjacent bottom cells at different levels, whenever the densest water is 
-located above the less dense water. The communication can be by a diffusive 
-(diffusive BBL), advective fluxes (advective BBL), or both. In the current 
+located above the less dense water. The communication can be by a diffusive flux
+(diffusive BBL), an advective flux (advective BBL), or both. In the current 
 implementation of the BBL, only the tracers are modified, not the velocities. 
 Furthermore, it only connects ocean bottom cells, and therefore does not include 
-the improvment proposed by \citet{Campin_Goosse_Tel99}.
+the improvements introduced by \citet{Campin_Goosse_Tel99}.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Diffusive BBL
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Diffusive Bottom Boundary layer (\key{bbl\_diff})}
+\subsection{Diffusive Bottom Boundary layer (\np{nn\_bbl\_ldf}=1)}
 \label{TRA_bbl_diff}
 
-When applying sigma-diffusion (\key{trabbl} is defined), the diffusive flux between 
-two adjacent cells living at the ocean bottom is given by 
+When applying sigma-diffusion (\key{trabbl} defined and \np{nn\_bbl\_ldf} set to 1), 
+the diffusive flux between two adjacent cells at the ocean floor is given by 
 \begin{equation} \label{Eq_tra_bbl_diff}
 {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
 \end{equation} 
 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
-and  $A_l^\sigma $ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
-the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
+and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{Beckmann_Doscher1997}, 
+the latter is prescribed with a spatial dependence, $i.e.$ in the conditional form
 \begin{equation} \label{Eq_tra_bbl_coef}
 A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
@@ -909,48 +945,104 @@
 \end{equation} 
 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist 
-parameter \np{atrbbl}. $A_{bbl}$ is usually set to a value much larger 
-than the one used on lateral mixing in open ocean. 
-Note that in practice, \eqref{Eq_tra_bbl_coef} constraint is applied 
-separately in the two horizontal directions, and the density gradient in 
-\eqref{Eq_tra_bbl_coef} is evaluated at $\overline{H}^i$ ($\overline{H}^j$) 
-using the along bottom mean temperature and salinity. 
+parameter \np{rn\_ahtbbl} and usually set to a value much larger 
+than the one used for lateral mixing in the open ocean. The constraint in \eqref{Eq_tra_bbl_coef} 
+implies that sigma-like diffusion only occurs when the density above the sea floor, at the top of 
+the slope, is larger than in the deeper ocean (see green arrow in Fig.\ref{Fig_bbl}). 
+In practice, this constraint is applied separately in the two horizontal directions, 
+and the density gradient in \eqref{Eq_tra_bbl_coef} is evaluated with the log gradient formulation: 
+\begin{equation} \label{Eq_tra_bbl_Drho}
+	\nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S
+\end{equation} 
+where $\rho$, $\alpha$ and $\beta$ are functions of $\overline{T}^\sigma$, 
+$\overline{S}^\sigma$ and $\overline{H}^\sigma$, the along bottom mean temperature, 
+salinity and depth, respectively.
 
 % -------------------------------------------------------------------------------------------------------------
 %        Advective BBL
 % -------------------------------------------------------------------------------------------------------------
-\subsection   {Advective Bottom Boundary Layer (\key{bbl\_adv})}
+\subsection   {Advective Bottom Boundary Layer  (\np{nn\_bbl\_adv}= 1 or 2)}
 \label{TRA_bbl_adv}
 
+\sgacomment{"downsloping flow" has been replaced by "downslope flow" in the following
+if this is not what is meant then "downwards sloping flow" is also a possibility"}
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!t] \label{Fig_bbl}  \begin{center}
-\includegraphics[width=0.8\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf}
-\caption{Advective Bottom Boundary Layer.}
+\includegraphics[width=0.7\textwidth]{./TexFiles/Figures/Fig_BBL_adv.pdf}
+\caption{Advective/diffusive Bottom Boundary Layer. The BBL parameterisation is 
+activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$. 
+Red arrows indicate the additional overturning circulation due to the advective BBL. 
+The transport of the downslope flow is defined either as the transport of the bottom 
+ocean cell (black arrow), or as a function of the along slope density gradient. 
+The green arrow indicates the diffusive BBL flux directly connecting $kup$ and $kdwn$
+ocean bottom cells.
+connection}
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
+
+%!!      nn_bbl_adv = 1   use of the ocean velocity as bbl velocity
+%!!      nn_bbl_adv = 2   follow Campin and Goosse (1999) implentation
+%!!        i.e. transport proportional to the along-slope density gradient
+
 %%%gmcomment   :  this section has to be really written
 
-The advective BBL is in fact not only an advective one but include a diffusive 
-component as we chose an upstream scheme to perform the advection within 
-the BBL. The associated diffusion only act in the stream direction and is 
-proportional to the velocity.
-
-When applying sigma-advection (\key{trabbl\_adv} defined), the advective 
-flux between two adjacent cells living at the ocean bottom is given by 
-\begin{equation} \label{Eq_tra_bbl_fadv}
-{\rm {\bf F}}_\sigma={\rm {\bf U}}_h^\sigma \; \overline{T}^\sigma
-\end{equation} 
-with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells, 
-and  $A_l^\sigma$ the lateral diffusivity in the BBL. Following \citet{BeckDos1998}, 
-the latter is prescribed with a spatial dependence, $e.g.$ in the conditional form
-\begin{equation} \label{Eq_tra_bbl_Aadv}
-A_l^\sigma (i,j,t)=\left\{ {\begin{array}{l}
- A_{bbl} \quad \quad \mbox{if}     \quad    \nabla_\sigma \rho \cdot \nabla H<0 
-              \quad \quad \mbox{and} \quad         {\rm {\bf U}}_h  \cdot \nabla H<0 \\ 
- \\
- 0\quad \quad \;\,\mbox{otherwise} \\ 
- \end{array}} \right.
-\end{equation} 
+When applying an advective BBL (\np{nn\_bbl\_adv} = 1 or 2), an overturning 
+circulation is added which connects two adjacent bottom grid-points only if dense 
+water overlies less dense water on the slope. The density difference causes dense 
+water to move down the slope. 
+
+\np{nn\_bbl\_adv} = 1 : the downslope velocity is chosen to be the Eulerian
+ocean velocity just above the topographic step (see black arrow in Fig.\ref{Fig_bbl}) 
+\citep{Beckmann_Doscher1997}. It is a \textit{conditional advection}, that is, advection
+is allowed only if dense water overlies less dense water on the slope ($i.e.$ 
+$\nabla_\sigma \rho  \cdot  \nabla H<0$) and if the velocity is directed towards 
+greater depth ($i.e.$ $\vect{U}  \cdot  \nabla H>0$).
+
+\np{nn\_bbl\_adv} = 2 : the downslope velocity is chosen to be proportional to $\Delta \rho$,
+the density difference between the higher cell and lower cell densities \citep{Campin_Goosse_Tel99}.
+The advection is allowed only  if dense water overlies less dense water on the slope ($i.e.$ 
+$\nabla_\sigma \rho  \cdot  \nabla H<0$). For example, the resulting transport of the 
+downslope flow, here in the $i$-direction (Fig.\ref{Fig_bbl}), is simply given by the 
+following expression:
+\begin{equation} \label{Eq_bbl_Utr}
+ u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)
+\end{equation}
+where $\gamma$, expressed in seconds, is the coefficient of proportionality 
+provided as \np{rn\_gambbl}, a namelist parameter, and \textit{kup} and \textit{kdwn} 
+are the vertical index of the higher and lower cells, respectively.
+The parameter $\gamma$ should take a different value for each bathymetric 
+step, but for simplicity, and because no direct estimation of this parameter is 
+available, a uniform value has been assumed. The possible values for $\gamma$ 
+range between 1 and $10~s$ \citep{Campin_Goosse_Tel99}.  
+
+Scalar properties are advected by this additional transport $( u^{tr}_{bbl}, v^{tr}_{bbl} )$ 
+using the upwind scheme. Such a diffusive advective scheme has been chosen 
+to mimic the entrainment between the downslope plume and the surrounding 
+water at intermediate depths. The entrainment is replaced by the vertical mixing 
+implicit in the advection scheme. Let us consider as an example the 
+case displayed in Fig.\ref{Fig_bbl} where the density at level $(i,kup)$ is 
+larger than the one at level $(i,kdwn)$. The advective BBL scheme
+modifies the tracer time tendency of the ocean cells near the 
+topographic step by the downslope flow \eqref{Eq_bbl_dw}, 
+the horizontal \eqref{Eq_bbl_hor}  and the upward \eqref{Eq_bbl_up} 
+return flows as follows: 
+\begin{align} 
+\partial_t T^{do}_{kdw} &\equiv \partial_t T^{do}_{kdw}
+                                     +  \frac{u^{tr}_{bbl}}{{b_t}^{do}_{kdw}}  \left( T^{sh}_{kup} - T^{do}_{kdw} \right)  \label{Eq_bbl_dw} \\
+%
+\partial_t T^{sh}_{kup} &\equiv \partial_t T^{sh}_{kup} 
+				   + \frac{u^{tr}_{bbl}}{{b_t}^{sh}_{kup}}   \left( T^{do}_{kup} - T^{sh}_{kup} \right)   \label{Eq_bbl_hor} \\
+%
+\intertext{and for $k =kdw-1,\;..., \; kup$ :} 
+%
+\partial_t T^{do}_{k} &\equiv \partial_t S^{do}_{k}
+				   + \frac{u^{tr}_{bbl}}{{b_t}^{do}_{k}}   \left( T^{do}_{k+1} - T^{sh}_{k} \right)   \label{Eq_bbl_up}
+\end{align}
+where $b_t$ is the $T$-cell volume. 
+
+Note that the BBL transport, $( u^{tr}_{bbl}, v^{tr}_{bbl} )$, is available in 
+the model outputs. It has to be used to compute the effective velocity 
+as well as the effective overturning circulation.
 
