Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/abstract_foreword.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/abstract_foreword.tex	(revision 10377)
+++ 	(revision )
@@ -1,58 +1,0 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
-\begin{document}
-
-% ================================================================
-% Abstract
-% ================================================================
-
-\chapter*{Abstract}
-
-\vspace{-40pt}
-
-{\small
-  The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive equation model adapted to
-  regional and global ocean circulation problems.
-  It is intended to be a flexible tool for studying the ocean and its interactions with
-  the others components of the earth climate system over a wide range of space and time scales.
-  Prognostic variables are the three-dimensional velocity field, a non-linear sea surface height,
-  the \textit{Conservative} Temperature and the \textit{Absolute} Salinity.
-  In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction,
-  a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two.
-  The distribution of variables is a three-dimensional Arakawa C-type grid.
-  Various physical choices are available to describe ocean physics, including TKE, and GLS vertical physics.
-  Within NEMO, the ocean is interfaced with a sea-ice model (LIM or CICE),
-  passive tracer and biogeochemical models (TOP) and,
-  via the OASIS coupler, with several atmospheric general circulation models.
-  It also support two-way grid embedding via the AGRIF software.
-} 
-
-% ================================================================
-% Disclaimer
-% ================================================================
-\chapter*{Disclaimer}
-
-Like all components of NEMO,
-the ocean component is developed under the \href{http://www.cecill.info/}{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}
-NEMO reference in papers and other publications is as follows:
-\vspace{0.5cm}
-
-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.\\
-
-
-\vspace{0.5cm}
-Additional information can be found on \href{http://www.nemo-ocean.eu/}{www.nemo-ocean.eu}.
-\vspace{0.5cm}
-
-\end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_A.tex	(revision 10419)
@@ -1,14 +1,15 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
 % ================================================================
-% Chapter Ñ Appendix A : Curvilinear s-Coordinate Equations
+% Chapter Appendix A : Curvilinear s-Coordinate Equations
 % ================================================================
 \chapter{Curvilinear $s-$Coordinate Equations}
 \label{apdx:A}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -26,38 +27,40 @@
 Let us define a new vertical scale factor by $e_3 = \partial z / \partial s$ (which now depends on $(i,j,z,t)$) and
 the horizontal slope of $s-$surfaces by:
-\begin{equation} \label{apdx:A_s_slope}
-\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
-\quad \text{and} \quad 
-\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s 
+\begin{equation}
+  \label{apdx:A_s_slope}
+  \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
+  \quad \text{and} \quad
+  \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
 \end{equation}
 
 The chain rule to establish the model equations in the curvilinear $s-$coordinate system is:
-\begin{equation} \label{apdx:A_s_chain_rule}
-\begin{aligned}
-&\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
-\left. {\frac{\partial \bullet }{\partial t}} \right|_s 
+\begin{equation}
+  \label{apdx:A_s_chain_rule}
+  \begin{aligned}
+    &\left. {\frac{\partial \bullet }{\partial t}} \right|_z  =
+    \left. {\frac{\partial \bullet }{\partial t}} \right|_s
 	 -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial t} \\
-&\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
-  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
-  	  -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
-	  \left. {\frac{\partial \bullet }{\partial i}} \right|_s 
-	  -\frac{e_1 }{e_3 }\sigma _1 \frac{\partial \bullet }{\partial s} \\
-&\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
-\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
-	- \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
-\left. {\frac{\partial \bullet }{\partial j}} \right|_s 
-	- \frac{e_2 }{e_3 }\sigma _2 \frac{\partial \bullet }{\partial s} \\
-&\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s} \\
-\end{aligned}
+    &\left. {\frac{\partial \bullet }{\partial i}} \right|_z  =
+    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
+    -\frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial i}=
+    \left. {\frac{\partial \bullet }{\partial i}} \right|_s
+    -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial \bullet }{\partial s} \\
+    &\left. {\frac{\partial \bullet }{\partial j}} \right|_z  =
+    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
+    - \frac{\partial \bullet }{\partial s}\;\frac{\partial s}{\partial j}=
+    \left. {\frac{\partial \bullet }{\partial j}} \right|_s
+    - \frac{e_2 }{e_3 }\sigma_2 \frac{\partial \bullet }{\partial s} \\
+    &\;\frac{\partial \bullet }{\partial z}  \;\; = \frac{1}{e_3 }\frac{\partial \bullet }{\partial s}
+  \end{aligned}
 \end{equation}
 
 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 
-				= \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t} 
-			    = e_3 \, \frac{\partial s}{\partial t} 
-\end{equation}
-
+\begin{equation}
+  \label{apdx:A_w_in_s}
+  w_s   =  \left.   \frac{\partial z }{\partial t}   \right|_s
+  = \frac{\partial z}{\partial s} \; \frac{\partial s}{\partial t}
+  = e_3 \, \frac{\partial s}{\partial t}
+\end{equation}
 
 % ================================================================
@@ -72,58 +75,58 @@
 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 
-						+\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right]
-+ \frac{\partial w}{\partial z} 		\\
-\\
-&     = \frac{1}{e_1 \,e_2 }  \left[ 
-		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
-		  - \frac{e_1 }{e_3 } \sigma _1 \frac{\partial (e_2 \,u)}{\partial s}
-		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       
-		  - \frac{e_2 }{e_3 } \sigma _2 \frac{\partial (e_1 \,v)}{\partial s}	\right]
-	+ \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z}                        \\
-\\
-&     = \frac{1}{e_1 \,e_2 }   \left[ 
-		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s       
-		+ \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       	\right]
-	+ \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
-						-  \sigma _1 \frac{\partial u}{\partial s}
-						-  \sigma _2 \frac{\partial v}{\partial s}      \right]          \\
-\\
-&     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[ 
-		  \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s  
-		  -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s      
-		+ \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
-		  - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right]          \\
-& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
-	+ \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
-						-  \sigma _1 \frac{\partial u}{\partial s}
-						-  \sigma _2 \frac{\partial v}{\partial s}      \right]      \\
-%
-\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 \quad
- +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma _1 }{\partial s}-v\frac{\partial \sigma _2 }{\partial s}-\sigma _1 \frac{\partial u}{\partial s}-\sigma _2 \frac{\partial v}{\partial s}} \right] \\ 
-\\
-& = \frac{1}{e_1 \,e_2 \,e_3 }  \left[   
-		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
-		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
-   + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma _1  - v\;\sigma _2  \right]
-\end{array} } 		
-\end{align*}
+\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
+        +\left. {\frac{\partial(e_1 \,v)}{\partial j}} \right|_z  \right]
+        + \frac{\partial w}{\partial z} \\ \\
+      &     = \frac{1}{e_1 \,e_2 }  \left[
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
+		  - \frac{e_1 }{e_3 } \sigma_1 \frac{\partial (e_2 \,u)}{\partial s}
+        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s
+		  - \frac{e_2 }{e_3 } \sigma_2 \frac{\partial (e_1 \,v)}{\partial s}	\right]
+        + \frac{\partial w}{\partial s} \; \frac{\partial s}{\partial z} \\ \\
+      &     = \frac{1}{e_1 \,e_2 }   \left[
+		  \left.   \frac{\partial (e_2 \,u)}{\partial i}    \right|_s
+        + \left.   \frac{\partial (e_1 \,v)}{\partial j}    \right|_s       	\right]
+        + \frac{1}{e_3 }\left[        \frac{\partial w}{\partial s}
+        -  \sigma_1 \frac{\partial u}{\partial s}
+        -  \sigma_2 \frac{\partial v}{\partial s}      \right] \\ \\
+      &     = \frac{1}{e_1 \,e_2 \,e_3 }   \left[
+		  \left.   \frac{\partial (e_2 \,e_3 \,u)}{\partial i}    \right|_s
+		  -\left.    e_2 \,u    \frac{\partial e_3 }{\partial i}     \right|_s
+        + \left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j}    \right|_s
+		  - \left.    e_1 v      \frac{\partial e_3 }{\partial j}    \right|_s   \right] \\
+      & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
+        + \frac{1}{e_3 } \left[        \frac{\partial w}{\partial s}
+        -  \sigma_1 \frac{\partial u}{\partial s}
+        -  \sigma_2 \frac{\partial v}{\partial s}      \right]      \\
+      %
+      \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 \quad
+        +\frac{1}{e_3 }\left[ {\frac{\partial w}{\partial s}-u\frac{\partial \sigma_1 }{\partial s}-v\frac{\partial \sigma_2 }{\partial s}-\sigma_1 \frac{\partial u}{\partial s}-\sigma_2 \frac{\partial v}{\partial s}} \right] \\
+      \\
+      & = \frac{1}{e_1 \,e_2 \,e_3 }  \left[
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
+        + \frac{1}{e_3 } \; \frac{\partial}{\partial s}   \left[  w -  u\;\sigma_1  - v\;\sigma_2  \right]
+    \end{array}
+        }
+  \end{align*}
 \end{subequations}
 
@@ -131,47 +134,49 @@
 Introducing the dia-surface velocity component,
 $\omega $, defined as the volume flux across the moving $s$-surfaces per unit horizontal area:
-\begin{equation} \label{apdx:A_w_s}
-\omega  = w - w_s - \sigma _1 \,u - \sigma _2 \,v    \\
+\begin{equation}
+  \label{apdx:A_w_s}
+  \omega  = w - w_s - \sigma_1 \,u - \sigma_2 \,v    \\
 \end{equation}
 with $w_s$ given by \autoref{apdx:A_w_in_s},
 we obtain the expression for 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[ 
-		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
-		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
-+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
-+ \frac{1}{e_3 } \frac{\partial w_s       }{\partial s}    \\
-\\
-&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
-		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
-		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
-+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
-+ \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right)   \\
-\\
-&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
-		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
-		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
-+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
-+ \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
-+ \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s}     \\
-\\
-&= \frac{1}{e_1 \,e_2 \,e_3 }    \left[ 
-		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s 
-		+\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]      
-+ \frac{1}{e_3 } \frac{\partial \omega }{\partial s} 
-+ \frac{1}{e_3 } \frac{\partial e_3}{\partial t}     \\
-\end{array} } 		
-\end{align*}
+\begin{subequations}
+  \begin{align*}
+    {
+    \begin{array}{*{20}l}
+      \nabla \cdot {\rm {\bf U}}
+      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
+        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
+        + \frac{1}{e_3 } \frac{\partial w_s       }{\partial s} \\ \\
+      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
+        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
+        + \frac{1}{e_3 } \frac{\partial}{\partial s}  \left(  e_3 \; \frac{\partial s}{\partial t}   \right) \\ \\
+      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
+        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
+        + \frac{\partial}{\partial s} \frac{\partial s}{\partial t}
+        + \frac{1}{e_3 } \frac{\partial s}{\partial t} \frac{\partial e_3}{\partial s} \\ \\
+      &= \frac{1}{e_1 \,e_2 \,e_3 }    \left[
+		  \left.  \frac{\partial (e_2 \,e_3 \,u)}{\partial i} \right|_s
+        +\left.  \frac{\partial (e_1 \,e_3 \,v)}{\partial j} \right|_s        \right]
+        + \frac{1}{e_3 } \frac{\partial \omega }{\partial s}
+        + \frac{1}{e_3 } \frac{\partial e_3}{\partial t}
+    \end{array}
+        }
+  \end{align*}
 \end{subequations}
 
 As a result, the continuity equation \autoref{eq:PE_continuity} in the $s-$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[ 
-			{\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s 
-		    +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
- +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0   
+\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[
+    {\left. {\frac{\partial (e_2 \,e_3 \,u)}{\partial i}} \right|_s
+      +  \left. {\frac{\partial (e_1 \,e_3 \,v)}{\partial j}} \right|_s } \right]
+  +\frac{1}{e_3 }\frac{\partial \omega }{\partial s} = 0
 \end{equation}
 A additional term has appeared that take into account
@@ -187,6 +192,4 @@
 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}
@@ -197,80 +200,85 @@
 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 (\autoref{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 (\autoref{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{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 (\autoref{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 (\autoref{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*}
 \end{subequations}
 %
 Applying the time derivative chain rule (first equation of (\autoref{apdx:A_s_chain_rule})) to $u$ and
 using (\autoref{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*}
+\[
+  {
+    \begin{array}{*{20}l}
+      w_s  \;\frac{\partial u}{\partial s}
+      = \frac{\partial s}{\partial t} \;  \frac{\partial u }{\partial s}
+      = \left. {\frac{\partial u }{\partial t}} \right|_s  - \left. {\frac{\partial u }{\partial t}} \right|_z \quad ,
+    \end{array}
+  }
+\]
 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       
+\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}   
+  + \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
@@ -278,84 +286,80 @@
 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 (\autoref{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} } 		
+% \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*}
 %
 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 (\autoref{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}    
+\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 (\autoref{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} } 		
+      -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)
+\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.
@@ -365,29 +369,28 @@
 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|_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*}
+\[
+  \begin{split}
+    -\frac{1}{\rho_o \, e_1 }\left. {\frac{\partial p}{\partial i}} \right|_z
+    & =-\frac{1}{\rho_o e_1 }\left[ {\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{e_1 }{e_3 }\sigma_1 \frac{\partial p}{\partial s}} \right] \\
+    & =-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s +\frac{\sigma_1 }{\rho_o \,e_3 }\left( {-g\;\rho \;e_3 } \right) \\
+    &=-\frac{1}{\rho_o \,e_1 }\left. {\frac{\partial p}{\partial i}} \right|_s -\frac{g\;\rho }{\rho_o }\sigma_1
+  \end{split}
+\]
 Applying similar manipulation to the second component and
-replacing $\sigma _1$ and $\sigma _2$ by their expression \autoref{apdx:A_s_slope}, it comes:
-\begin{equation} \label{apdx:A_grad_p_1}
-\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}
+replacing $\sigma_1$ and $\sigma_2$ by their expression \autoref{apdx:A_s_slope}, it comes:
+\begin{equation}
+  \label{apdx:A_grad_p_1}
+  \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}
 
@@ -400,31 +403,33 @@
 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*}
+\[
+  \begin{split}
+    p &= g\; \int_z^\eta \rho \; e_3 \; dk = g\; \int_z^\eta \left(  \rho_o \, d + 1 \right) \; e_3 \; dk   \\
+    &= g \, \rho_o \; \int_z^\eta d \; e_3 \; dk + g \, \int_z^\eta e_3 \; dk
+  \end{split}
+\]
 Therefore, $p$ and $p_h'$ are linked through:
-\begin{equation} \label{apdx:A_pressure}
-   p = \rho_o \; p_h' + g \, ( z + \eta )
+\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*}
+\[
+  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\]
 
 Substituing \autoref{apdx:A_pressure} in \autoref{apdx:A_grad_p_1} and
 using the definition of the density anomaly it comes the expression in two parts:
-\begin{equation} \label{apdx:A_grad_p_2}
-\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}
+\begin{equation}
+  \label{apdx:A_grad_p_2}
+  \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
@@ -437,7 +442,4 @@
 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}
 
@@ -446,7 +448,4 @@
 The form of the lateral physics is discussed in \autoref{apdx:B}.
 