 % ================================================================
@@ -960,6 +1052,6 @@
 		{Tracer damping (\mdl{tradmp})}
 \label{TRA_dmp}
-%--------------------------------------------namtdp-----------------------------------------------------
-\namdisplay{namtdp}
+%--------------------------------------------namtra_dmp-------------------------------------------------
+\namdisplay{namtra_dmp}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -969,5 +1061,4 @@
 \begin{split}
  \frac{\partial T}{\partial t}=\;\cdots \;-\gamma \,\left( {T-T_o } \right)  \\
-\\ 
  \frac{\partial S}{\partial t}=\;\cdots \;-\gamma \,\left( {S-S_o } \right) 
  \end{split}
@@ -976,7 +1067,7 @@
 are given temperature and salinity fields (usually a climatology). 
 The restoring term is added when \key{tradmp} is defined. 
-It also requires that both \key{temdta} and \key{saldta} are defined 
+It also requires that both \key{dtatem} and \key{dtasal} are defined 
 ($i.e.$ that $T_o$ and $S_o$ are read). The restoring coefficient 
-$S_o$ is a three-dimensional array initialized by the user in routine 
+$\gamma$ is a three-dimensional array initialized by the user in routine 
 \rou{dtacof} also located in module \mdl{tradmp}. 
 
@@ -988,22 +1079,28 @@
 field for a passive tracer study). The first case applies to regional 
 models that have artificial walls instead of open boundaries. 
-In the vicinity of these walls, $S_o$ takes large values (equivalent to 
+In the vicinity of these walls, $\gamma$ takes large values (equivalent to 
 a time scale of a few days) whereas it is zero in the interior of the 
 model domain. The second case corresponds to the use of the robust 
 diagnostic method \citep{Sarmiento1982}. It allows us to find the velocity 
-field consistent with the model dynamics whilst having a $T$-$S$ field 
-close to a given climatological field ($T_o -S_o$). The time scale 
+field consistent with the model dynamics whilst having a $T$, $S$ field 
+close to a given climatological field ($T_o$, $S_o$). The time scale 
 associated with $S_o$ is generally not a constant but spatially varying 
 in order to respect other properties. For example, it is usually set to zero 
 in the mixed layer (defined either on a density or $S_o$ criterion) 
-\citep{Madec1996} and in the equatorial region 
-\citep{Reverdin1991, Fujio1991, MartiTh1992} since these two regions 
-have a short time scale of adjustment; while smaller $S_o$ are used 
+\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 $\gamma$ 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{??} and \np{nn\_zdmp}. The former allows us to specify: the 
+width of the equatorial band in which no damping is applied; a decrease 
+in the vicinity of the coast; and a damping everywhere in the Red and Med Seas.
+The latter sets whether damping should act 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 +1110,5 @@
 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 
@@ -1029,18 +1126,24 @@
 %--------------------------------------------------------------------------------------------------------------
 
-The general framework for tracer time stepping is a leap-frog scheme, 
-$i.e.$ a three level centred time scheme associated with a Asselin time 
-filter (cf. \S\ref{DOM_nxt}):
+The general framework for tracer time stepping is a modified leap-frog scheme 
+\citep{Leclair_Madec_OM09}, $i.e.$ a three level centred time scheme associated 
+with a Asselin time filter (cf. \S\ref{STP_mLF}):
 \begin{equation} \label{Eq_tra_nxt}
-\begin{split}
-T^{t+\Delta t} &= T^{t-\Delta t} + 2 \, \Delta t  \ \text{RHS}_T^t 	\\
+\begin{aligned}
+(e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t &	\\
 \\
-T_f^t  \;\ \quad &= T^t \;\quad +\gamma \,\left[ {T_f^{t-\Delta t} -2T^t+T^{t+\Delta t}} \right]
-\end{split}
+(e_{3t}T)_f^t  \;\ \quad &= (e_{3t}T)^t \;\quad 
+                                    &+\gamma \,\left[ {(e_{3t}T)_f^{t-\rdt} -2(e_{3t}T)^t+(e_{3t}T)^{t+\rdt}} \right] &  \\
+                                 & &- \gamma\,\rdt \, \left[ Q^{t+\rdt/2} -  Q^{t-\rdt/2} \right]  &                      
+\end{aligned}
 \end{equation} 
-where $\text{RHS}_T$ is the right hand side of the temperature equation, 
-the subscript $f$ denotes filtered values and $\gamma$ is the Asselin 
-coefficient. $\gamma$ is initialized as \np{atfp} (\textbf{namelist} parameter). 
-Its default value is \np{atfp=0.1}. 
+where RHS is the right hand side of the temperature equation, 
+the subscript $f$ denotes filtered values, $\gamma$ is the Asselin coefficient,
+and $S$ is the total forcing applied on $T$ ($i.e.$ fluxes plus content in mass exchanges). 
+$\gamma$ is initialized as \np{rn\_atfp} (\textbf{namelist} parameter). 
+Its default value is \np{rn\_atfp}=$10^{-3}$. Note that the forcing correction term in the filter
+is not applied in linear free surface (\jp{lk\_vvl}=false) (see \S\ref{TRA_sbc}.
+Not also that in constant volume case, the time stepping is performed on $T$, 
+not on its content, $e_{3t}T$.
 
 When the vertical mixing is solved implicitly, the update of the \textit{next} tracer 
@@ -1049,5 +1152,5 @@
 
 In order to prepare for the computation of the \textit{next} time step, 
-a swap of tracer arrays is performed: $T^{t-\Delta t} = T^t$ and $T^t = T_f$. 
+a swap of tracer arrays is performed: $T^{t-\rdt} = T^t$ and $T^t = T_f$. 
 
 % ================================================================
@@ -1064,54 +1167,59 @@
 %        Equation of State
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Equation of State (\np{neos} = 0, 1 or 2)}
+\subsection{Equation of State (\np{nn\_eos} = 0, 1 or 2)}
 \label{TRA_eos}
 
 It is necessary to know the equation of state for the ocean very accurately 
 to determine stability properties (especially the Brunt-Vais\"{a}l\"{a} frequency), 
-particularly in the deep ocean. The ocean density is a non linear empirical 
-function of \textit{in situ }temperature, salinity and pressure. The reference 
-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 +1227,5 @@
 %        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 +1236,7 @@
  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 +1245,7 @@
 	   \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 +1253,5 @@
 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 +1264,5 @@
 %        Specific Heat
 % -------------------------------------------------------------------------------------------------------------
-\subsection   [Specific Heat (\textit{phycst})]
+\subsection   	[Specific Heat (\textit{phycst})]
 			{Specific Heat (\mdl{phycst})}
 \label{TRA_adv_ldf}
@@ -1171,14 +1277,10 @@
 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 +1296,7 @@
 \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}.  
 