-
-$\ $\newline    % force a new ligne
-
 $\bullet$ \textbf{Full momentum equation}
 
@@ -454,50 +453,57 @@
 the vector invariant momentum equation solved by the model has the same mathematical expression as
 the one in a curvilinear $z-$coordinate, except for the pressure gradient term:
-\begin{subequations} \label{apdx:A_dyn_vect}
-\begin{multline} \label{apdx:A_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} \omega \frac{\partial u}{\partial k}       \\
-        -   \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{\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 } \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}
+\begin{subequations}
+  \label{apdx:A_dyn_vect}
+  \begin{multline}
+    \label{apdx:A_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} \omega \frac{\partial u}{\partial k}       \\
+    -   \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{\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 } \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}
 whereas the flux form momentum equation differs 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}
+\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 anomalies, $p_h'$ and $d=( \frac{\rho}{\rho_o}-1 )$:
-\begin{equation} \label{apdx:A_dyn_zph}
-\frac{\partial p_h'}{\partial k} = - d \, g \, e_3
+\begin{equation}
+  \label{apdx:A_dyn_zph}
+  \frac{\partial p_h'}{\partial k} = - d \, g \, e_3
 \end{equation}
 
@@ -519,16 +525,18 @@
 regrouping the time derivative terms in the left hand side :
 
-\begin{multline} \label{apdx:A_tracer}
- \frac{1}{e_3} \frac{\partial \left(  e_3 T  \right)}{\partial t} 
- 	= -\frac{1}{e_1 \,e_2 \,e_3} 
-		\left[           \frac{\partial }{\partial i} \left( {e_2 \,e_3 \;Tu} \right) 
-		             +   \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}
+\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)
+    +   \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. 
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_B.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_B.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_B.tex	(revision 10419)
@@ -1,14 +1,14 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter Ñ Appendix B : Diffusive Operators
+% Chapter Appendix B : Diffusive Operators
 % ================================================================
 \chapter{Appendix B : Diffusive Operators}
 \label{apdx:B}
+
 \minitoc
 
-
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -19,39 +19,46 @@
 
 \subsubsection*{In z-coordinates}
+
 In $z$-coordinates, the horizontal/vertical second order tracer diffusion operator is given by:
-\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{eqnarray}
+\begin{align}
+  \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{align}
 
 \subsubsection*{In generalized vertical coordinates}
+
 In $s$-coordinates, we defined the slopes of $s$-surfaces, $\sigma_1$ and $\sigma_2$ by \autoref{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}
-D^T = \left. \nabla \right|_s \cdot
-           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
-\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-\sigma _1 } \hfill \\
- 0 \hfill & 1 \hfill & {-\sigma _2 } \hfill \\
- {-\sigma _1 } \hfill & {-\sigma _2 } \hfill & {\varepsilon +\sigma _1
-^2+\sigma _2 ^2} \hfill \\
-\end{array} }} \right)
+\begin{equation}
+  \label{apdx:B2}
+  D^T = \left. \nabla \right|_s \cdot
+  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
+  \;\;\text{where} \;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill & {-\sigma_1 } \hfill \\
+          0 \hfill & 1 \hfill & {-\sigma_2 } \hfill \\
+          {-\sigma_1 } \hfill & {-\sigma_2 } \hfill & {\varepsilon +\sigma_1
+                                                      ^2+\sigma_2 ^2} \hfill \\
+        \end{array}
+      }} \right)
 \end{equation}
 or in expanded form:
-\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} }
+\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 \autoref{apdx:B2} is obtained from \autoref{apdx:B1} without any additional assumption.
@@ -64,73 +71,83 @@
 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)     \\
- \\
-%
-&=\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 \\
- \\
-%
-&=\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}
-\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]}
-\end{array} } \\
-{\begin{array}{*{20}l}
-%
-\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] }
- \end{array} } \\
-{\begin{array}{*{20}l}
-%
-\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:}
-%
-& =\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]}
- \\
-%
-\intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
-\end{array} } \\
-%
-{\frac{1}{e_1\,e_2\,e_3}}
-\left( {{\begin{array}{*{30}c}
-{\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
-{\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
-\end{array}}}\right)
-\cdot \left[ {A^{lT}
-\left( {{\begin{array}{*{30}c}
- {1} \hfill & {-\sigma_1 } \hfill \\
- {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
-\end{array} }} \right)
-\cdot
-\left( {{\begin{array}{*{30}c}
-{\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
-{\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
-\end{array}}}       \right) \left( T \right)} \right]
+\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) \\ \\
+         %
+       &=\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 \\ \\
+         %
+       &=\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}
+    \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]}
+  \end{array}
+      } \\
+  {
+  \begin{array}{*{20}l}
+    %
+    \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] }
+  \end{array}
+      } \\
+  {
+  \begin{array}{*{20}l}
+    %
+    \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:}
+  %
+    & =\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]} \\
+    %
+    \intertext{in other words, the horizontal/vertical Laplacian operator in the ($i$,$s$) plane takes the following form:}
+  \end{array}
+  } \\
+  %
+  {\frac{1}{e_1\,e_2\,e_3}}
+  \left( {{
+  \begin{array}{*{30}c}
+    {\left. {\frac{\partial \left( {e_2 e_3 \bullet } \right)}{\partial i}} \right|_s } \hfill \\
+    {\frac{\partial \left( {e_1 e_2 \bullet } \right)}{\partial s}} \hfill \\
+  \end{array}}}
+  \right)
+  \cdot \left[ {A^{lT}
+  \left( {{
+  \begin{array}{*{30}c}
+    {1} \hfill & {-\sigma_1 } \hfill \\
+    {-\sigma_1} \hfill & {\varepsilon + \sigma_1^2} \hfill \\
+  \end{array}
+  }} \right)
+  \cdot
+  \left( {{
+  \begin{array}{*{30}c}
+    {\frac{1}{e_1 }\;\left. {\frac{\partial \bullet }{\partial i}} \right|_s } \hfill \\
+    {\frac{1}{e_3 }\;\frac{\partial \bullet }{\partial s}} \hfill \\
+  \end{array}
+  }}       \right) \left( T \right)} \right]
 \end{align*}
-\end{subequations}
-\addtocounter{equation}{-2}
+%\addtocounter{equation}{-2}
 
 % ================================================================
@@ -147,38 +164,45 @@
 takes the following form \citep{Redi_JPO82}:
 
-\begin{equation} \label{apdx:B3}
-\textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
-\left[ {{\begin{array}{*{20}c}
- {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
- {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
- {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
-\end{array} }} \right]
+\begin{equation}
+  \label{apdx:B3}
+  \textbf {A}_{\textbf I} = \frac{A^{lT}}{\left( {1+a_1 ^2+a_2 ^2} \right)}
+  \left[ {{
+        \begin{array}{*{20}c}
+          {1+a_1 ^2} \hfill & {-a_1 a_2 } \hfill & {-a_1 } \hfill \\
+          {-a_1 a_2 } \hfill & {1+a_2 ^2} \hfill & {-a_2 } \hfill \\
+          {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+        \end{array}
+      }} \right]
 \end{equation}
 where ($a_1$, $a_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions, relative to geopotentials:
-\begin{equation*}
-a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
-\qquad , \qquad
-a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
-\right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
-\end{equation*}
+\[
+  a_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+  \qquad , \qquad
+  a_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
+  \right)\left( {\frac{\partial \rho }{\partial k}} \right)^{-1}
+\]
 
 In practice, isopycnal slopes are generally less than $10^{-2}$ in the ocean,
 so $\textbf {A}_{\textbf I}$ can be simplified appreciably \citep{Cox1987}:
-\begin{subequations} \label{apdx:B4}
-\begin{equation} \label{apdx:B4a}
-{\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
-\left[ {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-a_1 } \hfill \\
- 0 \hfill & 1 \hfill & {-a_2 } \hfill \\
- {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
-\end{array} }} \right],
-\end{equation}
-and the iso/dianeutral diffusive operator in $z$-coordinates is then
-\begin{equation}\label{apdx:B4b}
- D^T = \left. \nabla \right|_z \cdot
-           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
-\end{equation}
+\begin{subequations}
+  \label{apdx:B4}
+  \begin{equation}
+    \label{apdx:B4a}
+    {\textbf{A}_{\textbf{I}}} \approx A^{lT}\;\Re\;\text{where} \;\Re =
+    \left[ {{
+          \begin{array}{*{20}c}
+            1 \hfill & 0 \hfill & {-a_1 } \hfill \\
+            0 \hfill & 1 \hfill & {-a_2 } \hfill \\
+            {-a_1 } \hfill & {-a_2 } \hfill & {\varepsilon +a_1 ^2+a_2 ^2} \hfill \\
+          \end{array}
+        }} \right],
+  \end{equation}
+  and the iso/dianeutral diffusive operator in $z$-coordinates is then
+  \begin{equation}
+    \label{apdx:B4b}
+    D^T = \left. \nabla \right|_z \cdot
+    \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_z T  \right]. \\
+  \end{equation}
 \end{subequations}
-
 
 Physically, the full tensor \autoref{apdx:B3} represents strong isoneutral diffusion on a plane parallel to
@@ -192,11 +216,11 @@
 Written out explicitly,
 
-\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]}
- {+\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+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\
+\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]}
+    {+\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+\varepsilon} \right)}{e_3 }\frac{\partial T}{\partial k}} \right)} \right]}. \\
 \end{multline}
-
 
 The isopycnal diffusion operator \autoref{apdx:B4},
@@ -204,26 +228,27 @@
 The demonstration of the first property is trivial as \autoref{apdx:B4} is the divergence of fluxes.
 Let us demonstrate the second one:
-\begin{equation*}
-\iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
-          = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
-\end{equation*}
+\[
+  \iiint\limits_D T\;\nabla .\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv
+  = -\iiint\limits_D \nabla T\;.\left( {\textbf{A}}_{\textbf{I}} \nabla T \right)\,dv,
+\]
 and since
-\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
-{ \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \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}
-  +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
-& \geq 0
-\end{array} }
+\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
+             { \left. -\,{2a_2 \frac{\partial T}{\partial j}\frac{\partial T}{\partial k}+\left( {a_1 ^2+a_2 ^2+\varepsilon} \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}
+             +\varepsilon \left(\frac{\partial T}{\partial k}\right) ^2\right]      \\
+           & \geq 0
+  \end{array}
+             }
 \end{align*}
-\end{subequations}
-\addtocounter{equation}{-1}
+%\addtocounter{equation}{-1}
 the property becomes obvious.
 
@@ -236,23 +261,26 @@
 The resulting operator then takes the simple form
 
-\begin{equation} \label{apdx:B_ldfiso_s}
-D^T = \left. \nabla \right|_s \cdot
-           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
-\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-r _1 } \hfill \\
- 0 \hfill & 1 \hfill & {-r _2 } \hfill \\
- {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
-^2+r _2 ^2} \hfill \\
-\end{array} }} \right),
+\begin{equation}
+  \label{apdx:B_ldfiso_s}
+  D^T = \left. \nabla \right|_s \cdot
+  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_s T  \right] \\
+  \;\;\text{where} \;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill & {-r _1 } \hfill \\
+          0 \hfill & 1 \hfill & {-r _2 } \hfill \\
+          {-r _1 } \hfill & {-r _2 } \hfill & {\varepsilon +r _1
+                                              ^2+r _2 ^2} \hfill \\
+        \end{array}
+      }} \right),
 \end{equation}
 
 where ($r_1$, $r_2$) are the isopycnal slopes in ($\textbf{i}$, $\textbf{j}$) directions,
 relative to $s$-coordinate surfaces:
-\begin{equation*}
-r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
-\qquad , \qquad
-r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
-\right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
-\end{equation*}
+\[
+  r_1 =\frac{e_3 }{e_1 }\left( {\frac{\partial \rho }{\partial i}} \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}
+  \qquad , \qquad
+  r_2 =\frac{e_3 }{e_2 }\left( {\frac{\partial \rho }{\partial j}}
+  \right)\left( {\frac{\partial \rho }{\partial s}} \right)^{-1}.
+\]
 
 To prove \autoref{apdx:B5} by direct re-expression of \autoref{apdx:B_ldfiso} is straightforward, but laborious.
@@ -260,12 +288,15 @@
 the weak-slope operator may be \emph{exactly} reexpressed in non-orthogonal $i,j,\rho$-coordinates as
 
-\begin{equation} \label{apdx:B5}
-D^T = \left. \nabla \right|_\rho \cdot
-           \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
-\;\;\text{where} \;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill &0 \hfill \\
- 0 \hfill & 1 \hfill & 0 \hfill \\
-0 \hfill & 0 \hfill & \varepsilon \hfill \\
-\end{array} }} \right).
+\begin{equation}
+  \label{apdx:B5}
+  D^T = \left. \nabla \right|_\rho \cdot
+  \left[ A^{lT} \;\Re \cdot \left. \nabla \right|_\rho T  \right] \\
+  \;\;\text{where} \;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill &0 \hfill \\
+          0 \hfill & 1 \hfill & 0 \hfill \\
+          0 \hfill & 0 \hfill & \varepsilon \hfill \\
+        \end{array}
+      }} \right).
 \end{equation}
 Then direct transformation from $i,j,\rho$-coordinates to $i,j,s$-coordinates gives
@@ -289,43 +320,49 @@
 to the horizontal velocity vector:
 \begin{align*}
-\Delta {\textbf{U}}_h
-&=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
-\nabla \times \left( {\nabla \times {\textbf{U}}_h } \right)    \\
-\\
-&=\left( {{\begin{array}{*{20}c}
- {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
- {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
- {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
-\end{array} }} \right)-\left( {{\begin{array}{*{20}c}
- {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
-}\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
-u}{\partial k}} \right)} \hfill \\
- {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
-}\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
-}{\partial i}} \hfill \\
- {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
-}{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
-j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
-\hfill \\
-\end{array} }} \right)
-\\
-\\
-&=\left( {{\begin{array}{*{20}c}
-{\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
-{\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
-0 \\
-\end{array} }} \right)
-+\frac{1}{e_3 }
-\left( {{\begin{array}{*{20}c}
-{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
-{\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
-{\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
-\end{array} }} \right)
+  \Delta {\textbf{U}}_h
+  &=\nabla \left( {\nabla \cdot {\textbf{U}}_h } \right)-
+    \nabla \times \left( {\nabla \times {\textbf{U}}_h } \right) \\ \\
+  &=\left( {{
+    \begin{array}{*{20}c}
+      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}} \hfill \\
+      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}} \hfill \\
+      {\frac{1}{e_3 }\frac{\partial \chi }{\partial k}} \hfill \\
+    \end{array}
+  }} \right)
+  -\left( {{
+  \begin{array}{*{20}c}
+    {\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}-\frac{1}{e_3
+    }\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial
+    u}{\partial k}} \right)} \hfill \\
+    {\frac{1}{e_3 }\frac{\partial }{\partial k}\left( {-\frac{1}{e_3
+    }\frac{\partial v}{\partial k}} \right)-\frac{1}{e_1 }\frac{\partial \zeta
+    }{\partial i}} \hfill \\
+    {\frac{1}{e_1 e_2 }\left[ {\frac{\partial }{\partial i}\left( {\frac{e_2
+    }{e_3 }\frac{\partial u}{\partial k}} \right)-\frac{\partial }{\partial
+    j}\left( {-\frac{e_1 }{e_3 }\frac{\partial v}{\partial k}} \right)} \right]}
+    \hfill \\
+  \end{array}
+  }} \right) \\ \\
+  &=\left( {{
+    \begin{array}{*{20}c}
+      {\frac{1}{e_1 }\frac{\partial \chi }{\partial i}-\frac{1}{e_2 }\frac{\partial \zeta }{\partial j}} \\
+      {\frac{1}{e_2 }\frac{\partial \chi }{\partial j}+\frac{1}{e_1 }\frac{\partial \zeta }{\partial i}} \\
+      0 \\
+    \end{array}
+  }} \right)
+  +\frac{1}{e_3 }
+  \left( {{
+  \begin{array}{*{20}c}
+    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial u}{\partial k}} \right)} \\
+    {\frac{\partial }{\partial k}\left( {\frac{1}{e_3 }\frac{\partial v}{\partial k}} \right)} \\
+    {\frac{\partial \chi }{\partial k}-\frac{1}{e_1 e_2 }\left( {\frac{\partial ^2\left( {e_2 \,u} \right)}{\partial i\partial k}+\frac{\partial ^2\left( {e_1 \,v} \right)}{\partial j\partial k}} \right)} \\
+  \end{array}
+  }} \right)
 \end{align*}
 Using \autoref{eq:PE_div}, the definition of the horizontal divergence,
 the third componant of the second vector is obviously zero and thus :
-\begin{equation*}
-\Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
-\end{equation*}
+\[
+  \Delta {\textbf{U}}_h = \nabla _h \left( \chi \right) - \nabla _h \times \left( \zeta \right) + \frac {1}{e_3 } \frac {\partial }{\partial k} \left( {\frac {1}{e_3 } \frac{\partial {\textbf{ U}}_h }{\partial k}} \right)
+\]
 
 Note that this operator ensures a full separation between
@@ -335,21 +372,23 @@
 The horizontal/vertical second order (Laplacian type) operator used to diffuse horizontal momentum in
 the $z$-coordinate therefore takes the following form:
-\begin{equation} \label{apdx:B_Lap_U}
- {\textbf{D}}^{\textbf{U}} =
- 	  \nabla _h \left( {A^{lm}\;\chi } \right)
-	- \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
-	+ \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
-				\frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
+\begin{equation}
+  \label{apdx:B_Lap_U}
+  {
+    \textbf{D}}^{\textbf{U}} =
+  \nabla _h \left( {A^{lm}\;\chi } \right)
+  - \nabla _h \times \left( {A^{lm}\;\zeta \;{\textbf{k}}} \right)
+  + \frac{1}{e_3 }\frac{\partial }{\partial k}\left( {\frac{A^{vm}\;}{e_3 }
+      \frac{\partial {\rm {\bf U}}_h }{\partial k}} \right) \\
 \end{equation}
 that is, in expanded form:
 \begin{align*}
-D^{\textbf{U}}_u
-& = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
-     -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
-     +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
-D^{\textbf{U}}_v
-& = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
-     +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
-     +\frac{1}{e_3} \frac{\partial v}{\partial k}
+  D^{\textbf{U}}_u
+  & = \frac{1}{e_1} \frac{\partial \left( {A^{lm}\chi   } \right)}{\partial i}
+    -\frac{1}{e_2} \frac{\partial \left( {A^{lm}\zeta } \right)}{\partial j}
+    +\frac{1}{e_3} \frac{\partial u}{\partial k}      \\
+  D^{\textbf{U}}_v
+  & = \frac{1}{e_2 }\frac{\partial \left( {A^{lm}\chi   } \right)}{\partial j}
+    +\frac{1}{e_1 }\frac{\partial \left( {A^{lm}\zeta } \right)}{\partial i}
+    +\frac{1}{e_3} \frac{\partial v}{\partial k}
 \end{align*}
 
@@ -360,3 +399,6 @@
 Generally, \autoref{apdx:B_Lap_U} is used in both $z$- and $s$-coordinate systems,
 that is a Laplacian diffusion is applied on momentum along the coordinate directions.
+
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_C.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,4 +7,5 @@
 \chapter{Discrete Invariants of the Equations}
 \label{apdx:C}
+
 \minitoc
 
@@ -14,5 +16,4 @@
 
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -25,12 +26,12 @@
 
 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*}
+\[
+  U = e_{2u}\,e_{3u}\; u  \qquad  V = e_{1v}\,e_{3v}\; v  \qquad W = e_{1w}\,e_{2w}\; \omega
+\]
 
 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*}
+\[
+  b_u = e_{1u}\,e_{2u}\,e_{3u}  \qquad  b_v = e_{1v}\,e_{2v}\,e_{3v}  \qquad b_t = e_{1t}\,e_{2t}\,e_{3t}
+\]
 
 partial derivative notation: $\partial_\bullet = \frac{\partial}{\partial \bullet}$
@@ -42,35 +43,36 @@
 ($i.e.$ $s(k_s) = \eta$ 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} 
+  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*}
+\[
+  \frac{1}{e_{3t}} \partial_t (e_{3t})+ \frac{1}{b_t}  \biggl\{  \delta_i [U] + \delta_j [V] + \delta_k [W] \biggr\} = 0
+\]
 
 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*}
+\[
+  \partial_t \left( \int_D{ Q\;dv } \right) =0
+\]
 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*}
+\[
+  \partial_t \left( \int_D {Q\;dv} \right) =  \int_D { \partial_t \left( e_3 \, Q \right) e_1e_2\;di\,dj\,dk }
+  =  \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0
+\]
 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 } \\
+  \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 }  
+\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}
@@ -78,16 +80,16 @@
 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 } \\
+  \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 }
+\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}
-
 
 % ================================================================
@@ -97,12 +99,11 @@
 \label{sec:C.1}
 
-
 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}
+\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}
@@ -118,104 +119,108 @@
 \autoref{eq:Tot_Energy} for the latter form leads to:
 
-\begin{subequations} \label{eq:E_tot}
-
+% \label{eq:E_tot}
 advection term (vector invariant form):
-\begin{equation} \label{eq:E_tot_vect_vor_1}
-\int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
-\end{equation}
+\[
+  % \label{eq:E_tot_vect_vor_1}
+  \int\limits_D  \zeta \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
+\]
 %
-\begin{equation} \label{eq:E_tot_vect_adv_1}
-   \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}
-
+\[
+  % \label{eq:E_tot_vect_adv_1}
+  \int\limits_D  \textbf{U}_h \cdot \nabla_h \left( \frac{{\textbf{U}_h}^2}{2} \right)     dv
+  + \int\limits_D  \textbf{U}_h \cdot \nabla_z \textbf{U}_h  \;dv
+  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0
+\]
 advection term (flux form):
-\begin{equation} \label{eq: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{eq: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}
-
+\[
+  % \label{eq:E_tot_flux_metric}
+  \int\limits_D  \frac{1} {e_1 e_2 } \left( v \,\partial_i e_2 - u \,\partial_j e_1  \right)\;
+  \left(  \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0
+\]
+\[
+  % \label{eq:E_tot_flux_adv}
+  \int\limits_D \textbf{U}_h \cdot     \left(                 {{
+        \begin{array} {*{20}c}
+          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+          \nabla \cdot \left( \textbf{U}\,v \right) \hfill
+        \end{array}}
+    }           \right)   \;dv
+  +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3 \;dv } =\;0
+\]
 coriolis term
-\begin{equation} \label{eq:E_tot_cor}
-\int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0   \\
-\end{equation}
-
+\[
+  % \label{eq:E_tot_cor}
+  \int\limits_D  f   \; \left( \textbf{k} \times \textbf{U}_h  \right) \cdot \textbf{U}_h  \;  dv   = 0
+\]
 pressure gradient:
-\begin{equation} \label{eq:E_tot_pg_1}
-   - \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}
+\[
+  % \label{eq:E_tot_pg_1}
+  - \int\limits_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
+  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+  + \int\limits_D g\, \rho \; \partial_t z  \;dv
+\]
+
+where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.
+
+blah blah....
+
+The prognostic ocean dynamics equation can be summarized as follows:
+\[
+  \text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
+  {\text{COR} + \text{ADV}                       }
+  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
+\]
+
+Vector invariant form:
+% \label{eq:E_tot_vect}
+\[
+  % \label{eq:E_tot_vect_vor_2}
+  \int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0
+\]
+\[
+  % \label{eq:E_tot_vect_adv_2}
+  \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv
+  + \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv
+  -  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0
+\]
+\[
+  % \label{eq:E_tot_pg_2}
+  - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv
+  = - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
+  + \int\limits_D g\, \rho \; \partial_t z  \;dv
+\]
+
+Flux form:
+\begin{subequations}
+  \label{eq:E_tot_flux}
+  \[
+    % \label{eq:E_tot_flux_metric_2}
+    \int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0
+  \]
+  \[
+    % \label{eq:E_tot_flux_adv_2}
+    \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv
+    +   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0
+  \]
+  \begin{equation}
+    \label{eq:E_tot_pg_3}
+    - \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}
-
-where $\nabla_h = \left. \nabla \right|_k$ is the gradient along the $s$-surfaces.
-
-blah blah....
-$\ $\newline    % force a new ligne
-The prognostic ocean dynamics equation can be summarized as follows:
-\begin{equation*}
-\text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
-						{\text{COR} + \text{ADV}                       }
-			+ \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
-\end{equation*}
-$\ $\newline    % force a new ligne
-
-Vector invariant form:
-\begin{subequations} \label{eq:E_tot_vect}
-\begin{equation} \label{eq:E_tot_vect_vor_2}
-\int\limits_D   \textbf{U}_h \cdot \text{VOR} \;dv   = 0   \\
-\end{equation}
-\begin{equation} \label{eq:E_tot_vect_adv_2}
-   \int\limits_D  \textbf{U}_h \cdot \text{KEG}  \;dv 
-+ \int\limits_D  \textbf{U}_h \cdot \text{ZAD}  \;dv    
--  \int\limits_D { \frac{{\textbf{U}_h}^2}{2} \frac{1}{e_3} \partial_t e_3 \;dv }   = 0   \\
-\end{equation}
-\begin{equation} \label{eq:E_tot_pg_2}
-   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
-= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
-   + \int\limits_D g\, \rho \; \partial_t z  \;dv   \\
-\end{equation}
-\end{subequations}
-
-Flux form:
-\begin{subequations} \label{eq:E_tot_flux}
-\begin{equation} \label{eq:E_tot_flux_metric_2}
-\int\limits_D  \textbf{U}_h \cdot \text {COR} \;  dv   = 0   \\
-\end{equation}
-\begin{equation} \label{eq:E_tot_flux_adv_2}
-   \int\limits_D \textbf{U}_h \cdot \text{ADV}   \;dv 
-+   \frac{1}{2} \int\limits_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \partial_t e_3  \;dv } =\;0  \\
-\end{equation}
-\begin{equation} \label{eq:E_tot_pg_3}
-   - \int\limits_D  \textbf{U}_h \cdot (\text{HPG}+ \text{SPG}) \;dv 
-= - \int\limits_D \nabla \cdot \left( \rho \,\textbf {U} \right)\;g\;z\;\;dv
-   + \int\limits_D g\, \rho \; \partial_t  z  \;dv   \\
-\end{equation}
-\end{subequations}
-
-
-$\ $\newline    % force a new ligne
-
 
 \autoref{eq:E_tot_pg_3} is the balance between the conversion KE to PE and PE to KE. 
 Indeed the left hand side of \autoref{eq:E_tot_pg_3} 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   &&&\\
+  \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 $w$, 
@@ -224,36 +229,35 @@
 The left hand side of \autoref{eq:E_tot_pg_3} 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*}
-
+  - \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
@@ -262,56 +266,55 @@
 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,  \\ 
+  \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 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 \, \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       &&&\\
+  %
+  \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  \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   
+  \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     
+    - \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
@@ -319,5 +322,4 @@
 %
 
-
 % ================================================================
 % Discrete Total energy Conservation : vector invariant form
@@ -334,15 +336,18 @@
 The discrete form of the total energy conservation, \autoref{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  \\
+  \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{eq: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}
+\begin{equation}
+  \label{eq: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,
@@ -365,13 +370,15 @@
 
 For the ENE scheme, the two components of the vorticity term are given by:
-\begin{equation*}
-- e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv 
-	\left( {{  \begin{array} {*{20}c}
-		+ \frac{1} {e_{1u}} \; 
-		\overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j}        \hfill \\
-		- \frac{1} {e_{2v}} \; 
-		\overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill \\
-	\end{array}} }    \right)
-\end{equation*}
+\[
+  - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h    \equiv
+  \left( {{
+        \begin{array} {*{20}c}
+          + \frac{1} {e_{1u}} \;
+          \overline {\, q \ \overline {\left( e_{1v}\,e_{3v}\,v \right)}^{\,i+1/2}} ^{\,j}        \hfill \\
+          - \frac{1} {e_{2v}} \;
+          \overline {\, q \ \overline {\left( e_{2u}\,e_{3u}\,u \right)}^{\,j+1/2}} ^{\,i}       \hfill
+        \end{array}
+      } }    \right)
+\]
 
 This formulation does not conserve the enstrophy but it does conserve the total kinetic energy.
@@ -379,18 +386,20 @@
 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} } 		
+  &\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.
-
 
 % -------------------------------------------------------------------------------------------------------------
@@ -401,17 +410,21 @@
 
 With the EEN scheme, the vorticity terms are represented as: 
-\begin{equation} \tag{\ref{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.
+\begin{equation}
+  \tag{\ref{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 $j_p$ 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} \tag{\ref{eq: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)
+\begin{equation}
+  \tag{\ref{eq: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}
 
@@ -419,48 +432,46 @@
 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\{ 
-	    \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} &  \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*}
-
+  &\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
+  \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} &  \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*}
 
 % -------------------------------------------------------------------------------------------------------------
@@ -471,9 +482,9 @@
 
 The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~:
-\begin{equation*}
-    \int_D \textbf{U}_h \cdot \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*}
+\[
+  \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv
+  =  -   \int_D \textbf{U}_h \cdot \nabla_h \left( \frac{1}{2}\;{\textbf{U}_h}^2 \right)\;dv
+  +   \frac{1}{2} \int_D {  \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv }
+\]
 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator)
 and the continuity equation, then \autoref{eq:DOM_di_adj} again,
@@ -482,58 +493,58 @@
 applied in the horizontal and vertical directions, it becomes:
 \begin{flalign*}
-& - \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 &     \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 \autoref{eq: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 }    &&&\\
+  & - \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 &     \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 \autoref{eq: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*}
 
@@ -544,13 +555,15 @@
 For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$.
 This leads to the following expression for the vertical advection:
-\begin{equation*}
-\frac{1} {e_3 }\; \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} \,\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} \,\omega \;\delta_{k+1/2}
-\left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill \\
-\end{array}} } \right)
-\end{equation*}
+\[
+  \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h
+  \equiv \left( {{
+        \begin{array} {*{20}c}
+          \frac{1} {e_{1u}\,e_{2u}\,e_{3u}} \;  \overline{\overline {e_{1t}\,e_{2t} \,\omega\;\delta_{k+1/2}
+          \left[ \overline u^{\,i+1/2} \right]}}^{\,i+1/2,k}  \hfill \\
+          \frac{1} {e_{1v}\,e_{2v}\,e_{3v}} \;   \overline{\overline {e_{1t}\,e_{2t} \,\omega \;\delta_{k+1/2}
+          \left[ \overline v^{\,j+1/2} \right]}}^{\,j+1/2,k} \hfill
+        \end{array}
+      } } \right)
+\]
 a formulation that requires an additional horizontal mean in contrast with the one used in NEMO.
 Nine velocity points have to be used instead of 3.
@@ -560,12 +573,13 @@
 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}
+  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{eq: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}
+\begin{flalign*}
+  % \label{eq: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.....
@@ -588,9 +602,9 @@
 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
+\[
+  - \int_D  \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv
+  = - \int_D \nabla \cdot \left( \rho \,\textbf {U} \right) \,g\,z \;dv
   + \int_D g\, \rho \; \partial_t (z)  \;dv
-\end{equation*}
+\]
 
 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates.
@@ -599,57 +613,57 @@
 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 \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of 
-the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{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\}              &&&\\	
-%
+  &- \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 \autoref{eq:DOM_di_adj}, $i.e.$ the skew symmetry property of
+    the $\delta$ operator, \autoref{eq:wzv}, the continuity equation, \autoref{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 \autoref{eq:KE+PE_vect_discrete}.
@@ -659,6 +673,6 @@
 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\}                                
+  \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*}
 
@@ -667,22 +681,22 @@
 
 \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\}  
+  \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\}          
+  &\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\}  
+  \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\}  
+  \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 $p_w = \overline{p_t}^{\,k+1/2}$
@@ -690,28 +704,26 @@
 %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*}
-%
+  \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
@@ -719,10 +731,4 @@
 Nevertheless, it is almost never satisfied since a linear equation of state is rarely used.
 