 % ================================================================
@@ -1214,5 +1309,5 @@
 \gmcomment{STEVEN: to be consistent with earlier discussion of differencing and averaging operators, I've changed "derivative" to "difference" and "mean" to "average"}
 
-With partial bottom cells (\np{ln\_zps}=.true.), in general, tracers in horizontally 
+With partial bottom cells (\np{ln\_zps}=true), in general, tracers in horizontally 
 adjacent cells live at different depths. Horizontal gradients of tracers are needed 
 for horizontal diffusion (\mdl{traldf} module) and for the hydrostatic pressure 
@@ -1228,5 +1323,5 @@
 \begin{figure}[!p] \label{Fig_Partial_step_scheme}  \begin{center}
 \includegraphics[width=0.9\textwidth]{./TexFiles/Figures/Partial_step_scheme.pdf}
-\caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=.true.) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
+\caption{ Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate (\np{ln\_zps}=true) in the case $( e3w_k^{i+1} - e3w_k^i  )>0$. A linear interpolation is used to estimate $\widetilde{T}_k^{i+1}$, the tracer value at the depth of the shallower tracer point of the two adjacent bottom $T$-points. The horizontal difference is then given by: $\delta _{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.  }
 \end{center}   \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_ZDF.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Chap_ZDF.tex	(revision 2282)
@@ -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
 
 
@@ -38,7 +40,7 @@
 general trend in the \mdl{dynzdf} and \mdl{trazdf} modules, respectively. 
 These trends can be computed using either a forward time stepping scheme 
-(namelist parameter \np{np\_zdfexp}=true) or a backward time stepping 
-scheme (\np{np\_zdfexp}=false) depending on the magnitude of the mixing 
-coefficients, and thus of the formulation used (see \S\ref{DOM_nxt}).
+(namelist parameter \np{ln\_zdfexp}=true) or a backward time stepping 
+scheme (\np{ln\_zdfexp}=false) depending on the magnitude of the mixing 
+coefficients, and thus of the formulation used (see \S\ref{STP}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -57,9 +59,8 @@
 \begin{align*} 
 A_u^{vm} = A_v^{vm} &= 1.2\ 10^{-4}~m^2.s^{-1} 	\\
-\\
 A^{vT} = A^{vS} &= 1.2\ 10^{-5}~m^2.s^{-1}
 \end{align*}
 
-These values are set through the \np{avm0} and \np{avt0} namelist parameters. 
+These values are set through the \np{rn\_avm0} and \np{rn\_avt0} namelist parameters. 
 In all cases, do not use values smaller that those associated with the molecular 
 viscosity and diffusivity, that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, 
@@ -74,5 +75,5 @@
 
 %--------------------------------------------namric---------------------------------------------------------
-\namdisplay{namric}
+\namdisplay{namzdf_ric}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -84,10 +85,9 @@
 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}
    \left\{      \begin{aligned}
          A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
-         \\
          A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
    \end{aligned}    \right.
@@ -99,5 +99,5 @@
 is the maximum value that can be reached by the coefficient when $Ri\leq 0$, 
 $a=5$ and $n=2$. The last three values can be modified by setting the 
-\np{avmri}, \np{alp} and \np{nric} namelist parameters, respectively.
+\np{rn\_avmri}, \np{rn\_alp} and \np{nn\_ric} namelist parameters, respectively.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -107,24 +107,24 @@
 \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
-							+\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
--A^{vT}\,N^2
+\frac{K_m}{{e_3}^2 }\;\left[ {\left( {\frac{\partial u}{\partial k}} \right)^2
+				        +\left( {\frac{\partial v}{\partial k}} \right)^2} \right]
+-K_\rho\,N^2
 +\frac{1}{e_3}	\;\frac{\partial }{\partial k}\left[ {\frac{A^{vm}}{e_3 }
 				\;\frac{\partial \bar{e}}{\partial k}} \right]
@@ -133,15 +133,16 @@
 \begin{equation} \label{Eq_zdftke_kz}
    \begin{split}
-         A^{vm} &= C_k\  l_k\  \sqrt {\bar{e}}  	\\
-         A^{vT} &= A^{vm} / P_{rt}
+         K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }  	\\
+         K_\rho &= A^{vm} / P_{rt}
    \end{split}
 \end{equation}
 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, $K_m$ and $K_\rho$ are the vertical eddy viscosity 
+and diffusivity coefficients. 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 +152,22 @@
             \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 +185,5 @@
 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 +195,16 @@
 \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}, 
-$i.e.$ $l^{(k)} = \sqrt {2 \bar e^{(k)} / N^{(k)} }$.
-
-In the \np{nmxl}=2 case, the dissipation and mixing length scales take the same 
+$i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
+
+In the \np{nn\_mxl}=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{nn\_mxl}=2 case, the dissipation and mixing turbulent length scales are give 
 as in \citet{Gaspar1990}:
 \begin{equation} \label{Eq_tke_mxl_gaspar}
@@ -217,18 +216,220 @@
 