-
-
-
-
-
-
 % ================================================================
 % Discrete Total energy Conservation : flux form
@@ -739,14 +745,14 @@
 The discrete form of the total energy conservation, \autoref{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  \\
+  \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\}                                     \\
+  \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*}
 
@@ -771,10 +777,10 @@
 This altered 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)\;
-\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{equation*}
+\[
+  f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\;
+  \equiv \;
+  f+\frac{1} {e_{1f}\,e_{2f}} \left( \overline v^{\,i+1/2} \delta_{i+1/2} \left[ e_{2u} \right]
+    -\overline u^{\,j+1/2} \delta_{j+1/2} \left[ e_{1u}  \right] \right)
+\]
 
 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form.
@@ -793,9 +799,13 @@
 Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is:
 
-\begin{equation} \label{eq: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 
--   \frac{1}{2} \int_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } =\;0
+\begin{equation}
+  \label{eq: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
+  -   \frac{1}{2} \int_D {  {\textbf{U}_h}^2 \frac{1}{e_3} \frac{\partial  e_3 }{\partial t} \;dv } =\;0
 \end{equation}
 
@@ -803,53 +813,55 @@
 ($i.e.$ 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 {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 {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\}    &&& \\
+  &  - \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 {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 {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*}
+\[
+  -  \int_D \textbf{U}_h \cdot     \left(                 {{
+        \begin{array} {*{20}c}
+          \nabla \cdot \left( \textbf{U}\,u \right) \hfill \\
+          \nabla \cdot \left( \textbf{U}\,v \right) \hfill \\
+        \end{array}
+      } }           \right)   \;dv
+  \equiv + \sum\limits_{i,j,k}  \frac{1}{2}  \left( \overline {u^2}^{\,i} + \overline {v^2}^{\,j} \right)
+  \biggl\{     \left(   \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t}   \right) \; b_t     \biggr\}
+\]
 which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left(   \textbf{U}\,u   \right) \; dv $.
 \autoref{eq:C_ADV_KE_flux} is thus satisfied.
-
 
 When the UBS scheme is used to evaluate the flux form momentum advection,
@@ -857,13 +869,4 @@
 The horizontal kinetic energy is not conserved, but forced to decay ($i.e.$ the scheme is diffusive). 
 
-
-
-
-
-
-
-
-
-
 % ================================================================
 % Discrete Enstrophy Conservation
@@ -872,5 +875,4 @@
 \label{sec:C.4}
 
-
 % -------------------------------------------------------------------------------------------------------------
 %       Vorticity Term with ENS scheme
@@ -880,9 +882,12 @@
 
 In the ENS scheme, the vorticity term is descretized as follows:
-\begin{equation} \tag{\ref{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.
+\begin{equation}
+  \tag{\ref{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} 
 
@@ -892,45 +897,50 @@
 ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}),
 it can be shown that:
-\begin{equation} \label{eq: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
+\begin{equation}
+  \label{eq: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 \autoref{eq:dynvor_ens},
 the discrete form of the right hand side of \autoref{eq:C_1.1} 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       \\
-%
-& \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} } 		
+\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 \\
+  %
+  & \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, $i.e.$ $\chi$=$0$. 
-
 
 % -------------------------------------------------------------------------------------------------------------
@@ -941,20 +951,23 @@
 
 With the EEN scheme, the vorticity terms are represented as: 
-\begin{equation} \tag{\ref{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.
+\begin{equation}
+  \tag{\ref{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 values: 
 $i_p = -1/2$ 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} \tag{\ref{eq: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)
+\begin{equation}
+  \tag{\ref{eq: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 ($i.e.$ $\chi=0$). 
@@ -965,46 +978,42 @@
 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& - \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*}
-
-
-
-
+\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& - \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*}
 
 % ================================================================
@@ -1013,5 +1022,4 @@
 \section{Conservation properties on tracers}
 \label{sec:C.5}
-
 
 All the numerical schemes used in NEMO are written such that the tracer content is conserved by
@@ -1032,16 +1040,15 @@
 
 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*}
+\[
+  \frac{\partial }{\partial t} \left(   \int_D {T\;dv}   \right)
+  =  \int_D { \frac{1}{e_3}\frac{\partial \left( e_3 \, T \right)}{\partial t} \;dv }=0
+\]
 
 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*}
-
+\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
@@ -1051,15 +1058,15 @@
 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]
+  &\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 &&&
+  &\equiv 0 &&&
 \end{flalign*}
 
@@ -1068,25 +1075,25 @@
 It can be demonstarted as follows:
 \begin{flalign*}
-&\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[ 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\}      &&& \\
+  &\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[ 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 }$.
@@ -1097,5 +1104,4 @@
 \section{Conservation properties on lateral momentum physics}
 \label{sec:dynldf_properties}
-
 
 The discrete formulation of the horizontal diffusion of momentum ensures
@@ -1122,28 +1128,28 @@
 The lateral momentum diffusion term conserves the potential vorticity:
 \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   \\ 
-%\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  \\ 
-%\end{flalign*}
-%\begin{flalign*}
-\equiv& \sum\limits_{i,j}
-	\left\{
-	  \delta_{i+1/2} \left[  \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
- 	+ \delta_{j+1/2} \left[  \frac {e_{1u}} {e_{2u}\,e_{3u}}  \delta_j \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
-	\right\} 	 \\ 
-%
-\intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
-%
-\equiv& \sum\limits_{i,j,k} 
-	-\,\left\{
-	   \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i	\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right]
+  &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times
+  \Bigl[    \nabla_h  \left( A^{\,lm}\;\chi  \right)
+  - \nabla_h \times \left( A^{\,lm}\;\zeta \; \textbf{k} \right)    \Bigr]\;dv   \\
+  % \end{flalign*}
+  %%%%%%%%%% 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  \\
+  % \end{flalign*}
+  % \begin{flalign*}
+  \equiv& \sum\limits_{i,j}
+  \left\{
+    \delta_{i+1/2} \left[  \frac {e_{2v}} {e_{1v}\,e_{3v}}  \delta_i \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
+    + \delta_{j+1/2} \left[  \frac {e_{1u}} {e_{2u}\,e_{3u}}  \delta_j \left[ A_f^{\,lm} e_{3f} \zeta  \right]  \right]
+  \right\} 	 \\
+  %
+  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
+  %
+  \equiv& \sum\limits_{i,j,k}
+  -\,\left\{
+    \frac{e_{2v}} {e_{1v}\,e_{3v}}  \delta_i	\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_i \left[ 1\right]
 	 + \frac{e_{1u}} {e_{2u}\,e_{3u}}  \delta_j 	\left[ A_f^{\,lm} e_{3f} \zeta  \right]\;\delta_j \left[ 1\right]
-	\right\} \quad \equiv 0 
-	 \\ 
+  \right\} \quad \equiv 0
+  \\
 \end{flalign*}
 
@@ -1156,47 +1162,47 @@
 The lateral momentum diffusion term dissipates the horizontal kinetic energy:
 %\begin{flalign*}
-\begin{equation*}
-\begin{split}
-\int_D \textbf{U}_h \cdot 
-	\left[ \nabla_h 		\right.   &     \left.       \left( A^{\,lm}\;\chi \right)      
-	- \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\
-\\  %%%
-\equiv& \sum\limits_{i,j,k} 
-	\left\{
-	  \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right]
-	- \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right]
-	\right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\
-&\;\; + 	\left\{
-	   \frac{1} {e_{2u}}             \delta_{j+1/2}	\left[ A_T^{\,lm}          \chi    \right] 
-	+ \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right] 
-	\right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\ 
-\\  %%%
-\equiv& \sum\limits_{i,j,k} 
- 	\Bigl\{
-	  e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right]
-	- e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right]
-	 \Bigl\} 
-	 \\ 
-&\;\; + \Bigl\{
-	   e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right]
-	+ e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
-	\Bigl\}      \\ 
-\\  %%%
-\equiv& \sum\limits_{i,j,k} 
-	- \Bigl(
-	  \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right]
-	+ \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right] 
-	     \Bigr) \;  A_T^{\,lm} \chi   \\ 
-&\;\; - \Bigl(
-	  \delta_{i+1/2}  \left[  e_{2v}  \;v  \right]
-	- \delta_{j+1/2}  \left[  e_{1u} \;u  \right] 
-	     \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\ 
-\\  %%%
-\equiv& \sum\limits_{i,j,k} 
-	- A_T^{\,lm}  \,\chi^2   \;e_{1t}\,e_{2t}\,e_{3t}
-	- A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}  
-\quad \leq 0       \\
-\end{split}
-\end{equation*}
+\[
+  \begin{split}
+    \int_D \textbf{U}_h \cdot
+    \left[ \nabla_h 		\right.   &     \left.       \left( A^{\,lm}\;\chi \right)
+      - \nabla_h \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)     \right] \; dv    \\
+    \\  %%%
+    \equiv& \sum\limits_{i,j,k}
+    \left\{
+      \frac{1} {e_{1u}}               \delta_{i+1/2} \left[  A_T^{\,lm}          \chi     \right]
+      - \frac{1} {e_{2u}\,e_{3u}}  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta   \right]
+    \right\} \; e_{1u}\,e_{2u}\,e_{3u} \;u     \\
+    &\;\; + 	\left\{
+	   \frac{1} {e_{2u}}             \delta_{j+1/2}	\left[ A_T^{\,lm}          \chi    \right]
+      + \frac{1} {e_{1v}\,e_{3v}} \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+    \right\} \; e_{1v}\,e_{2u}\,e_{3v} \;v     \qquad \\
+    \\  %%%
+    \equiv& \sum\limits_{i,j,k}
+    \Bigl\{
+    e_{2u}\,e_{3u} \;u\;  \delta_{i+1/2} \left[ A_T^{\,lm}           \chi    \right]
+    - e_{1u}             \;u\;  \delta_j           \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+	 \Bigl\}
+	 \\
+    &\;\; + \Bigl\{
+    e_{1v}\,e_{3v} \;v\;  \delta_{j+1/2}  \left[ A_T^{\,lm}           \chi    \right]
+    + e_{2v}             \;v\;  \delta_i            \left[ A_f^{\,lm} e_{3f} \zeta  \right]
+    \Bigl\}      \\
+    \\  %%%
+    \equiv& \sum\limits_{i,j,k}
+    - \Bigl(
+    \delta_i   \left[  e_{2u}\,e_{3u} \;u  \right]
+    + \delta_j  \left[  e_{1v}\,e_{3v}  \;v  \right]
+    \Bigr) \;  A_T^{\,lm} \chi   \\
+    &\;\; - \Bigl(
+    \delta_{i+1/2}  \left[  e_{2v}  \;v  \right]
+    - \delta_{j+1/2}  \left[  e_{1u} \;u  \right]
+    \Bigr)\;  A_f^{\,lm} e_{3f} \zeta      \\
+    \\  %%%
+    \equiv& \sum\limits_{i,j,k}
+    - A_T^{\,lm}  \,\chi^2   \;e_{1t}\,e_{2t}\,e_{3t}
+    - A_f ^{\,lm}  \,\zeta^2 \;e_{1f }\,e_{2f }\,e_{3f}
+    \quad \leq 0       \\
+  \end{split}
+\]
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1208,18 +1214,18 @@
 The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform:
 \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 &&&\\
-&\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times 
- 	\left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\
-&\quad \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\}   &&&\\ 
-%
-\intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
-%
-&\quad \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   b_v
-	         + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}  \quad \leq \;0    &&&\\
+  &\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 &&&\\
+  &\quad = A^{\,lm} \int \limits_D \zeta \textbf{k} \cdot \nabla \times
+  \left[    \nabla_h \times \left( \zeta \; \textbf{k} \right)   \right]\;dv &&&\\
+  &\quad \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\}   &&&\\
+  %
+  \intertext{Using \autoref{eq:DOM_di_adj}, it follows:}
+  %
+  &\quad \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   b_v
+    + \left(  \frac{1} {e_{2u}\,e_{3u}}  \delta_j \left[ e_{3f} \zeta  \right] \right)^2   b_u  \right\}  \quad \leq \;0    &&&\\
 \end{flalign*}
 
@@ -1234,19 +1240,19 @@
 The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform.
 \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   \\
-%
-&\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\}    \\ 
-%
-\intertext{Using \autoref{eq: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\} 
-	\quad \equiv 0      \\ 
+  & \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   \\
+  %
+  &\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\}    \\
+  %
+  \intertext{Using \autoref{eq: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\}
+  \quad \equiv 0
 \end{flalign*}
 
@@ -1258,21 +1264,21 @@
 
 \begin{flalign*}
-&\int\limits_D \chi \;\nabla_h \cdot 
-	\left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right)
-	        - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv
- = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    \\ 
-%
-&\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}    \\ 
-%
-\intertext{Using \autoref{eq: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  b_u
-            + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\}    
-\quad \leq 0             \\
+  &\int\limits_D \chi \;\nabla_h \cdot
+  \left[    \nabla_h              \left( A^{\,lm}\;\chi                    \right)
+    - \nabla_h   \times  \left( A^{\,lm}\;\zeta \;\textbf{k} \right)    \right]\;  dv
+  = A^{\,lm}   \int\limits_D \chi \;\nabla_h \cdot \nabla_h \left( \chi \right)\;  dv    \\
+  %
+  &\equiv A^{\,lm}  \sum\limits_{i,j,k}  \frac{1} {e_{1t}\,e_{2t}\,e_{3t}}  \chi
+  \left\{
+    \delta_i  \left[   \frac{e_{2u}\,e_{3u}} {e_{1u}}  \delta_{i+1/2} \left[ \chi \right]   \right]
+    + \delta_j  \left[   \frac{e_{1v}\,e_{3v}} {e_{2v}}   \delta_{j+1/2} \left[ \chi \right]   \right]
+  \right\} \;   e_{1t}\,e_{2t}\,e_{3t}    \\
+  %
+  \intertext{Using \autoref{eq: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  b_u
+    + \left(  \frac{1} {e_{2v}}  \delta_{j+1/2}  \left[ \chi \right]  \right)^2  b_v    \right\}
+  \quad \leq 0
 \end{flalign*}
 
@@ -1287,13 +1293,13 @@
 The first two are associated 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}}    \\
-%
-\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     \\
+  \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
 \end{align*}
 
@@ -1301,20 +1307,20 @@
 The second results from:
 \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    &&&\\
-\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)   &&&\\ 
-%
-\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   &&&\\
+  \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    &&&\\
+\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)   &&&
+  %
+  \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   &&&
 \end{flalign*}
 
@@ -1322,52 +1328,52 @@
 Indeed:
 \begin{flalign*}
-\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   &&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
-	\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)   &&\\
-	\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)
-	\biggr\} \;
-	e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   && \\
+  \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   &&&
+\end{flalign*}
+\begin{flalign*}
+  \equiv \sum\limits_{i,j,k}  \frac{1} {e_{3f}}\; \frac{1} {e_{1f}\,e_{2f}}
+  \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)   &&\\
+  \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)
+  \biggr\} \;
+  e_{1f}\,e_{2f}\,e_{3f} \; \equiv 0   &&
 \end{flalign*}
 
 If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$
 \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   &&&\\
+  \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   &&&
 \end{flalign*}
 
 This property is only satisfied in $z$-coordinates:
 \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   &&& \\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
-	\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}[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} [u]  \right]    \right) \biggr\}   &&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} \zeta \;e_{3f} 
-	\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]    &&\\ 
- 	 \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\}  &&\\ 
+  \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   &&&
+\end{flalign*}
+\begin{flalign*}
+  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f} \;
+  \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}[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} [u]  \right]    \right) \biggr\}   &&
+\end{flalign*}
+\begin{flalign*}
+  \equiv \sum\limits_{i,j,k} \zeta \;e_{3f}
+  \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]    &&\\
+  \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\}  &&
 \end{flalign*}
 Using the fact that the vertical diffusion coefficients are uniform,
@@ -1375,18 +1381,18 @@
 $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]    &&&\\
-\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    &&&\\
+  \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]    &&&
+\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    &&&
 \end{flalign*}
 Similarly, the horizontal divergence is obviously conserved:
 
 \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    &&&\\
+  \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    &&&
 \end{flalign*}
 and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if
@@ -1394,56 +1400,56 @@
 
 \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  &&&\\
+  \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  &&&
 \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}}
-	\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} [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} [v] \right]   \right)
-	\Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   &&\\ 
-\end{flalign*}
-
-\begin{flalign*}
-\equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \,
-	\biggl\{	\biggr.	\quad
-	\delta_{i+1/2}
-		&\left( 
-		   \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\}    && \\ 
-\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\}    &&&\\
-\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]   &&&\\
-\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    &&&\\
+  \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}}
+  \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} [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} [v] \right]   \right)
+  \Biggr\} \;  e_{1t}\,e_{2t}\,e_{3t}   &&
+\end{flalign*}
+
+\begin{flalign*}
+  \equiv A^{\,vm} \sum\limits_{i,j,k}  \chi \,
+  \biggl\{	\biggr.	\quad
+  \delta_{i+1/2}
+  &\left(
+    \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\}    &&
+\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\}    &&&
+\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]   &&&
+\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    &&&
 \end{flalign*}
 