 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 :  $(A^{vT} = 10^{-3} / N)$. 
-In addition, a cut-off is applied on $A^{vm}$ and $A^{vT}$ to avoid numerical 
+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 :  $K_\rho = 10^{-3} / N$. 
+In addition, a cut-off is applied on $K_m$ and $K_\rho$ 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{rn\_avm0} and \np{rn\_avt0} (namzdf namelist, see \S\ref{ZDF_cst}).
+
+% -------------------------------------------------------------------------------------------------------------
+%        TKE Turbulent Closure Scheme : new organization to energetic considerations
+% -------------------------------------------------------------------------------------------------------------
+\subsection{TKE discretization considerations (\key{zdftke})}
+\label{ZDF_tke_ene}
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\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}). To do so a special care 
+have to be taken for both the time and space discretization of the TKE equation \citep{Burchard_OM02}.
+
+Let us first address the time stepping issue. Fig.~\ref{Fig_TKE_time_scheme} shows 
+how the two-level Leap-Frog time stepping of the momentum and tracer equations interplays 
+with the one-level forward time stepping of TKE equation. With this framework, the total loss 
+of kinetic energy (in 1D for the demonstration) due to the vertical momentum diffusion is 
+obtained by multiplying this quantity by $u^t$ and summing the result vertically:   
+\begin{equation} \label{Eq_energ1}
+\begin{split}
+\int_{-H}^{\eta}  u^t \,\partial_z &\left( {K_m}^t \,(\partial_z u)^{t+\rdt}  \right) \,dz   \\
+&= \Bigl[  u^t \,{K_m}^t \,(\partial_z u)^{t+\rdt} \Bigr]_{-H}^{\eta}          
+ - \int_{-H}^{\eta}{ {K_m}^t \,\partial_z{u^t} \,\partial_z u^{t+\rdt} \,dz }
+\end{split}
+\end{equation}
+Here, the vertical diffusion of momentum is discretized backward in time 
+with a coefficient, $K_m$, known at time $t$ (Fig.~\ref{Fig_TKE_time_scheme}), 
+as it is required when using the TKE scheme (see \S\ref{STP_forward_imp}). 
+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. It is the dissipation rate of kinetic energy, 
+and thus minus the shear production rate of $\bar{e}$. \eqref{Eq_energ1} 
+implies that, to be energetically consistent, the production rate of $\bar{e}$ 
+used to compute $(\bar{e})^t$ (and thus ${K_m}^t$) should be expressed as 
+${K_m}^{t-\rdt}\,(\partial_z u)^{t-\rdt} \,(\partial_z u)^t$ (and not by the more straightforward 
+$K_m \left( \partial_z u \right)^2$ expression taken at time $t$ or $t-\rdt$).
+
+A similar consideration applies on the destruction rate of $\bar{e}$ due to stratification 
+(second term of the right hand side of \eqref{Eq_zdftke_e}). This term 
+must balance the input of potential energy resulting from vertical mixing. 
+The rate of change of potential energy (in 1D for the demonstration) due vertical 
+mixing is obtained by multiplying vertical density diffusion 
+tendency by $g\,z$ and and summing the result vertically:
+\begin{equation} \label{Eq_energ2}
+\begin{split}
+\int_{-H}^{\eta} g\,z\,\partial_z &\left( {K_\rho}^t \,(\partial_k \rho)^{t+\rdt}   \right) \,dz    \\
+&= \Bigl[  g\,z \,{K_\rho}^t \,(\partial_z \rho)^{t+\rdt} \Bigr]_{-H}^{\eta}  
+   - \int_{-H}^{\eta}{ g \,{K_\rho}^t \,(\partial_k \rho)^{t+\rdt} } \,dz   \\
+&= - \Bigl[  z\,{K_\rho}^t \,(N^2)^{t+\rdt} \Bigr]_{-H}^{\eta}
++ \int_{-H}^{\eta}{  \rho^{t+\rdt} \, {K_\rho}^t \,(N^2)^{t+\rdt} \,dz  }
+\end{split}
+\end{equation}
+where we use $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 
+The first term of the right hand side of \eqref{Eq_energ2}  is always zero 
+because there is no diffusive flux through the ocean surface and bottom). 
+The second term is minus the destruction rate of  $\bar{e}$ due to stratification. 
+Therefore \eqref{Eq_energ1} implies that, to be energetically consistent, the product 
+${K_\rho}^{t-\rdt}\,(N^2)^t$ should be used in \eqref{Eq_zdftke_e}, the TKE equation.
+
+Let us now address the space discretization issue. 
+The vertical eddy coefficients are defined at $w$-point whereas the horizontal velocity 
+components are in the centre of the side faces of a $t$-box in staggered C-grid 
+(Fig.\ref{Fig_cell}). A space averaging is thus required to obtain the shear TKE production term.
+By redoing the \eqref{Eq_energ1} in the 3D case, it can be shown that the product of 
+eddy coefficient by the shear at $t$ and $t-\rdt$ must be performed prior to the averaging.
+Furthermore, the possible time variation of $e_3$ (\key{vvl} case) have to be taken into 
+account.
+
+The above energetic considerations leads to 
+the following final discrete form for the TKE equation:
+\begin{equation} \label{Eq_zdftke_ene}
+\begin{split}
+\frac { (\bar{e})^t - (\bar{e})^{t-\rdt} } {\rdt}  \equiv  
+\Biggl\{ \Biggr.
+  &\overline{ \left( \left(\overline{K_m}^{\,i+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[u^{t+\rdt}]}{{e_3u}^{t+\rdt} } 
+                                                                              \ \frac{\delta_{k+1/2}[u^ t         ]}{{e_3u}^ t          }  \right) }^{\,i} \\
++&\overline{  \left( \left(\overline{K_m}^{\,j+1/2}\right)^{t-\rdt} \,\frac{\delta_{k+1/2}[v^{t+\rdt}]}{{e_3v}^{t+\rdt} } 
+                                                                               \ \frac{\delta_{k+1/2}[v^ t         ]}{{e_3v}^ t          }  \right) }^{\,j} 
+\Biggr. \Biggr\}   \\
+%
+- &{K_\rho}^{t-\rdt}\,{(N^2)^t}    \\
+%
++&\frac{1}{{e_3w}^{t+\rdt}}  \;\delta_{k+1/2} \left[   {K_m}^{t-\rdt} \,\frac{\delta_{k}[(\bar{e})^{t+\rdt}]} {{e_3w}^{t+\rdt}}   \right]   \\
+%
+- &c_\epsilon \; \left( \frac{\sqrt{\bar {e}}}{l_\epsilon}\right)^{t-\rdt}\,(\bar {e})^{t+\rdt}
+\end{split}
+\end{equation}
+where the last two terms in \eqref{Eq_zdftke_ene} (vertical diffusion and Kolmogorov dissipation) 
+are time stepped using a backward scheme (see\S\ref{STP_forward_imp}). 
+Note that the Kolmogorov term has been linearized in time in order to render 
+the implicit computation possible. The restart of the TKE scheme 
+requires the storage of $\bar {e}$, $K_m$, $K_\rho$ and $l_\epsilon$ as they all appear in 
+the right hand side of \eqref{Eq_zdftke_ene}. For the latter, it is in fact 
+the ratio $\sqrt{\bar{e}}/l_\epsilon$ which is stored. 
+
+% -------------------------------------------------------------------------------------------------------------
+%        GLS Generic Length Scale Scheme 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{GLS Generic Length Scale (\key{zdfgls})}
+\label{ZDF_gls}
+
+%--------------------------------------------namgls---------------------------------------------------------
+\namdisplay{namgls}
+%--------------------------------------------------------------------------------------------------------------
+
+The Generic Length Scale (GLS) scheme is a turbulent closure scheme based on 
+two prognostic equations: one for the turbulent kinetic energy $\bar {e}$, and another 
+for the generic length scale, $\psi$ \citep{Umlauf_Burchard_JMS03, Umlauf_Burchard_CSR05}. 
+This later variable is defined as : $\psi = {C_{0\mu}}^{p} \ {\bar{e}}^{m} \ l^{n}$, 
+where the triplet $(p, m, n)$ value given in Tab.\ref{Tab_GLS} allows to recover 
+a number of well-known turbulent closures ($k$-$kl$ \citep{Mellor_Yamada_1982}, 
+$k$-$\epsilon$ \citep{Rodi_1987}, $k$-$\omega$ \citep{Wilcox_1988} 
+among others \citep{Umlauf_Burchard_JMS03,Kantha_Carniel_CSR05}). 
+The GLS scheme is given by the following set of equations:
+\begin{equation} \label{Eq_zdfgls_e}
+\frac{\partial \bar{e}}{\partial t} = 
+\frac{K_m}{\sigma_e e_3 }\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
+                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right]
+-K_\rho \,N^2
++\frac{1}{e_3}\,\frac{\partial}{\partial k} \left[ \frac{K_m}{e_3}\,\frac{\partial \bar{e}}{\partial k} \right]
+- \epsilon
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_psi}
+   \begin{split}
+\frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
+\frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
+                                                                   +\left( \frac{\partial v}{\partial k} \right)^2} \right]
+- C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
+&+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
+                                  \;\frac{\partial \psi}{\partial k}} \right]\;
+   \end{split}
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_kz}
+   \begin{split}
+         K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
+         K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
+   \end{split}
+\end{equation}
+
+\begin{equation} \label{Eq_zdfgls_eps}
+{\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
+\end{equation}
+where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \S\ref{TRA_bn2}) 
+and $\epsilon$ the dissipation rate. 
+The constants $C_1$, $C_2$, $C_3$, ${\sigma_e}$, ${\sigma_{\psi}}$ and the wall function ($Fw$) 
+depends of the choice of the turbulence model. Four different turbulent models are pre-defined 
+(Tab.\ref{Tab_GLS}). They are made available through th \np{gls} namelist parameter. 
+
+%--------------------------------------------------TABLE--------------------------------------------------
+\begin{table}[htbp]  \label{Tab_GLS}
+\begin{center}
+%\begin{tabular}{cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}cp{70pt}c}
+\begin{tabular}{ccccc}
+          	             &   $k-kl$   & $k-\epsilon$ & $k-\omega$ &   generic   \\  
+%                        & \citep{Mellor_Yamada_1982} &  \citep{Rodi_1987}       & \citep{Wilcox_1988} &                 \\  
+\hline  \hline 
+\np{nn\_clo}     & \textbf{0} &   \textbf{1}  &   \textbf{2}   &    \textbf{3}   \\  
+\hline 
+$( p , n , m )$	       &   ( 0 , 1 , 1 )   & ( 3 , 1.5 , -1 )   & ( -1 , 0.5 , -1 )    &  ( 2 , 1 , -0.67 )  \\
+$\sigma_k$      &    2.44         &     1.              &      2.                &      0.8          \\
+$\sigma_\psi$  &    2.44         &     1.3            &      2.                 &       1.07       \\
+$C_1$              &      0.9         &     1.44          &      0.555          &       1.           \\
+$C_2$              &      0.5         &     1.92          &      0.833          &       1.22       \\
+$C_3$              &      1.           &     1.              &      1.                &       1.           \\
+$F_{wall}$        &      Yes        &       --             &     --                  &      --          \\
+\hline
+\hline
+\end{tabular}
+\caption {Set of predefined GLS parameters, or equivalently predefined turbulence models available with \key{gls} and controlled by the \np{nn\_clos} namelist parameter.}
+\end{center}
+\end{table}
+%--------------------------------------------------------------------------------------------------------------
+
+In the Mellor-Yamada model, the negativity of $n$ allows to use a wall function to force
+the convergence of the mixing length towards $K\,z_b$ ($K$: Kappa and $z_b$: rugosity length) 
+value near physical boundaries (logarithmic boundary layer law). $C_{\mu}$ and $C_{\mu'}$ 
+are calculated from stability function proposed by \citet{Galperin_al_JAS88}, or by \citet{Kantha_Clayson_1994} 
+or one of the two functions suggested by \citet{Canuto_2001}  (\np{nn\_stab\_func} = 0, 1, 2 or 3, resp.}). The value of $C_{0\mu}$ depends of the choice of the stability function.
+
+The surface and bottom boundary condition on both $\bar{e}$ and $\psi$ can be calculated 
+thanks to Dirichlet or Neumann condition through \np{nn\_tkebc\_surf} and \np{nn\_tkebc\_bot}, resp. 
+The wave effect on the mixing could be also being considered \citep{Craig_Banner_1994}.
+
+The $\psi$ equation is known to fail in stably stratified flows, and for this reason 
+almost all authors apply a clipping of the length scale as an \textit{ad hoc} remedy. 
+With this clipping, the maximum permissible length scale is determined by 
+$l_{max} = c_{lim} \sqrt{2\bar{e}}/ N$. A value of $c_{lim} = 0.53$ is often used 
+\citep{Galperin_al_JAS88}. \cite{Umlauf_Burchard_CSR05} show that the value of 
+the clipping factor is of crucial importance for the entrainment depth predicted in 
+stably stratified situations, and that its value has to be chosen in accordance 
+with the algebraic model for the turbulent ßuxes. The clipping is only activated 
+if \np{ln\_length\_lim}=true, and the $c_{lim}$ is set to the \np{clim\_galp} value.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -239,9 +440,8 @@
 