@@ -1468,26 +1474,25 @@
 constraint of conservation of tracers:
 \begin{flalign*}
-&\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
-\\
-&\equiv \sum\limits_{i,j,k} 
-	\biggl\{ 	\biggr.
-	\delta_i 
-		\left[ 
-		A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
-			\left[ T \right]
-		\right]
-	+ \delta_j 
-		\left[ 
-		A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
-			\left[ T \right] 
-		\right]
-	&&\\  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
-	+ \delta_k 
-		\left[ 
-		A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2} 
-			\left[ T \right] 
-		\right]
-	\biggr\}   \quad  \equiv 0
-	&&\\ 
+  &\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&& \\ \\
+  &\equiv \sum\limits_{i,j,k}
+  \biggl\{ 	\biggr.
+  \delta_i
+  \left[
+    A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2}
+    \left[ T \right]
+  \right]
+  + \delta_j
+  \left[
+    A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2}
+    \left[ T \right]
+  \right] && \\
+  & \qquad \qquad \qquad \qquad \qquad \qquad \quad \;\;\;
+  + \delta_k
+  \left[
+    A_w^{\,vT} \frac{e_{1t}\,e_{2t}} {e_{3t}} \delta_{k+1/2}
+    \left[ T \right]
+  \right]
+  \biggr\}   \quad  \equiv 0
+  &&
 \end{flalign*}
 
@@ -1502,25 +1507,26 @@
 constraint on the dissipation of tracer variance:
 \begin{flalign*}
-\int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv	&&&\\ 
-&\equiv   \sum\limits_{i,j,k} \; T
-\biggl\{  \biggr.
-     \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
-& + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 
-		\quad&& \\ 
- \biggl. 
-&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right]
-\biggr\} && 
-\end{flalign*}
-\begin{flalign*}
-\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      && \\ 
-\end{flalign*}
-
+  \int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv	&&&\\
+  &\equiv   \sum\limits_{i,j,k} \; T
+  \biggl\{  \biggr.
+  \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
+  & + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right]
+  \quad&& \\
+  \biggl.
+  &&+ \delta_k \left[A_w^{\,vT}\frac{e_{1t}\,e_{2t}} {e_{3t}}\delta_{k+1/2}\left[T\right]\right]
+  \biggr\} &&
+\end{flalign*}
+\begin{flalign*}
+  \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      &&
+\end{flalign*}
 
 %%%%  end of appendix in gm comment
 %}
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_D.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_D.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_D.tex	(revision 10419)
@@ -1,15 +1,14 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Appendix D Ñ Coding Rules
+% Appendix D Coding Rules
 % ================================================================
 \chapter{Coding Rules}
 \label{apdx:D}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
-$\ $\newline    % force a new ligne
-
 
 A "model life" is more than ten years.
@@ -102,6 +101,6 @@
 - use call to ctl\_stop routine instead of just a STOP.
 
-
 \newpage
+
 % ================================================================
 % Naming Conventions
@@ -116,77 +115,74 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]  \label{tab:VarName}
-\begin{center}
-\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\_ np\_}& 
-\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{ld}& 
-\textbf{cd}& 
-\textbf{sd}& 
-\textbf{dd}& 
-\textbf{yd} \\
-\hline
-local \par variable& 
-\textbf{i}& 
-\textbf{z}& 
-\textbf{ll}& 
-\textbf{cl}& 
-\textbf{sl}& 
-\textbf{dl}& 
-\textbf{yl} \\
-\hline
-loop \par control& 
-\textbf{j} \par \textit{but not} \par \textbf{jp}& 
-& 
-& 
-&
-& 
-& 
- \\
-\hline
-parameter& 
-\textbf{jp np\_}& 
-\textbf{pp}& 
-\textbf{lp}& 
-\textbf{cp}& 
-\textbf{sp}& 
-\textbf{dp}& 
-\textbf{yp} \\
-\hline
-
-namelist&
-\textbf{nn\_}& 
-\textbf{rn\_}& 
-\textbf{ln\_}& 
-\textbf{cn\_}& 
-\textbf{sn\_}& 
-\textbf{dn\_}& 
-\textbf{yn\_}
-\\
-\hline
-CPP \par macro& 
-\textbf{kf}& 
-\textbf{fs} \par & 
-& 
-&
-& 
-& 
- \\
-\hline
-\end{tabular}
-\label{tab:tab1}
-\end{center}
+\begin{table}[htbp]
+  \label{tab:VarName}
+  \begin{center}
+    \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\_ np\_}
+      & \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{ld}
+      & \textbf{cd}
+      & \textbf{sd}
+      & \textbf{dd}
+      & \textbf{yd} \\
+      \hline
+      local \par variable
+      & \textbf{i}
+      & \textbf{z}
+      & \textbf{ll}
+      & \textbf{cl}
+      & \textbf{sl}
+      & \textbf{dl}
+      & \textbf{yl} \\
+      \hline
+      loop \par control
+      & \textbf{j} \par \textit{but not} \par \textbf{jp} &&&&&& \\
+      \hline
+      parameter
+      & \textbf{jp np\_}
+      & \textbf{pp}
+      & \textbf{lp}
+      & \textbf{cp}
+      & \textbf{sp}
+      & \textbf{dp}
+      & \textbf{yp} \\
+      \hline
+      namelist
+      & \textbf{nn\_}
+      & \textbf{rn\_}
+      & \textbf{ln\_}
+      & \textbf{cn\_}
+      & \textbf{sn\_}
+      & \textbf{dn\_}
+      & \textbf{yn\_}
+      \\
+      \hline
+      CPP \par macro
+      & \textbf{kf}
+      & \textbf{fs} \par &&&&& \\
+      \hline
+    \end{tabular}
+    \label{tab:tab1}
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -197,10 +193,13 @@
 
 \newpage
+
 % ================================================================
 % The program structure
 % ================================================================
 %\section{Program structure}
-%abel{sec:Apdx_D_structure}
+%\label{sec:Apdx_D_structure}
 
 %To be done....
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_E.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_E.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_E.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,12 +7,10 @@
 \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
 
 % -------------------------------------------------------------------------------------------------------------
@@ -25,24 +24,27 @@
 It is also known as Cell Averaged QUICK scheme (Quadratic Upstream Interpolation for Convective Kinematics).
 For example, in the $i$-direction:
-\begin{equation} \label{eq:tra_adv_ubs2}
-\tau _u^{ubs} = \left\{	 \begin{aligned}
-  & \tau _u^{cen4} + \frac{1}{12} \,\tau"_i	   & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
-  & \tau _u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0
-  						 \end{aligned}    \right.
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \tau_u^{ubs} = \left\{
+	 \begin{aligned}
+      & \tau_u^{cen4} + \frac{1}{12} \,\tau"_i	   & \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+      & \tau_u^{cen4} - \frac{1}{12} \,\tau"_{i+1} & \quad \text{if }\ u_{i+1/2}       <       0
+    \end{aligned}
+  \right.
 \end{equation}
 or equivalently, the advective flux is
-\begin{equation} \label{eq:tra_adv_ubs2}
-U_{i+1/2} \ \tau _u^{ubs} 
-=U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2}
-- \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  U_{i+1/2} \ \tau_u^{ubs}
+  =U_{i+1/2} \ \overline{ T_i - \frac{1}{6}\,\tau"_i }^{\,i+1/2}
+  - \frac{1}{2}\, |U|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
 \end{equation}
 where $U_{i+1/2} = e_{1u}\,e_{3u}\,u_{i+1/2}$ and
-$\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+$\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$.
 By choosing this expression for $\tau "$ we consider a fourth order approximation of $\partial_i^2$ with
 a constant i-grid spacing ($\Delta i=1$).
 
 Alternative choice: introduce the scale factors:  
-$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta _{i+1/2}[\tau] \right]$.
-
+$\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} }\delta_{i+1/2}[\tau] \right]$.
 
 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
@@ -76,5 +78,5 @@
 
 NB 2: In a forthcoming release four options will be proposed for the vertical component used in the UBS scheme.
-$\tau _w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme,
+$\tau_w^{ubs}$ will be evaluated using either \textit{(a)} a centered $2^{nd}$ order scheme,
 or \textit{(b)} a TVD scheme, or \textit{(c)} an interpolation based on conservative parabolic splines following
 \citet{Shchepetkin_McWilliams_OM05} implementation of UBS in ROMS, or \textit{(d)} an UBS.
@@ -82,17 +84,21 @@
 
 NB 3: It is straight forward to rewrite \autoref{eq:tra_adv_ubs} as follows:
-\begin{equation} \label{eq:tra_adv_ubs2}
-\tau _u^{ubs} = \left\{	 \begin{aligned}
-	& \tau _u^{cen4} + \frac{1}{12} \tau"_i		& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
-	& \tau _u^{cen4} - \frac{1}{12} \tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
-  						 \end{aligned}    \right.
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \tau_u^{ubs} = \left\{
+	 \begin{aligned}
+      & \tau_u^{cen4} + \frac{1}{12} \tau"_i		& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+      & \tau_u^{cen4} - \frac{1}{12} \tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
+    \end{aligned}
+  \right.
 \end{equation}
 or equivalently 
-\begin{equation} \label{eq:tra_adv_ubs2}
-\begin{split}
-e_{2u} e_{3u}\,u_{i+1/2} \ \tau _u^{ubs} 
-&= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\
-& - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
-\end{split}
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \begin{split}
+    e_{2u} e_{3u}\,u_{i+1/2} \ \tau_u^{ubs}
+    &= e_{2u} e_{3u}\,u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\tau"_i }^{\,i+1/2} \\
+    & - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+  \end{split}
 \end{equation}
 \autoref{eq:tra_adv_ubs2} has several advantages.
@@ -105,78 +111,80 @@
 
 laplacian diffusion:
-\begin{equation} \label{eq:tra_ldf_lap}
-\begin{split}
-D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta _i 
-\left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta _{i+1/2} 
-\left[ T \right]} \right]} \right.
-\\
-&\ \left. {+\; \delta _j \left[ 
-{A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta _{j+1/2} \left[ T 
-\right]} \right)} \right]\quad } \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_ldf_lap}
+  \begin{split}
+    D_T^{lT} =\frac{1}{e_{1T} \; e_{2T}\;  e_{3T} } &\left[ {\quad \delta_i
+        \left[ {A_u^{lT} \frac{e_{2u} e_{3u} }{e_{1u} }\;\delta_{i+1/2}
+            \left[ T \right]} \right]} \right. \\
+    &\ \left. {+\; \delta_j \left[
+          {A_v^{lT} \left( {\frac{e_{1v} e_{3v} }{e_{2v} }\;\delta_{j+1/2} \left[ T
+                \right]} \right)} \right]\quad } \right]
+  \end{split}
 \end{equation}
 
 bilaplacian:
-\begin{equation} \label{eq:tra_ldf_lap}
-\begin{split}
-D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
-& \delta _i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 
-		  \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
-    \delta _i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta _{i+1/2} 
-    		  [T] \right] \right] \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_ldf_lap}
+  \begin{split}
+    D_T^{lT} =&-\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
+    & \delta_i \left[  \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}
+      \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+        \delta_i \left[ \sqrt{A_u^{lT}}\ \frac{e_{2u}\,e_{3u}}{e_{1u}}\;\delta_{i+1/2}
+          [T] \right] \right] \right]
+  \end{split}
 \end{equation}
 with ${A_u^{lT}}^2 = \frac{1}{12} {e_{1u}}^3\ |u|$, 
 $i.e.$ $A_u^{lT} = \frac{1}{\sqrt{12}} \,e_{1u}\ \sqrt{ e_{1u}\,|u|\,}$
 it comes:
-\begin{equation} \label{eq:tra_ldf_lap}
-\begin{split}
-D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
-& \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 
-		 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
-    \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta _{i+1/2} 
-    		[T] \right] \right] \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_ldf_lap}
+  \begin{split}
+    D_T^{lT} =&-\frac{1}{12}\,\frac{1}{e_{1T} \; e_{2T}\;  e_{3T}} \\
+    & \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}
+      \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+        \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}\,|u|\,}\;\delta_{i+1/2}
+          [T] \right] \right] \right]
+  \end{split}
 \end{equation}
 if the velocity is uniform ($i.e.$ $|u|=cst$) then the diffusive flux is
-\begin{equation} \label{eq:tra_ldf_lap}
-\begin{split}
-F_u^{lT} = - \frac{1}{12}
- e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta _{i+1/2} 
-		 \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}} 
-    \delta _i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta _{i+1/2} 
-    		[T] \right] \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_ldf_lap}
+  \begin{split}
+    F_u^{lT} = - \frac{1}{12}
+    e_{2u}\,e_{3u}\,|u| \;\sqrt{ e_{1u}}\,\delta_{i+1/2}
+    \left[ \frac{1}{e_{1T}\,e_{2T}\, e_{3T}}
+      \delta_i \left[ e_{2u}\,e_{3u}\,\sqrt{ e_{1u}}\:\delta_{i+1/2}
+        [T] \right] \right]
+  \end{split}
 \end{equation}
 beurk....  reverte the logic: starting from the diffusive part of the advective flux it comes:
 
-\begin{equation} \label{eq:tra_adv_ubs2}
-\begin{split}
-F_u^{lT}
-&= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
-\end{split}
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \begin{split}
+    F_u^{lT} &= - \frac{1}{2} e_{2u} e_{3u}\,|u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+  \end{split}
 \end{equation}
 if the velocity is uniform ($i.e.$ $|u|=cst$) and
-choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right]$
+choosing $\tau "_i =\frac{e_{1T}}{e_{2T}\,e_{3T}}\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right]$
 
 sol 1 coefficient at T-point ( add $e_{1u}$ and $e_{1T}$ on both side of first $\delta$):
-\begin{equation} \label{eq:tra_adv_ubs2}
-\begin{split}
-F_u^{lT}
-&= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta _i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \begin{split}
+    F_u^{lT} &= - \frac{1}{12} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{e_{1T}^3\,|u|}{e_{1T}e_{2T}\,e_{3T}}\,\delta_i \left[ \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right]
+  \end{split}
 \end{equation}
 which leads to ${A_T^{lT}}^2 = \frac{1}{12} {e_{1T}}^3\ \overline{|u|}^{\,i+1/2}$
 
 sol 2 coefficient at u-point: split $|u|$ into $\sqrt{|u|}$ and $e_{1T}$ into $\sqrt{e_{1u}}$
-\begin{equation} \label{eq:tra_adv_ubs2}
-\begin{split}
-F_u^{lT}
-&= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta _i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta _{i+1/2}[\tau] \right] \right] \\
-&= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta _i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta _{i+1/2}[\tau] \right] \right]
-\end{split}
+\begin{equation}
+  \label{eq:tra_adv_ubs2}
+  \begin{split}
+    F_u^{lT} &= - \frac{1}{12} {e_{1u}}^1 \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{2T}\,e_{3T}}\,\delta_i \left[ \sqrt{e_{1u}|u|} \frac{e_{2u} e_{3u} }{e_{1u} } \delta_{i+1/2}[\tau] \right] \right] \\
+    &= - \frac{1}{12} e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u}}{e_{1u}}\;\delta_{i+1/2}\left[ \frac{1}{e_{1T}\,e_{2T}\,e_{3T}}\,\delta_i \left[ e_{1u} \sqrt{e_{1u}|u|\,} \frac{e_{2u} e_{3u} }{e_{1u}} \delta_{i+1/2}[\tau] \right] \right]
+  \end{split}
 \end{equation}
 which leads to ${A_u^{lT}} = \frac{1}{12} {e_{1u}}^3\ |u|$
-
 
 % -------------------------------------------------------------------------------------------------------------
@@ -189,10 +197,11 @@
 Given the values of a variable $q$ at successive time step,
 the time derivation and averaging operators at the mid time step are:
-\begin{subequations} \label{eq:dt_mt}
-\begin{align}
- \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}
+\[
+  % \label{eq:dt_mt}
+  \begin{split}
+    \delta_{t+\rdt/2} [q]     &=  \  \ \,   q^{t+\rdt}  - q^{t}		\\
+    \overline q^{\,t+\rdt/2} &= \left\{ q^{t+\rdt} + q^{t} \right\} \; / \; 2
+  \end{split}
+\]
 As for space operator,
 the adjoint of the derivation and averaging time operators are $\delta_t^*=\delta_{t+\rdt/2}$ and
@@ -200,39 +209,38 @@
 