 %--------------------------------------------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.}
@@ -274,8 +474,7 @@
 \label{ZDF_npc}
 
-%--------------------------------------------namnpc--------------------------------------------------------
-\namdisplay{namnpc}
-%--------------------------------------------------------------------------------------------------------------
-
+%--------------------------------------------namzdf--------------------------------------------------------
+\namdisplay{namzdf}
+%--------------------------------------------------------------------------------------------------------------
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -295,8 +494,8 @@
 
 The non-penetrative convective adjustment is used when \np{ln\_zdfnpc}=true. 
-It is applied at each \np{nnpc1} time step and mixes downwards instantaneously 
+It is applied at each \np{nn\_npc} time step and mixes downwards instantaneously 
 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 +519,5 @@
 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 +540,5 @@
 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}.
 
 % -------------------------------------------------------------------------------------------------------------
@@ -347,5 +546,5 @@
 % -------------------------------------------------------------------------------------------------------------
 \subsection   [Enhanced Vertical Diffusion (\np{ln\_zdfevd})]
-			{Enhanced Vertical Diffusion (\np{ln\_zdfevd}=.true.)}
+			{Enhanced Vertical Diffusion (\np{ln\_zdfevd}=true)}
 \label{ZDF_evd}
 
@@ -357,16 +556,20 @@
 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}. 
-This is done either on tracers only (\np{n\_evdm}=0) or on both momentum and 
-tracers (\np{n\_evdm}=1).
+($i.e.$ when $N^2$ the Brunt-Vais\"{a}l\"{a} frequency is negative) 
+\citep{Lazar_PhD97, Lazar_al_JPO99}. This is done either on tracers only 
+(\np{nn\_evdm}=0) or on both momentum and tracers (\np{nn\_evdm}=1).
 
 In practice, where $N^2\leq 10^{-12}$, $A_T^{vT}$ and $A_T^{vS}$, and 
-if \np{n\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ 
-values also, are set equal to the namelist parameter \np{avevd}. A typical value 
-for $avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of 
+if \np{nn\_evdm}=1, the four neighbouring $A_u^{vm} \;\mbox{and}\;A_v^{vm}$ 
+values also, are set equal to the namelist parameter \np{rn\_avevd}. A typical value 
+for $rn\_avevd$ is between 1 and $100~m^2.s^{-1}$. This parameterisation of 
 convective processes is less time consuming than the convective adjustment 
 algorithm presented above when mixing both tracers and momentum in the 
 case of static instabilities. It requires the use of an implicit time stepping on 
 vertical diffusion terms (i.e. \np{ln\_zdfexp}=false). 
+
+Note that the stability test is performed on both \textit{before} and \textit{now} 
+values of $N^2$. This removes a potential source of divergence of odd and
+even time step in a leapfrog environment \citep{Leclair_PhD2010} (see \S\ref{STP_mLF}).
 
 % -------------------------------------------------------------------------------------------------------------
@@ -394,5 +597,5 @@
 The KPP turbulent closure scheme already includes enhanced vertical diffusion 
 in the case of convection, as governed by the variables $bvsqcon$ and $difcon$ 
-found in \mdl{zdfkpp}, therefore \np{np\_zdfevd} should not be used with the KPP 
+found in \mdl{zdfkpp}, therefore \np{ln\_zdfevd}=false should be used with the KPP 
 scheme. %gm%  + one word on non local flux with KPP scheme trakpp.F90 module...
 
@@ -404,6 +607,6 @@
 \label{ZDF_ddm}
 
-%-------------------------------------------namddm--------------------------------------------------------
-\namdisplay{namddm}
+%-------------------------------------------namzdf_ddm-------------------------------------------------
+\namdisplay{namzdf_ddm}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -422,6 +625,5 @@
 \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 +656,5 @@
 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}
@@ -481,6 +682,5 @@
 % Bottom Friction
 % ================================================================
-\section  [Bottom Friction (\textit{zdfbfr})]
-		{Bottom Friction (\mdl{zdfbfr} module)}
+\section  [Bottom Friction (\textit{zdfbfr})]   {Bottom Friction (\mdl{zdfbfr} module)}
 \label{ZDF_bfr}
 
@@ -493,9 +693,9 @@
 diffusive flux. For the bottom boundary layer, one has:
 \begin{equation} \label{Eq_zdfbfr_flux}
-A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h
-\end{equation}
-where $\textbf{F}_h$ is supposed to represent the horizontal momentum flux 
+A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
+\end{equation}
+where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum 
 outside the logarithmic turbulent boundary layer (thickness of the order of 
-1~m in the ocean). How $\textbf{F}_h$ influences the interior depends on the 
+1~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the 
 vertical resolution of the model near the bottom relative to the Ekman layer 
 depth. For example, in order to obtain an Ekman layer depth 
@@ -509,66 +709,78 @@
 this, consider the equation for $u$ at $k$, the last ocean level:
 \begin{equation} \label{Eq_zdfbfr_flux2}
-\frac{\partial u \; (k)}{\partial t} = \frac{1}{e_{3u}} \left[ A^{vm} \; (k) \frac{U \; (k-1) - U \; (k)}{e_{3uw} \; (k-1)} - F_u \right] \approx - \frac{F_u}{e_{3u}}
-\end{equation}
-For example, if the bottom layer thickness is 200~m, the Ekman transport will 
+\frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}}
+\end{equation}
+If the bottom layer thickness is 200~m, the Ekman transport will 
 be distributed over that depth. On the other hand, if the vertical resolution 
 is high (1~m or less) and a turbulent closure model is used, the turbulent 
 Ekman layer will be represented explicitly by the model. However, the 
 logarithmic layer is never represented in current primitive equation model 
-applications: it is \emph{necessary} to parameterize the flux $\textbf{F}_h $. 
+applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $. 
 Two choices are available in \NEMO: a linear and a quadratic bottom friction. 
 Note that in both cases, the rotation between the interior velocity and the 
 bottom friction is neglected in the present release of \NEMO.
 
+In the code, the bottom friction is imposed by adding the trend due to the bottom 
+friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface 
+pressure gradient algorithm, the momentum trend due to the barotropic component 
+needs to be handled separately. For this purpose it is convenient to compute and 
+store coefficients which can be simply combined with bottom velocities and geometric 
+values to provide the momentum trend due to bottom friction. 
+These coefficients are computed in \mdl{zdfbfr} and generally take the form 
+$c_b^{\textbf U}$ where:
+\begin{equation} \label{Eq_zdfbfr_bdef}
+\frac{\partial {\textbf U_h}}{\partial t} = 
+  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b
+\end{equation}
+where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity.
+
 % -------------------------------------------------------------------------------------------------------------
 %       Linear Bottom Friction
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Linear Bottom Friction (\np{nbotfr} = 1) }
+\subsection{Linear Bottom Friction (\np{nn\_botfr} = 0 or 1) }
 \label{ZDF_bfr_linear}
 