 The Leap-frog time stepping given by \autoref{eq:DOM_nxt} can be defined as:
-\begin{equation} \label{eq:LF}
-   \frac{\partial q}{\partial t} 
-   		\equiv \frac{1}{\rdt} \overline{ \delta _{t+\rdt/2}[q]}^{\,t} 
-		=         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt}
-\end{equation} 
+\[
+  % \label{eq:LF}
+  \frac{\partial q}{\partial t}
+  \equiv \frac{1}{\rdt} \overline{ \delta_{t+\rdt/2}[q]}^{\,t}
+  =         \frac{q^{t+\rdt}-q^{t-\rdt}}{2\rdt}
+\] 
 Note that \autoref{chap:LF} shows that the leapfrog time step is $\rdt$,
 not $2\rdt$ as it can be found sometimes in literature.
 The leap-Frog time stepping is a second order centered scheme.
 As such it respects the quadratic invariant in integral forms, $i.e.$ the following continuous property,
-\begin{equation} \label{eq:Energy}
-\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
-	=\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt} 
-	=  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) ,
-\end{equation}
+\[
+  % \label{eq:Energy}
+  \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
+  =\int_{t_0}^{t_1} {\frac{1}{2}\, \frac{\partial q^2}{\partial t} \;dt}
+  =  \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right) ,
+\]
 is satisfied in discrete form.
 Indeed, 
-\begin{equation} \begin{split}
-\int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt} 
-	&\equiv \sum\limits_{0}^{N} 
-			{\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}
+\[
+  \begin{split}
+    \int_{t_0}^{t_1} {q\, \frac{\partial q}{\partial t} \;dt}
+    &\equiv \sum\limits_{0}^{N}
+    {\frac{1}{\rdt} q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} \ \rdt}
+    \equiv \sum\limits_{0}^{N}  { q^t \ \overline{ \delta_{t+\rdt/2}[q]}^{\,t} } \\
+    &\equiv \sum\limits_{0}^{N}  { \overline{q}^{\,t+\Delta/2}{ \delta_{t+\rdt/2}[q]}}
+    \equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }\\
+    &\equiv \sum\limits_{0}^{N}  { \frac{1}{2} \delta_{t+\rdt/2}[q^2] }
+    \equiv \frac{1}{2} \left( {q_{t_1}}^2 - {q_{t_0}}^2 \right)
+  \end{split}
+\]
 NB here pb of boundary condition when applying the adjoint!
 In space, setting to 0 the quantity in land area is sufficient to get rid of the boundary condition 
 (equivalently of the boundary value of the integration by part).
 In time this boundary condition is not physical and \textbf{add something here!!!}
-
-
-
-
-
 
 % ================================================================
@@ -269,10 +277,11 @@
 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{eq: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)
+\begin{equation}
+  \label{eq: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:
@@ -281,9 +290,10 @@
 $A_i^k$ 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{eq: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 ] }
+\begin{equation}
+  \label{eq: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 \autoref{eq:Gf_slopes} we use the ratio $\alpha / \beta$ instead of
@@ -296,10 +306,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht] \begin{center}
-\includegraphics[width=0.70\textwidth]{Fig_ISO_triad}
-\caption{  \protect\label{fig:ISO_triad}
-  Triads used in the Griffies's like iso-neutral diffision scheme for
-  $u$-component (upper panel) and $w$-component (lower panel).}
-\end{center}
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.70\textwidth]{Fig_ISO_triad}
+    \caption{
+      \protect\label{fig:ISO_triad}
+      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}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -307,50 +320,52 @@
 The four iso-neutral fluxes associated with the triads are defined at $T$-point. 
 They take the following expression:
-\begin{flalign} \label{eq: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}
+\begin{flalign*}
+  % \label{eq: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 (\autoref{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}      
+\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{eq: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\}
+\begin{equation}
+  \label{eq: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. 
@@ -361,10 +376,11 @@
   The discretization of the diffusion operator recovers the traditional five-point Laplacian in
   the limit of flat iso-neutral direction:
-\begin{equation} \label{eq: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}
+  \[
+    % \label{eq:Gf_property1a}
+    D_l^T = \frac{1}{b_T}  \ \delta_{i}
+    \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \; \overline{A}^{\,i} \; \delta_{i+1/2}[T] \right]
+    \qquad  \text{when} \quad
+    { _i^k \mathbb{R}_{i_p}^{k_p} }=0
+  \]
 
 \item[$\bullet$ implicit treatment in the vertical]
@@ -374,39 +390,42 @@
   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\{  
+  It is a necessity since the vertical eddy diffusivity associated with this term,
+  \[
+	 \sum_{\substack{i_p, \,k_p}} \left\{
 		A_i^k \; \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
-	\right\} 
-\end{equation}
-can be quite large.
+    \right\}
+  \]
+  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{eq: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 \autoref{eq:Gf_triads} applied to $T$ and $S$ and
-the definition of the triads' slopes \autoref{eq:Gf_slopes}.
+  \begin{align*}
+    % \label{eq: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 \autoref{eq:Gf_triads} applied to $T$ and $S$ and
+  the definition of the triads' slopes \autoref{eq:Gf_slopes}.
 
 \item[$\bullet$ conservation of tracer]
   The iso-neutral diffusion term conserve the total tracer content, $i.e.$
-\begin{equation} \label{eq:Gf_property1}
-\sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0
-\end{equation}
+  \[
+    % \label{eq:Gf_property1}
+    \sum_{i,j,k} \left\{ D_l^T \ b_T \right\} = 0
+  \]
 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{eq:Gf_property1}
-\sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0
-\end{equation}
+  \[
+    % \label{eq:Gf_property1}
+    \sum_{i,j,k} \left\{ T \ D_l^T \ b_T \right\} \leq 0
+  \]
 The property is demonstrated in the \autoref{apdx:Gf_operator}.
 It is a key property for a diffusion term.
@@ -418,7 +437,8 @@
 \item[$\bullet$ self-adjoint operator]
   The iso-neutral diffusion operator is self-adjoint, $i.e.$
-\begin{equation} \label{eq: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}
+  \[
+    % \label{eq:Gf_property1}
+    \sum_{i,j,k} \left\{ S \ D_l^T \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
+  \]
 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.
@@ -427,6 +447,4 @@
 \end{description}
 
-
-$\ $\newline      %force an empty line
 % ================================================================
 % Skew flux formulation for Eddy Induced Velocity : 
@@ -443,13 +461,14 @@
 
 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}
+\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,
@@ -475,15 +494,17 @@
 %\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}
+\[
+  % \label{eq:eiv_vd}
+  \textbf{F}_{eiv}^T   \equiv   \left(
+    \begin{aligned}
+      \sum_{\substack{i_p,\,k_p}} &
+      +{e_{2u}}_{i+1/2-i_p}^{k}                                  \ \ {A_{e}}_{i+1/2-i_p}^{k}
+      \ \ \ { _{i+1/2-i_p}^k \mathbb{R}_{i_p}^{k_p} }    \ \ \delta_{k+k_p}[T_{i+1/2-i_p}] \\ \\
+      \sum_{\substack{i_p,\,k_p}} &
+      - {e_{2u}}_i^{k+1/2-k_p}                                      \ {A_{e}}_i^{k+1/2-k_p}
+      \ \ { _i^{k+1/2-k_p} \mathbb{R}_{i_p}^{k_p} }    \ \delta_{i+i_p}[T^{k+1/2-k_p}]
+    \end{aligned}
+  \right)
+\]
 
 \citep{Griffies_JPO98} introduces another way to implement the eddy induced advection, the so-called skew form.
@@ -491,34 +512,38 @@
 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}
+  \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}
+\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
@@ -528,20 +553,23 @@
 Another interesting property of \autoref{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}
+\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 keeps the same complexity, but not more.
@@ -552,14 +580,16 @@
 Using the slopes \autoref{eq: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)
+\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 \autoref{eq:eiv_skew} is valid in $z$-coordinate with or without partial cells. 
@@ -572,7 +602,4 @@
 $i.e.$ it does not include a diffusive component but is a "pure" advection term.
 
-
-
-
 $\ $\newpage      %force an empty line
 % ================================================================
@@ -587,125 +614,130 @@
 
 The continuous property to be demonstrated is:
+\[
+  \int_D  D_l^T \; T \;dv   \leq 0
+\]
+The discrete form of its left hand side is obtained using \autoref{eq:iso_flux}
+
 \begin{align*}
-\int_D  D_l^T \; T \;dv   \leq 0
-\end{align*}
-The discrete form of its left hand side is obtained using \autoref{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,
+  &\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
+  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 \autoref{eq: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
+  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.
@@ -714,27 +746,28 @@
 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\}   
+  \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
+\newpage
+
 % ================================================================
 % Discrete Invariants of the skew flux formulation
@@ -743,5 +776,4 @@
 \label{subsec:eiv_skew}
 
-
 Demonstration for the conservation of the tracer variance in the (\textbf{i},\textbf{j}) plane. 
 
@@ -750,53 +782,53 @@
 The continuous property to be demonstrated is:
 \begin{align*}
-\int_D \nabla \cdot \textbf{F}_{eiv}(T) \; T \;dv  \equiv 0
+  \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 \autoref{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\}   
+  \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\}       
+  \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\ \ \ \:}       
+  \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}   
+    %
+    &&-{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,
@@ -810,3 +842,5 @@
 $i.e.$ the variance of the tracer is preserved by the discretisation of the skew fluxes.
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_iso.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_iso.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/annex_iso.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -7,6 +8,9 @@
 			{\texorpdfstring{Iso-Neutral Diffusion and\\ Eddy Advection using Triads}{Iso-Neutral Diffusion and Eddy Advection using Triads}}
 \label{apdx:triad}
+
 \minitoc
-\pagebreak
+
+\newpage
+
 \section{Choice of \protect\ngn{namtra\_ldf} namelist parameters}
 %-----------------------------------------nam_traldf------------------------------------------------------
@@ -59,11 +63,14 @@
 \section{Triad formulation of iso-neutral diffusion}
 \label{sec:iso}
+
 We have implemented into \NEMO a scheme inspired by \citet{Griffies_al_JPO98},
 but formulated within the \NEMO framework, using scale factors rather than grid-sizes.
 
 \subsection{Iso-neutral diffusion operator}
+
 The iso-neutral second order tracer diffusive operator for small angles between
 iso-neutral surfaces and geopotentials is given by \autoref{eq:iso_tensor_1}:
-\begin{subequations} \label{eq:iso_tensor_1}
+\begin{subequations}
+  \label{eq:iso_tensor_1}
   \begin{equation}
     D^{lT}=-\Div\vect{f}^{lT}\equiv
@@ -79,6 +86,6 @@
     \mbox{with}\quad \;\;\Re =
     \begin{pmatrix}
-       1   &  0   & -r_1           \mystrut \\
-       0   &  1   & -r_2           \mystrut \\
+      1   &  0   & -r_1           \mystrut \\
+      0   &  1   & -r_2           \mystrut \\
       -r_1 & -r_2 &  r_1 ^2+r_2 ^2 \mystrut
     \end{pmatrix}
@@ -88,5 +95,6 @@
       \frac{1}{e_2} \pd[T]{j} \mystrut \\
       \frac{1}{e_3} \pd[T]{k} \mystrut
-    \end{pmatrix}.
+    \end{pmatrix}
+    .
   \end{equation}
 \end{subequations}
@@ -99,10 +107,10 @@
 \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}
+        \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 iso-neutral surface relative to the computational surface,
@@ -110,8 +118,8 @@
 
 We will find it useful to consider the fluxes per unit area in $i,j,k$ space; we write
-\begin{equation}
-  \label{eq:Fijk}
+\[
+  % \label{eq:Fijk}
   \vect{F}_{\mathrm{iso}}=\left(f_1^{lT}e_2e_3, f_2^{lT}e_1e_3, f_3^{lT}e_1e_2\right).
-\end{equation}
+\]
 Additionally, we will sometimes write the contributions towards the fluxes $\vect{f}$ and
 $\vect{F}_{\mathrm{iso}}$ from the component $R_{ij}$ of $\Re$ as $f_{ij}$, $F_{\mathrm{iso}\: ij}$,
@@ -124,7 +132,7 @@
   \label{eq:i13c}
   f_{13}=&+\Alt r_1\frac{1}{e_3}\frac{\partial T}{\partial k},\qquad f_{23}=+\Alt r_2\frac{1}{e_3}\frac{\partial T}{\partial k}\\
-\intertext{and in the k-direction resulting from the lateral tracer gradients}
+  \intertext{and in the k-direction resulting from the lateral tracer gradients}
   \label{eq:i31c}
- f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i}
+  f_{31}+f_{32}=& \Alt r_1\frac{1}{e_1}\frac{\partial T}{\partial i}+\Alt r_2\frac{1}{e_1}\frac{\partial T}{\partial i}
 \end{align}
 
@@ -147,4 +155,5 @@
 
 \subsection{Standard discretization}
+
 The straightforward approach to discretize the lateral skew flux
 \autoref{eq:i13c} from tracer cell $i,k$ to $i+1,k$, introduced in 1995 into OPA,
@@ -155,15 +164,15 @@
 noting that the $e_{{3}_{i+1/2}^k}$ in the area $e{_{3}}_{i+1/2}^k{e_{2}}_{i+1/2}i^k$ at the $u$-point cancels out with
 the $1/{e_{3}}_{i+1/2}^k$ associated with the vertical tracer gradient, is then \autoref{eq:tra_ldf_iso}
-\begin{equation*}
+\[
   \left(F_u^{13} \right)_{i+\hhalf}^k = \Alts_{i+\hhalf}^k
   {e_{2}}_{i+1/2}^k \overline{\overline
     r_1} ^{\,i,k}\,\overline{\overline{\delta_k T}}^{\,i,k},
-\end{equation*}
+\]
 where
-\begin{equation*}
+\[
   \overline{\overline
-   r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k}
+    r_1} ^{\,i,k} = -\frac{{e_{3u}}_{i+1/2}^k}{{e_{1u}}_{i+1/2}^k}
   \frac{\delta_{i+1/2} [\rho]}{\overline{\overline{\delta_k \rho}}^{\,i,k}},
-\end{equation*}
+\]
 and here and in the following we drop the $^{lT}$ superscript from $\Alt$ for simplicity.
 Unfortunately the resulting combination $\overline{\overline{\delta_k\bullet}}^{\,i,k}$ of a $k$ average and
@@ -177,4 +186,5 @@
 
 \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.
@@ -182,12 +192,16 @@
 % the mean vertical gradient at the $u$-point,
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[tb] \begin{center}
+\begin{figure}[tb]
+  \begin{center}
     \includegraphics[width=1.05\textwidth]{Fig_GRIFF_triad_fluxes}
-    \caption{ \protect\label{fig:ISO_triad}
+    \caption{
+      \protect\label{fig:ISO_triad}
       (a) Arrangement of triads $S_i$ and tracer gradients to
-           give lateral tracer flux from box $i,k$ to $i+1,k$
+      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 $i,k+1$.}
- \end{center} \end{figure}
+      box $i,k$ to $i,k+1$.
+    }
+  \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
@@ -200,9 +214,9 @@
   \label{eq:i13}
   \left( F_u^{13}  \right)_{i+\frac{1}{2}}^k = \Alts_{i+1}^k a_1 s_1
-  \delta _{k+\frac{1}{2}} \left[ T^{i+1}
+  \delta_{k+\frac{1}{2}} \left[ T^{i+1}
   \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}}  + \Alts _i^k a_2 s_2 \delta
   _{k+\frac{1}{2}} \left[ T^i
   \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}} \\
-   +\Alts _{i+1}^k a_3 s_3 \delta _{k-\frac{1}{2}} \left[ T^{i+1}
+  +\Alts _{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}}  +\Alts _i^k a_4 s_4 \delta
   _{k-\frac{1}{2}} \left[ T^i \right]/e_{{3w}_{i+1}}^{k+\frac{1}{2}},
@@ -218,8 +232,8 @@
   \label{eq:i31}
   \left( F_w^{31} \right) _i ^{k+\frac{1}{2}} =  \Alts_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}
-   +\Alts_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}\\
-  + \Alts_i^k a_{3}' s_{3}' \delta _{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
-  +\Alts_i^k a_{4}' s_{4}' \delta _{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
+  s_{1}' \delta_{i-\frac{1}{2}} \left[ T^{k+1} \right]/{e_{3u}}_{i-\frac{1}{2}}^{k+1}
+  +\Alts_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} \\
+  + \Alts_i^k a_{3}' s_{3}' \delta_{i-\frac{1}{2}} \left[ T^k\right]/{e_{3u}}_{i-\frac{1}{2}}^k
+  +\Alts_i^k a_{4}' s_{4}' \delta_{i+\frac{1}{2}} \left[ T^k \right]/{e_{3u}}_{i+\frac{1}{2}}^k.
 \end{multline}
 