-The linear bottom friction parameterisation assumes that the bottom friction 
-is proportional to the interior velocity (i.e. the velocity of the last model level):
+The linear bottom friction parameterisation (including the special case 
+of a free-slip condition) assumes that the bottom friction 
+is proportional to the interior velocity (i.e. the velocity of the last 
+model level):
 \begin{equation} \label{Eq_zdfbfr_linear}
-\textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
-\end{equation}
-where $\textbf{U}_h^b$ is the horizontal velocity vector of the bottom ocean 
-layer and $r$ is a friction coefficient expressed in m.s$^{-1}$. This coefficient 
-is generally estimated by setting a typical decay time $\tau$ in the deep ocean, 
+{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
+\end{equation}
+where $r$ is a friction coefficient expressed in ms$^{-1}$. 
+This coefficient 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 
 approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, 
 Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed 
-of tidal currents of $U_{av} =0.1$~m.s$^{-1}$, and assuming an ocean depth 
-$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m.s$^{-1}$. 
+of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth 
+$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. 
 This is the default value used in \NEMO. It corresponds to a decay time scale 
-of 115~days. It can be changed by specifying \np{bfric1} (namelist parameter).
-
-In the code, the bottom friction is imposed by updating the value of the 
-vertical eddy coefficient at the bottom level. Indeed, the discrete formulation 
-of (\ref{Eq_zdfbfr_linear}) at the last ocean $T-$level, using the fact that 
-$\textbf {U}_h =0$ below the ocean floor, leads to
-\begin{equation} \label{Eq_zdfbfr_linKz}
+of 115~days. It can be changed by specifying \np{rn\_bfric1} (namelist parameter).
+
+For the linear friction case the coefficients defined in the general 
+expression \eqref{Eq_zdfbfr_bdef} are: 
+\begin{equation} \label{Eq_zdfbfr_linbfr_b}
 \begin{split}
-A_u^{vm} &= r\;e_{3uw}\\
-A_v^{vm} &= r\;e_{3vw}\\
+ c_b^u &= - r\\
+ c_b^v &= - r\\
 \end{split}
 \end{equation}
-
-This update is done in \mdl{zdfbfr} when \np{nbotfr}=1. The value of $r$ 
-used is \np{bfric1}. Setting \np{nbotfr}=3 is equivalent to setting $r=0$ and 
-leads to a free-slip bottom boundary condition. Setting \np{nbotfr}=0 sets 
-$r=2\;A_{vb}^{\rm {\bf U}} $, where $A_{vb}^{\rm {\bf U}} $ is the background 
-vertical eddy coefficient, and a no-slip boundary condition is imposed. 
-Note that this latter choice generally leads to an underestimation of the 
-bottom friction: for example with a deepest level thickness of $200~m$ 
-and $A_{vb}^{\rm {\bf U}} =10^{-4}$m$^2$.s$^{-1}$, the friction coefficient 
-is only $r=10^{-6}$m.s$^{-1}$.
+When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfric1}. 
+Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip 
+bottom boundary condition. These values are assigned in \mdl{zdfbfr}. 
+From v3.2 onwards there is support for local enhancement of these values 
+via an externally defined 2D mask array (\np{ln\_bfr2d}=true) given
+in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1. 
+Locations with a non-zero mask value will have the friction coefficient increased 
+by $mask\_value$*\np{rn\_bfrien}*\np{rn\_bfric1}.
 
 % -------------------------------------------------------------------------------------------------------------
 %       Non-Linear Bottom Friction
 % -------------------------------------------------------------------------------------------------------------
-\subsection{Non-Linear Bottom Friction (\np{nbotfr} = 2)}
+\subsection{Non-Linear Bottom Friction (\np{nn\_botfr} = 2)}
 \label{ZDF_bfr_nonlinear}
 
@@ -576,31 +788,254 @@
 friction is quadratic: 
 \begin{equation} \label{Eq_zdfbfr_nonlinear}
-\textbf {F}_h = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
+{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h 
 }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b 
 \end{equation}
-
-with $\textbf{U}_h^b = (u_b\;,\;v_b)$ the horizontal interior velocity ($i.e.$ 
-the horizontal velocity of the bottom ocean layer), $C_D$ a drag coefficient, 
-and $e_b $ a bottom turbulent kinetic energy due to tides, internal waves 
-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$^{-2}$, 
-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 
-set as default values (\np{bfric2} and \np{bfeb2} namelist parameters).
+where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy 
+due to tides, internal waves 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{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 $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. 
+The CME choices have been set as default values (\np{rn\_bfric2} and \np{rn\_bfeb2} 
+namelist parameters).
 
 As for the linear case, the bottom friction is imposed in the code by 
-updating the value of the vertical eddy coefficient at the bottom level:
-\begin{equation} \label{Eq_zdfbfr_nonlinKz}
+adding the trend due to the bottom friction to the general momentum trend 
+in \mdl{dynbfr}.
+For the non-linear friction case the terms
+computed in \mdl{zdfbfr}  are: 
+\begin{equation} \label{Eq_zdfbfr_nonlinbfr}
 \begin{split}
-A_u^{vm} &=C_D\; e_{3uw} \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^
-{1/2}\\
-A_v^{vm} &=C_D\; e_{3uw} \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\
+ c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\
+ c_b^v &= - \; C_D\;\left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\
 \end{split}
 \end{equation}
 
-This update is done in \mdl{zdfbfr}. The coefficients that control the strength of the 
-non-linear bottom friction are initialized as namelist parameters: $C_D$= \np{bfri2}, 
-and $e_b$ =\np{bfeb2}.
-
-% ================================================================
+The coefficients that control the strength of the non-linear bottom friction are 
+initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. 
+Note for applications which treat tides explicitly a low or even zero value of 
+\np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ 
+is possible via an externally defined 2D mask array (\np{ln\_bfr2d}=true). 
+See previous section for details.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Bottom Friction stability
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Bottom Friction stability considerations}
+\label{ZDF_bfr_stability}
+
+Some care needs to exercised over the choice of parameters to ensure that the
+implementation of bottom friction does not induce numerical instability. For 
+the purposes of stability analysis, an approximation to \eqref{Eq_zdfbfr_flux2}
+is:
+\begin{equation} \label{Eqn_bfrstab}
+\begin{split}
+ \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
+               &= -\frac{ru}{e_{3u}}\;2\rdt\\
+\end{split}
+\end{equation}
+\noindent where linear bottom friction and a leapfrog timestep have been assumed. 
+To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have:
+\begin{equation}
+ |\Delta u| < \;|u| 
+\end{equation}
+\noindent which, using \eqref{Eqn_bfrstab}, gives:
+\begin{equation}
+r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
+\end{equation}
+This same inequality can also be derived in the non-linear bottom friction case 
+if a velocity of 1 m.s$^{-1}$ is assumed. Alternatively, this criterion can be 
+rearranged to suggest a minimum bottom box thickness to ensure stability:
+\begin{equation}
+e_{3u} > 2\;r\;\rdt
+\end{equation}
+\noindent which it may be necessary to impose if partial steps are being used. 
+For example, if $|u| = 1$ m.s$^{-1}$, $rdt = 1800$ s, $r = 10^{-3}$ then
+$e_{3u}$ should be greater than 3.6 m. For most applications, with physically
+sensible parameters these restrictions should not be of concern. But 
+caution may be necessary if attempts are made to locally enhance the bottom
+friction parameters. 
+To ensure stability limits are imposed on the bottom friction coefficients both during 
+initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr} 
+(assuming a 1 m.s$^{-1}$ velocity in the non-linear case).
+The number of breaches of the stability criterion are reported as well as the minimum 
+and maximum values that have been set. The criterion is also checked at each time step, 
+using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are 
+reduced as necessary to ensure stability; these changes are not reported.
+
+% -------------------------------------------------------------------------------------------------------------
+%       Bottom Friction with split-explicit time splitting
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Bottom Friction with split-explicit time splitting}
+\label{ZDF_bfr_ts}
+
+When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the
+bottom velocity at the before time step is used. This velocity includes both the
+baroclinic and barotropic components which is appropriate when using either the
+explicit or filtered surface pressure gradient algorithms (\key{dynspg\_exp} or 
+{\key{dynspg\_flt}). Extra attention is required, however, when using 
+split-explicit time stepping (\key{dynspg\_ts}). In this case the free surface 
+equation is solved with a small time step \np{nn\_baro}*\np{rn\_rdt}, while the three 
+dimensional prognostic variables are solved with a longer time step that is a 
+multiple of \np{rn\_rdt}. The trend in the barotropic momentum due to bottom 
+friction appropriate to this method is that given by the selected parameterisation 
+($i.e.$ linear or non-linear bottom friction) computed with the evolving velocities 
+at each barotropic timestep. 
+
+In the case of non-linear bottom friction, we have elected to partially linearise 
+the problem by keeping the coefficients fixed throughout the barotropic 
+time-stepping to those computed in \mdl{zdfbfr} using the now timestep. 
+This decision allows an efficient use of the $c_b^{\vect{U}}$ coefficients to:
+
+\begin{enumerate}
+\item On entry to \rou{dyn\_spg\_ts}, remove the contribution of the before
+barotropic velocity to the bottom friction component of the vertically
+integrated momentum trend. Note the same stability check that is carried out 
+on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to
+ensure that the trend removed matches that which was added in \mdl{dynbfr}.
+\item At each barotropic step, compute the contribution of the current barotropic
+velocity to the trend due to bottom friction. Add this contribution to the
+vertically integrated momentum trend. This contribution is handled implicitly which
+eliminates the need to impose a stability criteria on the values of the bottom friction
+coefficient within the barotropic loop. 
+\end{enumerate}
+
+Note that the use of an implicit formulation
+for the bottom friction trend means that any limiting of the bottom friction coefficient 
+in \mdl{dynbfr} does not adversely affect the solution when using split-explicit time 
+splitting. This is because the major contribution to bottom friction is likely to come from 
+the barotropic component which uses the unrestricted value of the coefficient.
+
+The implicit formulation takes the form:
+\begin{equation} \label{Eq_zdfbfr_implicitts}
+ \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ]  
+\end{equation}
+where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height), 
+$c_b^u$ is the bottom friction coefficient as calculated in \rou{zdf\_bfr} and $RHS$ represents 
+all the components to the vertically integrated momentum trend except for that due to bottom friction.
+
+
+
+
+% ================================================================
+% Tidal Mixing
+% ================================================================
+\section{Tidal Mixing}
+\label{ZDF_tmx}
+
+%--------------------------------------------namzdf_tmx--------------------------------------------------
+\namdisplay{namzdf_tmx}
+%--------------------------------------------------------------------------------------------------------------
+
+
+% -------------------------------------------------------------------------------------------------------------
+%        Bottom intensified tidal mixing 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Bottom intensified tidal mixing}
+\label{ZDF_tmx_bottom}
+
+The parameterization of tidal mixing follows the general formulation for 
+the vertical eddy diffusivity proposed by \citet{St_Laurent_al_GRL02} and 
+first introduced in an OGCM by \citep{Simmons_al_OM04}. 
+In this formulation an additional vertical diffusivity resulting from internal tide breaking, 
+$A^{vT}_{tides}$ is expressed as a function of $E(x,y)$, the energy transfer from barotropic 
+tides to baroclinic tides : 
+\begin{equation} \label{Eq_Ktides}
+A^{vT}_{tides} =  q \,\Gamma \,\frac{ E(x,y) \, F(z) }{ \rho \, N^2 }
+\end{equation}
+where $\Gamma$ is the mixing efficiency, $N$ the Brunt-Vais\"{a}l\"{a} frequency 
+(see \S\ref{TRA_bn2}), $\rho$ the density, $q$ the tidal dissipation efficiency, 
+and $F(z)$ the vertical structure function. 
+
+The mixing efficiency of turbulence is set by $\Gamma$ (\np{rn\_me} namelist parameter)
+and is usually taken to be the canonical value of $\Gamma = 0.2$ (Osborn 1980). 
+The tidal dissipation efficiency is given by the parameter $q$ (\np{rn\_tfe} namelist parameter) 
+represents the part of the internal wave energy flux $E(x, y)$ that is dissipated locally, 
+with the remaining $1-q$ radiating away as low mode internal waves and 
+contributing to the background internal wave field. A value of $q=1/3$ is typically used  
+\citet{St_Laurent_al_GRL02}.
+The vertical structure function $F(z)$ models the distribution of the turbulent mixing in the vertical. 
+It is implemented as a simple exponential decaying upward away from the bottom, 
+with a vertical scale of $h_o$ (\np{rn\_htmx} namelist parameter, with a typical value of $500\,m$) \citep{St_Laurent_Nash_DSR04}, 
+\begin{equation} \label{Eq_Fz}
+F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) }
+\end{equation}
+and is normalized so that vertical integral over the water column is unity. 
+
+The associated vertical viscosity is calculated from the vertical 
+diffusivity assuming a Prandtl number of 1, $i.e.$ $A^{vm}_{tides}=A^{vT}_{tides}$. 
+In the limit of $N \rightarrow 0$ (or becoming negative), the vertical diffusivity 
+is capped at $300\,cm^2/s$ and impose a lower limit on $N^2$ of \np{rn\_n2min} 
+usually set to $10^{-8} s^{-2}$. These bounds are usually rarely encountered.
+
+The internal wave energy map, $E(x, y)$ in \eqref{Eq_Ktides}, is derived 
+from a barotropic model of the tides utilizing a parameterization of the 
+conversion of barotropic tidal energy into internal waves. 
+The essential goal of the parameterization is to represent the momentum 
+exchange between the barotropic tides and the unrepresented internal waves 
+induced by the tidal ßow over rough topography in a stratified ocean. 
+In the current version of \NEMO, the map is built from the output of 
+the barotropic global ocean tide model MOG2D-G \citep{Carrere_Lyard_GRL03}.
+This model provides the dissipation associated with internal wave energy for the M2 and K1 
+tides component (Fig.~\ref{Fig_ZDF_M2_K1_tmx}). The S2 dissipation is simply approximated
+as being $1/4$ of the M2 one. The internal wave energy is thus : $E(x, y) = 1.25 E_{M2} + E_{K1}$. 
+Its global mean value is $1.1$ TW, in agreement with independent estimates 
+\citep{Egbert_Ray_Nat00, Egbert_Ray_JGR01}. 
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!t] \label{Fig_ZDF_M2_K1_tmx}  \begin{center}
+\includegraphics[width=0.90\textwidth]{./TexFiles/Figures/Fig_ZDF_M2_K1_tmx.pdf}
+\caption {(a) M2 and (b) K2 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). }
+\end{center}  
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
+ 
+% -------------------------------------------------------------------------------------------------------------
+%        Indonesian area specific treatment 
+% -------------------------------------------------------------------------------------------------------------
+\subsection{Indonesian area specific treatment}
+\label{ZDF_tmx_itf}
+
+When the Indonesian Through Flow (ITF) area is included in the model domain,
+a specific treatment of tidal induced mixing in this area can be used. 
+It is activated through the namelist logical \np{ln\_tmx\_itf}, and the user must provide
+an input NetCDF file, \ifile{mask\_itf}, which contains a mask array defining the ITF area
+where the specific treatment is applied.
+
+When \np{ln\_tmx\_itf}=true, the two key parameters $q$ and $F(z)$ are adjusted following 
+the parameterisation developed by \ref{Koch-Larrouy_al_GRL07}:
+
+First, the Indonesian archipelago is a complex geographic region 
+with a series of large, deep, semi-enclosed basins connected via 
+numerous narrow straits. Once generated, internal tides remain 
+confined within this semi-enclosed area and hardly radiate away. 
+Therefore all the internal tides energy is consumed within this area. 
+So it is assumed that $q = 1$, $i.e.$ all the energy generated is available for mixing.
+Note that for test purposed, the ITF tidal dissipation efficiency is a 
+namelist parameter (\np{rn\_tfe\_itf}). A value of $1$ or close to is
+this recommended for this parameter.
+
+Second, the vertical structure function, $F(z)$, is no more associated
+with a bottom intensification of the mixing, but with a maximum of 
+energy available within the thermocline. \ref{Koch-Larrouy_al_GRL07} 
+have suggested that the vertical distribution of the energy dissipation 
+proportional to $N^2$ below the core of the thermocline and to $N$ above. 
+The resulting $F(z)$ is:
+\begin{equation} \label{Eq_Fz_itf}
+F(i,j,k) \sim     \left\{ \begin{aligned}
+\frac{q\,\Gamma E(i,j) } {\rho N \, \int N     dz}    \qquad \text{when $\partial_z N < 0$} \\
+\frac{q\,\Gamma E(i,j) } {\rho     \, \int N^2 dz}    \qquad \text{when $\partial_z N > 0$}
+                      \end{aligned} \right.
+\end{equation}
+
+Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$, 
+which agrees with the independent estimates inferred from observations. 
+Introduced in a regional OGCM, the parameterization improves the water mass 
+characteristics in the different Indonesian seas, suggesting that the horizontal 
+and vertical distributions of the mixing are adequately prescribed 
+\citep{Koch-Larrouy_al_GRL07, Koch-Larrouy_al_OD08a, Koch-Larrouy_al_OD08b}.
+Note also that such a parameterisation has a sugnificant impact on the behaviour 
+of global coupled GCMs \citep{Koch-Larrouy_al_CD10}.
+
+
+% ================================================================
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Introduction.tex
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Introduction.tex	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Chapters/Introduction.tex	(revision 2282)
@@ -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 
@@ -133,18 +133,18 @@
 coordinate and for the new options for horizontal pressure gradient computation with 
 a non-linear equation of state.}; \\ 
-(4) more choices for the treatment of the free surface: full explicit, split-explicit , filtered 
-and rigid-lid; \\
-(5) non linear free surface option (associated with the rescaled height coordinate  
-\textit{z*} or  \textit{s*}); \\
+(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; \\
 (7) additional advection schemes for tracers; \\
 (8) implementation of the AGRIF package (Adaptative Grid Refinement in \textsc{Fortran}) \citep{Debreu_al_CG2008}; \\
 (9) online diagnostics : tracers trend in the mixed layer and vorticity balance; \\
-(10) rewriting of the I/O management; \\
-(11) OASIS 3 and 4 couplers interfacing with atmospheric global circulation models. \\
+(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynadv
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynadv	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynhpg
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynhpg	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynldf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynldf	(revision 2281)
+++ 	(revision )
@@ -1,15 +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
-   rn_ahm_0         = 40000.    !     horizontal laplacian eddy viscosity   [m2/s]