@@ -242,11 +256,15 @@
 
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[tb] \begin{center}
+\begin{figure}[tb]
+  \begin{center}
     \includegraphics[width=0.80\textwidth]{Fig_GRIFF_qcells}
-    \caption{   \protect\label{fig:qcells}
+    \caption{
+      \protect\label{fig:qcells}
       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.}
-  \end{center} \end{figure}
+      the $i,k+\half$ $w$-cell is shaded in pink.
+    }
+  \end{center}
+\end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -266,4 +284,5 @@
 
 \subsection{Full triad fluxes}
+
 A key property of iso-neutral diffusion is that it should not affect the (locally referenced) density.
 In particular there should be no lateral or vertical density flux.
@@ -275,5 +294,5 @@
   - \left( \Alts_i^{k+1} a_{1} + \Alts_i^{k+1} a_{2} + \Alts_i^k
     a_{3} + \Alts_i^k a_{4} \right)
-  \frac{\delta _{i+1/2} \left[ T^k\right]}{{e_{1u}}_{\,i+1/2}^{\,k}},
+  \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 \autoref{eq:i13}.
@@ -306,5 +325,5 @@
   \label{eq:i33}
   \left( F_w^{33} \right) _i^{k+\frac{1}{2}} =
-    - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2
+  - \left( \Alts_i^{k+1} a_{1}' s_{1}'^2
     + \Alts_i^{k+1} a_{2}' s_{2}'^2
     + \Alts_i^k a_{3}' s_{3}'^2
@@ -318,11 +337,11 @@
   _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
   &= \Alts_i^k{\: }_i^k{\mathbb{A}_w}_{i_p}^{k_p}
-  \left(
+    \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) \\
+    \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}
+    {_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.
@@ -338,10 +357,10 @@
 the iso-neutral fluxes at $u$- and $w$-points as sums of the triad fluxes that cross the $u$- and $w$-faces:
 %(\autoref{fig:ISO_triad}):
-\begin{flalign} \label{eq:iso_flux} \vect{F}_{\mathrm{iso}}(T) &\equiv
+\begin{flalign}
+  \label{eq:iso_flux} \vect{F}_{\mathrm{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)      \\
+    {_{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}
@@ -369,14 +388,14 @@
   \quad {b_T}_{i+i_p+1/2}^k\left(\frac{\partial T}{\partial
       t}T\right)_{i+i_p+1/2}^k \\
- \begin{aligned}
-  &= -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{eq:dvar_iso_i}
- \end{aligned}
+  \begin{aligned}
+    &= -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{eq:dvar_iso_i}
+  \end{aligned}
 \end{multline}
 while the vertical flux similarly drives a net rate of change of variance summed over
 the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
 \begin{equation}
-\label{eq:dvar_iso_k}
+  \label{eq:dvar_iso_k}
   _i^k{\mathbb{F}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
 \end{equation}
@@ -385,17 +404,17 @@
 \autoref{eq:latflux-triad} and \autoref{eq:vertflux-triad}, it is
 \begin{multline*}
--\Alts_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 \}.
+  -\Alts_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
@@ -431,17 +450,19 @@
 where, within each triad volume $_i^k\mathbb{V}_{i_p}^{k_p}$, the lateral and vertical fluxes/unit area
 \[
-\mathbf{F}=\left(
-\left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p},
-\left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p}
- \right)
+  \mathbf{F}=\left(
+    \left.{}_i^k{\mathbb{F}_u}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_u}_{i_p}^{k_p},
+    \left.{\:}_i^k{\mathbb{F}_w}_{i_p}^{k_p} (T)\right/{}_i^k{\mathbb{A}_w}_{i_p}^{k_p}
+  \right)
 \]
 and the gradient
- \[\nabla T = \left(
-\left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k},
-\left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p}
-\right)
+\[
+  \nabla T = \left(
+    \left.\delta_{i+ i_p}[T^k] \right/ {e_{1u}}_{\,i + i_p}^{\,k},
+    \left.\delta_{k+ k_p}[T^i] \right/ {e_{3w}}_{\,i}^{\,k + k_p}
+  \right)
 \]
 
 \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} identifies these $_i^k\mathbb{V}_{i_p}^{k_p}$ as the volumes of the quarter cells,
@@ -460,8 +481,8 @@
 \begin{equation}
   \label{eq:lat-normal}
--\overline\Alts_{\,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\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
+  -\overline\Alts_{\,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\Alts_{\,i+1/2}^k\;\frac{{e_{1w}}_{\,i + 1/2}^{\,k}\:{e_{1v}}_{\,i + 1/2}^{\,k}\;\delta_{i+ 1/2}[T^k]}{{e_{1u}}_{\,i + 1/2}^{\,k}}.
 \end{equation}
 In fact if the diffusive coefficient is defined at $u$-points,
@@ -471,49 +492,53 @@
 
 \subsection{Summary of the scheme}
+
 The iso-neutral fluxes at $u$- and $w$-points are the sums of the triad fluxes that
 cross the $u$- and $w$-faces \autoref{eq:iso_flux}:
-\begin{subequations}\label{eq:alltriadflux}
-  \begin{flalign}\label{eq:vect_isoflux}
+\begin{subequations}
+  % \label{eq:alltriadflux}
+  \begin{flalign*}
+    % \label{eq:vect_isoflux}
     \vect{F}_{\mathrm{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) 
+      {_{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}
+  \end{flalign*}
   where \autoref{eq:latflux-triad}:
   \begin{align}
     \label{eq:triadfluxu}
     _i^k {\mathbb{F}_u}_{i_p}^{k_p} (T) &= - \Alts_i^k{
-      \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,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),\\
+                                          \:}\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{1u}}_{\,i + i_p}^{\,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),\\
     \intertext{and}
     _i^k {\mathbb{F}_w}_{i_p}^{k_p} (T)
-    &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+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),\label{eq:triadfluxw}
+                                        &= \Alts_i^k{\: }\frac{{{}_i^k\mathbb{V}}_{i_p}^{k_p}}{{e_{3w}}_{\,i}^{\,k+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),\label{eq:triadfluxw}
   \end{align}
   with \autoref{eq:V-NEMO}
-  \begin{equation}
-    \label{eq:V-NEMO2}
+  \[
+    % \label{eq:V-NEMO2}
     _i^k{\mathbb{V}}_{i_p}^{k_p}=\quarter {b_u}_{i+i_p}^k.
-  \end{equation}
+  \]
 \end{subequations}
 
 The divergence of the expression \autoref{eq:iso_flux} for the fluxes gives the iso-neutral diffusion tendency at
 each tracer point:
-\begin{equation} \label{eq:iso_operator} D_l^T = \frac{1}{b_T}
+\[
+  % \label{eq:iso_operator}
+  D_l^T = \frac{1}{b_T}
   \sum_{\substack{i_p,\,k_p}} \left\{ \delta_{i} \left[{_{i+1/2-i_p}^k
         {\mathbb{F}_u }_{i_p}^{k_p}} \right] + \delta_{k} \left[
       {_i^{k+1/2-k_p} {\mathbb{F}_w}_{i_p}^{k_p}} \right] \right\}
-\end{equation}
+\]
 where $b_T= e_{1T}\,e_{2T}\,e_{3T}$ is the volume of $T$-cells.
 The diffusion scheme satisfies the following six properties:
@@ -522,9 +547,11 @@
   The discretization of the diffusion operator recovers the traditional five-point Laplacian
   \autoref{eq:lat-normal} in the limit of flat iso-neutral direction:
-  \begin{equation} \label{eq:iso_property0} D_l^T = \frac{1}{b_T} \
+  \[
+    % \label{eq:iso_property0}
+    D_l^T = \frac{1}{b_T} \
     \delta_{i} \left[ \frac{e_{2u}\,e_{3u}}{e_{1u}} \;
       \overline\Alts^{\,i} \; \delta_{i+1/2}[T] \right] \qquad
     \text{when} \quad { _i^k \mathbb{R}_{i_p}^{k_p} }=0
-  \end{equation}
+  \]
 
 \item[$\bullet$ implicit treatment in the vertical]
@@ -534,5 +561,5 @@
   solve the vertical diffusion equation.
   This is necessary since the vertical eddy diffusivity associated with this term,
-  \begin{equation}
+  \[
     \frac{1}{b_w}\sum_{\substack{i_p, \,k_p}} \left\{
       {\:}_i^k\mathbb{V}_{i_p}^{k_p} \: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
@@ -541,5 +568,5 @@
       {b_u}_{i+i_p}^k\: \Alts_i^k \: \left(_i^k \mathbb{R}_{i_p}^{k_p}\right)^2
     \right\},
-  \end{equation}
+  \]
   (where $b_w= e_{1w}\,e_{2w}\,e_{3w}$ is the volume of $w$-cells) can be quite large.
 
@@ -550,14 +577,16 @@
 \item[$\bullet$ conservation of tracer]
   The iso-neutral diffusion conserves tracer content, $i.e.$
-  \begin{equation} \label{eq:iso_property1} \sum_{i,j,k} \left\{ D_l^T \
-      b_T \right\} = 0
-  \end{equation}
+  \[
+    % \label{eq:iso_property1}
+    \sum_{i,j,k} \left\{ D_l^T \      b_T \right\} = 0
+  \]
   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{eq:iso_property2} \sum_{i,j,k} \left\{ T \ D_l^T
-      \ b_T \right\} \leq 0
-  \end{equation}
+  \[
+    % \label{eq:iso_property2}
+    \sum_{i,j,k} \left\{ T \ D_l^T      \ b_T \right\} \leq 0
+  \]
   The property is demonstrated in \autoref{subsec:variance} above.
   It is a key property for a diffusion term.
@@ -569,6 +598,7 @@
 \item[$\bullet$ self-adjoint operator]
   The iso-neutral diffusion operator is self-adjoint, $i.e.$
-  \begin{equation} \label{eq:iso_property3} \sum_{i,j,k} \left\{ S \ D_l^T
-      \ b_T \right\} = \sum_{i,j,k} \left\{ D_l^S \ T \ b_T \right\}
+  \begin{equation}
+    \label{eq:iso_property3}
+    \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.
@@ -578,23 +608,25 @@
   can be found by replacing $\delta[T]$ by $\delta[S]$ in \autoref{eq:dvar_iso_i} and \autoref{eq:dvar_iso_k}.
   This results in a term similar to \autoref{eq:perfect-square},
-\begin{equation}
-  \label{eq:TScovar}
-  - \Alts_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}
+  \[
+    % \label{eq:TScovar}
+    - \Alts_i^k{\:} _i^k\mathbb{V}_{i_p}^{k_p}
+    \left(
+      \frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+    \right)
+    \left(
+      \frac{ \delta_{i+ i_p}[S^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }
+      -{\:}_i^k\mathbb{R}_{i_p}^{k_p}
+      \frac{ \delta_{k+k_p} [S^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }
+    \right).
+  \]
 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 \autoref{eq:iso_property3}.
 \end{description}
 
-\subsection{Treatment of the triads at the boundaries}\label{sec:iso_bdry}
+\subsection{Treatment of the triads at the boundaries}
+\label{sec:iso_bdry}
+
 The triad slope can only be defined where both the grid boxes centred at the end of the arms exist.
 Triads that would poke up through the upper ocean surface into the atmosphere,
@@ -618,7 +650,9 @@
 For setups with topography without bbl mixing, \np{ln\_botmix\_triad}\forcode{ = .true.} may be necessary.
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[h] \begin{center}
+\begin{figure}[h]
+  \begin{center}
     \includegraphics[width=0.60\textwidth]{Fig_GRIFF_bdry_triads}
-    \caption{  \protect\label{fig:bdry_triads}
+    \caption{
+      \protect\label{fig:bdry_triads}
       (a) Uppermost model layer $k=1$ with $i,1$ and $i+1,1$ tracer points (black dots),
       and $i+1/2,1$ $u$-point (blue square).
@@ -627,5 +661,6 @@
       However, the lateral $_{11}$ contributions towards $\triad[u]{i}{1}{F}{1/2}{-1/2}$ and
       $\triad[u]{i+1}{1}{F}{-1/2}{-1/2}$ (yellow line) are still applied,
-      giving diapycnal diffusive fluxes.\newline
+      giving diapycnal diffusive fluxes.
+      \newline
       (b) Both near bottom triad slopes $\triad{i}{k}{R}{1/2}{1/2}$ and
       $\triad{i+1}{k}{R}{-1/2}{1/2}$ are masked when either of the $i,k+1$ or $i+1,k+1$ tracer points is masked,
@@ -633,9 +668,13 @@
       The associated lateral fluxes (grey-black dashed line) are masked if
       \protect\np{botmix\_triad}\forcode{ = .false.}, but left unmasked,
-      giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}}
- \end{center} \end{figure}
+      giving bottom mixing, if \protect\np{botmix\_triad}\forcode{ = .true.}
+    }
+  \end{center}
+\end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-\subsection{ Limiting of the slopes within the interior}\label{sec:limit}
+\subsection{ Limiting of the slopes within the interior}
+\label{sec:limit}
+
 As discussed in \autoref{subsec:LDF_slp_iso},
 iso-neutral slopes relative to geopotentials must be bounded everywhere,
@@ -647,7 +686,7 @@
 (here the $\sigma_i$ are the slopes of the coordinate surfaces relative to geopotentials)
 \autoref{eq:PE_slopes_eiv} rather than the slope $r_i$ relative to coordinate surfaces, so we require
-\begin{equation*}
+\[
   |\tilde{r}_i|\leq \tilde{r}_\mathrm{max}=0.01.
-\end{equation*}
+\]
 and then recalculate the slopes $r_i$ relative to coordinates.
 Each individual triad slope
@@ -663,33 +702,37 @@
 and so acts to reduce gravitational potential energy.
 
-\subsection{Tapering within the surface mixed layer}\label{sec:taper}
+\subsection{Tapering within the surface mixed layer}
+\label{sec:taper}
+
 Additional tapering of the iso-neutral fluxes is necessary within the surface mixed layer.
 When the Griffies triads are used, we offer two options for this.
 