-   rn_ahmb_0        =     0.    !     background eddy viscosity for ldf_iso [m2/s]
-   rn_ahm_0_blp     =     0.    !     horizontal bilaplacian eddy viscosity [m4/s] 
-/
Index: anches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynspg
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynspg	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynvor
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_dynvor	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_lbc
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_lbc	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_mpp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_mpp	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_mpp_dyndist
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_mpp_dyndist	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_traadv
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_traadv	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_traldf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_traldf	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_zgr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_zgr	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_zgr_sco
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nam_zgr_sco	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namagrif
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namagrif	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namagrif	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbc
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbc	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbc	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbl
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbl	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambbl	(revision 2282)
@@ -2,6 +2,7 @@
 &nambbl        !   bottom boundary layer scheme
 !-----------------------------------------------------------------------
-!                          !  diffusive bbl                             ("key_trabbl")
-!                          !  advective bbl                             ("key_trabbl_adv")
-   atrbbl      =  10000.   !  lateral mixing coefficient in the bbl  [m2/s]
-/
+   nn_bbl_ldf  =  1      !  diffusive bbl (=1)   or not (=0)
+   nn_bbl_adv  =  0      !  advective bbl (=1/2) or not (=0)
+   rn_ahtbbl   =  1000.  !  lateral mixing coefficient in the bbl  [m2/s]
+   rn_gambbl   =  10.    !  advective bbl coefficient                 [s]
+   /
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy	(revision 2282)
@@ -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.)
@@ -6,4 +6,7 @@
     filbdy_data_U  = 'bdydata_grid_U.nc'  !  name of data file (U-points)
     filbdy_data_V  = 'bdydata_grid_V.nc'  !  name of data file (V-points)
+    filbdy_data_bt_T  = 'bdydata_bt_grid_T.nc'  !  name of data file for Flather condition (T-points)
+    filbdy_data_bt_U  = 'bdydata_bt_grid_U.nc'  !  name of data file for Flather condition (U-points)
+    filbdy_data_bt_V  = 'bdydata_bt_grid_V.nc'  !  name of data file for Flather condition (V-points)
     ln_bdy_clim    = .false.              !  contain 1 (T) or 12 (F) time dumps and be cyclic
     ln_bdy_vol     = .true.               !  total volume correction (see volbdy parameter)
@@ -19,2 +22,3 @@
                                           !  = 1, the total volume of the system is conserved
 /
+
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy_tide
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy_tide	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambdy_tide	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambfr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambfr	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nambfr	(revision 2282)
@@ -2,8 +2,10 @@
 &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)
+   ln_bfr2d    =   .false. !  horizontal variation of the bottom friction coef (read a 2D mask file )
+   rn_bfrien   =    50.    !  local multiplying factor of bfr (ln_bfr2d = .true.)
 /
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namcla
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namcla	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namcla	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namctl
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namctl	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namctl	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namddm
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namddm	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdom
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdom	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdom	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_adv
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_adv	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_adv	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_hpg
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_hpg	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_hpg	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_ldf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_ldf	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_ldf	(revision 2282)
@@ -0,0 +1,15 @@
+!-----------------------------------------------------------------------
+&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         =    40.e3  !  horizontal eddy viscosity   [m2/s]
+   rn_ahmb_0        =     0.    !  background eddy viscosity for ldf_iso [m2/s]
+   rn_ahm_0_blp     =     0.    !  horizontal bilaplacian eddy viscosity [m4/s] 
+/
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_spg
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_spg	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_spg	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_vor
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_vor	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namdyn_vor	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nameos
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nameos	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nameos	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namflg
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namflg	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namflo
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namflo	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namflo	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgap
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgap	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgap	(revision 2282)
@@ -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: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgls
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgls	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namgls	(revision 2282)
@@ -0,0 +1,18 @@
+!-----------------------------------------------------------------------
+&namzdf_gls                !   GLS vertical diffusion                   ("key_zdfgls")
+!-----------------------------------------------------------------------
+   rn_emin     =   1.e-6   !  minimum value of e   [m2/s2]
+   rn_epsmin   =   1.e-12  !  minimum value of eps [m2/s3]
+   ln_length_lim = .true.  !  limit on the dissipation rate under stable stratification (Galperin et al., 1988)
+   clim_galp   =   0.53    !  galperin limit
+   ln_crban = .TRUE.       !  Use Craig & Banner (1994) surface wave mixing parametrisation
+   ln_sigpsi = .TRUE.      !  Activate or not Burchard 2001 mods on psi schmidt number in the wb case
+   rn_crban = 100.         !  Craig and Banner 1994 constant for wb tke flux
+   rn_charn =  70000.      !  Charnock constant for wb induced roughness length
+   nn_tkebc_surf  =   1    !  surface tke condition (0/1/2=Dir/Neum/Dir Mellor-Blumberg)
+   nn_tkebc_bot   =   1    !  bottom tke condition (0/1=Dir/Neum)
+   nn_psibc_surf  =   1    !  surface psi condition (0/1/2=Dir/Neum/Dir Mellor-Blumberg)
+   nn_psibc_bot   =   1    !  bottom psi condition (0/1=Dir/Neum)
+   nn_stab_func   =   2    !  stability function (0=Galp, 1= KC94, 2=CanutoA, 3=CanutoB)
+   nn_clos        =   1    !  predefined closure type (0=MY82, 1=k-eps, 2=k-w, 3=Gen)
+/
Index: anches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namkpp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namkpp	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp_dyndist
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp_dyndist	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/nammpp_dyndist	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namnpc
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namnpc	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namobc
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namobc	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namobc	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namptr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namptr	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namptr	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namqsr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namqsr	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namric
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namric	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namrun
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namrun	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namrun	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_alb
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_alb	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_alb	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_clio
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_clio	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_clio	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_core
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_core	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_core	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl_co2
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl_co2	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_cpl_co2	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_flx
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_flx	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_flx	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_qsr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_qsr	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_rnf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_rnf	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_rnf	(revision 2282)
@@ -2,13 +2,20 @@
 &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     , ''       , ''
+!               !   file name        ! frequency (h)   ! variable   ! time interp. !  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     , ''   , ''
+   sn_s_rnf   = 'runoffs'            ,  24             , 'rosaline' ,   .true.     , .true. ,  'yearly', ''   , ''
+   sn_t_rnf   = 'runoffs'            ,  24             , 'rotemper' ,   .true.     , .true. ,  'yearly', ''   , ''
+   sn_dep_rnf = 'runoffs'            ,   0             , 'rodepth'  ,   .false.    , .true. ,  'yearly', ''   , ''
 !
-   cn_dir       = './'       ! directory in which the model is executed
+   cn_dir       = './'      ! directory in which the model is executed
    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
+   ln_rnf_depth =  .false.  !  read in depth information for runoff
+   ln_rnf_temp  =  .false.  !  read in temperature information for runoff
+   ln_rnf_sal   =  .false.  !  read in salinity information for runoff
 /
Index: /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_ssr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_ssr	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsbc_ssr	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsol
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsol	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namsol	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namspr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namspr	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtdp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtdp	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtide
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtide	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtke
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtke	(revision 2281)
+++ 	(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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_adv
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_adv	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_adv	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_dmp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_dmp	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_dmp	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_ldf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_ldf	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_ldf	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_qsr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_qsr	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtra_qsr	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtrd
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtrd	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namtrd	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf	(revision 2281)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ddm
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ddm	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ddm	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_kpp
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_kpp	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_kpp	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ric
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ric	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_ric	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tke
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tke	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tke	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tmx
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tmx	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzdf_tmx	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr	(revision 2282)
@@ -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/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr_sco
===================================================================
--- /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr_sco	(revision 2282)
+++ /branches/nemo_v3_3_beta/DOC/TexFiles/Namelist/namzgr_sco	(revision 2282)
@@ -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 
+/