-\subsubsection{Linear slope tapering within the surface mixed layer}\label{sec:lintaper}
+\subsubsection{Linear slope tapering within the surface mixed layer}
+\label{sec:lintaper}
+
 This is the option activated by the default choice \np{ln\_triad\_iso}\forcode{ = .false.}.
 Slopes $\tilde{r}_i$ relative to geopotentials are tapered linearly from their value immediately below
 the mixed layer to zero at the surface, as described in option (c) of \autoref{fig:eiv_slp}, to values
-\begin{subequations}
-  \begin{equation}
-    \label{eq:rmtilde}
-    \rMLt =
-    -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
-  \end{equation}
-  and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to
-  \begin{equation}
-    \label{eq:rm}
-    \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
-  \end{equation}
-\end{subequations}
+\begin{equation}
+  \label{eq:rmtilde}
+  \rMLt = -\frac{z}{h}\left.\tilde{r}_i\right|_{z=-h}\quad \text{ for  } z>-h,
+\end{equation}
+and then the $r_i$ relative to vertical coordinate surfaces are appropriately adjusted to
+\[
+  % \label{eq:rm}
+  \rML =\rMLt -\sigma_i \quad \text{ for  } z>-h.
+\]
 Thus the diffusion operator within the mixed layer is given by:
-\begin{equation} \label{eq:iso_tensor_ML}
-D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
-\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\
- 0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\
- {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill
-\end{array} }} \right)
-\end{equation}
+\[
+  % \label{eq:iso_tensor_ML}
+  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+  \mbox{with}\quad \;\;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill & {-\rML[1]}\hfill \\
+          0 \hfill & 1 \hfill & {-\rML[2]} \hfill \\
+          {-\rML[1]}\hfill &   {-\rML[2]} \hfill & {\rML[1]^2+\rML[2]^2} \hfill
+        \end{array}
+      }} \right)
+\]
 
 This slope tapering gives a natural connection between tracer in the mixed-layer and
@@ -726,18 +769,19 @@
   these basal triad slopes ${\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}$ are representative of the thermocline.
   The four basal triads defined in the bottom part of \autoref{fig:MLB_triad} are then
-\begin{align}
-  {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &=
- {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}, \label{eq:Rbase}
-\\
-\intertext{with e.g.\ the green triad}
-{\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&=
-{\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}. \notag
-\end{align}
+  \begin{align*}
+    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p} &=
+                                                       {\:}^{k_{\mathrm{ML}}-k_p-1/2}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p},
+                                                       % \label{eq:Rbase}
+    \\
+    \intertext{with e.g.\ the green triad}
+    {\:}_i{\mathbb{R}_{\mathrm{base}}}_{1/2}^{-1/2}&=
+                                                     {\:}^{k_{\mathrm{ML}}}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}.
+  \end{align*}
 The vertical flux associated with each of these triads passes through
 the $w$-point $i,k_{\mathrm{ML}}-1/2$ lying \emph{below} the $i,k_{\mathrm{ML}}$ tracer point, so it is this depth
-\begin{equation}
-  \label{eq:zbase}
+\[
+  % \label{eq:zbase}
   {z_\mathrm{base}}_{\,i}={z_{w}}_{k_\mathrm{ML}-1/2}
-\end{equation}
+\]
 one gridbox deeper than the diagnosed ML depth $z_{\mathrm{ML}})$ that sets the $h$ used to taper the slopes in
 \autoref{eq:rmtilde}.
@@ -747,12 +791,12 @@
   the ratio of the depth of the $w$-point ${z_w}_{k+k_p}$ to ${z_{\mathrm{base}}}_{\,i}$.
   For instance the green triad centred on $i,k$
-\begin{align}
-  {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &=
-\frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2}
-\notag \\
-\intertext{and more generally}
- {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &=
-\frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.\label{eq:RML}
-\end{align}
+  \begin{align*}
+    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,1/2}^{-1/2} &=
+                                                        \frac{{z_w}_{k-1/2}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,1/2}^{-1/2} \\
+    \intertext{and more generally}
+    {\:}_i^k{\mathbb{R}_{\mathrm{ML}}}_{\,i_p}^{k_p} &=
+                                                       \frac{{z_w}_{k+k_p}}{{z_{\mathrm{base}}}_{\,i}}{\:}_i{\mathbb{R}_{\mathrm{base}}}_{\,i_p}^{k_p}.
+                                                       % \label{eq:RML}
+  \end{align*}
 \end{enumerate}
 
@@ -760,9 +804,11 @@
 \begin{figure}[h]
 %  \fcapside {
-  \caption{\protect\label{fig:MLB_triad}
+  \caption{
+    \protect\label{fig:MLB_triad}
     Definition of mixed-layer depth and calculation of linearly tapered triads.
     The figure shows a water column at a given $i,j$ (simplified to $i$), with the ocean surface at the top.
     Tracer points are denoted by bullets, and black lines the edges of the tracer cells;
-    $k$ increases upwards. \newline
+    $k$ increases upwards.
+    \newline
     \hspace{5 em}
     We define the mixed-layer by setting the vertical index of the tracer point immediately below the mixed layer,
@@ -776,6 +822,6 @@
     Triads with different $i_p,k_p$, denoted by different colours,
     (e.g. the green triad $i_p=1/2,k_p=-1/2$) are tapered to the appropriate basal triad.}
-%}
-  {\includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}}
+  % }
+  \includegraphics[width=0.60\textwidth]{Fig_GRIFF_MLB_triads}
 \end{figure}
 % >>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -783,4 +829,5 @@
 \subsubsection{Additional truncation of skew iso-neutral flux components}
 \label{subsec:Gerdes-taper}
+
 The alternative option is activated by setting \np{ln\_triad\_iso} = true.
 This retains the same tapered slope $\rML$  described above for the calculation of the $_{33}$ term of
@@ -792,17 +839,21 @@
 \end{equation}
 giving a ML diffusive operator
-\begin{equation} \label{eq:iso_tensor_ML2}
-D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
-\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\
- 0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\
- {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\
-\end{array} }} \right).
-\end{equation}
+\[
+  % \label{eq:iso_tensor_ML2}
+  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+  \mbox{with}\quad \;\;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill & {-\rML[1]^*}\hfill \\
+          0 \hfill & 1 \hfill & {-\rML[2]^*} \hfill \\
+          {-\rML[1]^*}\hfill &   {-\rML[2]^*} \hfill & {\rML[1]^2+\rML[2]^2} \hfill \\
+        \end{array}
+      }} \right).
+\]
 This operator
 \footnote{
   To ensure good behaviour where horizontal density gradients are weak,
   we in fact follow \citet{Gerdes1991} and
-  set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.}
+  set $\rML^*=\mathrm{sgn}(\tilde{r}_i)\min(|\rMLt^2/\tilde{r}_i|,|\tilde{r}_i|)-\sigma_i$.
+}
 then has the property it gives no vertical density flux, and so does not change the potential energy.
 This approach is similar to multiplying the iso-neutral diffusion coefficient by
@@ -821,7 +872,9 @@
 % Skew flux formulation for Eddy Induced Velocity :
 % ================================================================
-\section{Eddy induced advection formulated as a skew flux}\label{sec:skew-flux}
-
-\subsection{Continuous skew flux formulation}\label{sec:continuous-skew-flux}
+\section{Eddy induced advection formulated as a skew flux}
+\label{sec:skew-flux}
+
+\subsection{Continuous skew flux formulation}
+\label{sec:continuous-skew-flux}
 
 When Gent and McWilliams's [1990] diffusion is used, an additional advection term is added.
@@ -833,20 +886,23 @@
 
 The eddy induced velocity is given by:
-\begin{subequations} \label{eq:eiv}
-\begin{equation}\label{eq:eiv_v}
-\begin{split}
- u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\
- v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\
-w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i  \left( e_{2} \, \psi_1\right)
-							    + \partial_j  \left( e_{1} \, \psi_2\right) \right\},
-\end{split}
-\end{equation}
-where the streamfunctions $\psi_i$ are given by
-\begin{equation} \label{eq:eiv_psi}
-\begin{split}
-\psi_1 & = A_{e} \; \tilde{r}_1,   \\
-\psi_2 & = A_{e} \; \tilde{r}_2,
-\end{split}
-\end{equation}
+\begin{subequations}
+  % \label{eq:eiv}
+  \begin{equation}
+    \label{eq:eiv_v}
+    \begin{split}
+      u^* & = - \frac{1}{e_{3}}\;          \partial_i\psi_1,  \\
+      v^* & = - \frac{1}{e_{3}}\;          \partial_j\psi_2,    \\
+      w^* & =    \frac{1}{e_{1}e_{2}}\; \left\{ \partial_i  \left( e_{2} \, \psi_1\right)
+        + \partial_j  \left( e_{1} \, \psi_2\right) \right\},
+    \end{split}
+  \end{equation}
+  where the streamfunctions $\psi_i$ are given by
+  \begin{equation}
+    \label{eq:eiv_psi}
+    \begin{split}
+      \psi_1 & = A_{e} \; \tilde{r}_1,   \\
+      \psi_2 & = A_{e} \; \tilde{r}_2,
+    \end{split}
+  \end{equation}
 \end{subequations}
 with $A_{e}$ the eddy induced velocity coefficient,
@@ -868,42 +924,44 @@
 the tracer advective fluxes per unit area in $ijk$ space can be transformed as follows:
 \begin{flalign*}
-\begin{split}
-\textbf{F}_{\mathrm{eiv}}^T =
-\begin{pmatrix}
- 	        {e_{2}\,e_{3}\;  u^*} 	 	\\
- 		{e_{1}\,e_{2}\; w^*}	 \\
-\end{pmatrix}   \;   T
-&=
-\begin{pmatrix}
- 	        { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} 	 	\\
- 		{+ \partial_i  \left( e_{2} \, \psi_1 \right) \; T \;}	 \\
-\end{pmatrix} 			\\
-&=
-\begin{pmatrix}
- 	        { - \partial_k \left( e_{2} \, \psi_1  \; T \right) \;}  \\
- 		{+ \partial_i  \left( e_{2} \,\psi_1 \; T \right) \;}	 \\
-\end{pmatrix}
- +
-\begin{pmatrix}
- 	        {+ e_{2} \, \psi_1  \; \partial_k T}  \\
- 		{ - e_{2} \, \psi_1  \; \partial_i  T}	 \\
-\end{pmatrix}
-\end{split}
+  \begin{split}
+    \textbf{F}_{\mathrm{eiv}}^T =
+    \begin{pmatrix}
+      {e_{2}\,e_{3}\;  u^*} \\
+ 		{e_{1}\,e_{2}\; w^*}
+    \end{pmatrix}   \;   T
+    &=
+    \begin{pmatrix}
+      { - \partial_k \left( e_{2} \,\psi_1 \right) \; T \;} \\
+ 		{+ \partial_i  \left( e_{2} \, \psi_1 \right) \; T \;}
+    \end{pmatrix} 			\\
+    &=
+    \begin{pmatrix}
+      { - \partial_k \left( e_{2} \, \psi_1  \; T \right) \;} \\
+ 		{+ \partial_i  \left( e_{2} \,\psi_1 \; T \right) \;}
+    \end{pmatrix}
+    +
+    \begin{pmatrix}
+      {+ e_{2} \, \psi_1  \; \partial_k T} \\
+ 		{ - e_{2} \, \psi_1  \; \partial_i  T}
+    \end{pmatrix}
+  \end{split}
 \end{flalign*}
 and since the eddy induced velocity field is non-divergent,
 we end up with the skew form of the eddy induced advective fluxes per unit area in $ijk$ space:
-\begin{equation} \label{eq:eiv_skew_ijk}
-\textbf{F}_\mathrm{eiv}^T = \begin{pmatrix}
- 	        {+ e_{2} \, \psi_1  \; \partial_k T}   \\
- 		{ - e_{2} \, \psi_1  \; \partial_i  T}	 \\
-                                 \end{pmatrix}
+\begin{equation}
+  \label{eq:eiv_skew_ijk}
+  \textbf{F}_\mathrm{eiv}^T =
+  \begin{pmatrix}
+    {+ e_{2} \, \psi_1  \; \partial_k T}   \\
+    { - e_{2} \, \psi_1  \; \partial_i  T}
+  \end{pmatrix}
 \end{equation}
 The total fluxes per unit physical area are then
-\begin{equation}\label{eq:eiv_skew_physical}
-\begin{split}
- f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\
- f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\
- f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T
-   + e_{1} \psi_2 \partial_j T \right\}. \\
+\begin{equation}
+  \label{eq:eiv_skew_physical}
+  \begin{split}
+    f^*_1 & = \frac{1}{e_{3}}\; \psi_1 \partial_k T   \\
+    f^*_2 & = \frac{1}{e_{3}}\; \psi_2 \partial_k T   \\
+    f^*_3 & =  -\frac{1}{e_{1}e_{2}}\; \left\{ e_{2} \psi_1 \partial_i T + e_{1} \psi_2 \partial_j T \right\}.
 \end{split}
 \end{equation}
@@ -913,16 +971,18 @@
 The tendency associated with eddy induced velocity is then simply the convergence of the fluxes
 (\autoref{eq:eiv_skew_ijk}, \autoref{eq:eiv_skew_physical}), so
-\begin{equation} \label{eq:skew_eiv_conv}
-\frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[
-  \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right)
-  + \frac{\partial}{\partial j} \left( e_1  \;
-    \psi_2 \partial_k T\right)
- -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T
-   + e_{1} \psi_2 \partial_j T \right)  \right]
-\end{equation}
+\[
+  % \label{eq:skew_eiv_conv}
+  \frac{\partial T}{\partial t}= -\frac{1}{e_1 \, e_2 \, e_3 }      \left[
+    \frac{\partial}{\partial i} \left( e_2 \psi_1 \partial_k T\right)
+    + \frac{\partial}{\partial j} \left( e_1  \;
+      \psi_2 \partial_k T\right)
+    -  \frac{\partial}{\partial k} \left( e_{2} \psi_1 \partial_i T
+      + e_{1} \psi_2 \partial_j T \right)  \right]
+\]
 It naturally conserves the tracer content, as it is expressed in flux form.
 Since it has the same divergence as the advective form it also preserves the tracer variance.
 
 \subsection{Discrete skew flux formulation}
+
 The skew fluxes in (\autoref{eq:eiv_skew_physical}, \autoref{eq:eiv_skew_ijk}),
 like the off-diagonal terms (\autoref{eq:i13c}, \autoref{eq:i31c}) of the small angle diffusion tensor,
@@ -934,15 +994,14 @@
 defining $A_e$ at $T$-points is then given by:
 
-
-\begin{subequations}\label{eq:allskewflux}
-  \begin{flalign}\label{eq:vect_skew_flux}
-    \vect{F}_{\mathrm{eiv}}(T) &\equiv
-    \sum_{\substack{i_p,\,k_p}}
+\begin{subequations}
+  % \label{eq:allskewflux}
+  \begin{flalign*}
+    % \label{eq:vect_skew_flux}
+    \vect{F}_{\mathrm{eiv}}(T) &\equiv    \sum_{\substack{i_p,\,k_p}}
     \begin{pmatrix}
-      {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T)      \\
-      \\
+      {_{i+1/2-i_p}^k {\mathbb{S}_u}_{i_p}^{k_p} } (T)      \\      \\
       {_i^{k+1/2-k_p} {\mathbb{S}_w}_{i_p}^{k_p} } (T)      \\
     \end{pmatrix},
-  \end{flalign}
+  \end{flalign*}
   where the skew flux in the $i$-direction associated with a given triad is (\autoref{eq:latflux-triad},
   \autoref{eq:triadfluxu}):
@@ -950,13 +1009,14 @@
     \label{eq:skewfluxu}
     _i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) &= + \quarter {A_e}_i^k{
-      \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
-     \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \
-      \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} },
-   \\
-    \intertext{and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign
-      to be consistent with \autoref{eq:eiv_skew_ijk}:}
+                                          \:}\frac{{b_u}_{i+i_p}^k}{{e_{1u}}_{\,i + i_p}^{\,k}}
+                                          \ {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}} \
+                                          \frac{ \delta_{k+k_p} [T^i] }{{e_{3w}}_{\,i}^{\,k+k_p} }, \\
+    \intertext{
+    and \autoref{eq:triadfluxw} in the $k$-direction, changing the sign
+    to be consistent with \autoref{eq:eiv_skew_ijk}:
+    }
     _i^k {\mathbb{S}_w}_{i_p}^{k_p} (T)
-    &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}}
-     {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw}
+                                        &= -\quarter {A_e}_i^k{\: }\frac{{b_u}_{i+i_p}^k}{{e_{3w}}_{\,i}^{\,k+k_p}}
+                                          {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}\frac{ \delta_{i+ i_p}[T^k] }{ {e_{1u}}_{\,i + i_p}^{\,k} }.\label{eq:skewfluxw}
   \end{align}
 \end{subequations}
@@ -966,4 +1026,5 @@
 
 \subsubsection{No change in tracer variance}
+
 The discretization conserves tracer variance, $i.e.$ it does not include a diffusive component but is a `pure' advection term.
 This can be seen %either from Appendix \autoref{apdx:eiv_skew} or
@@ -973,5 +1034,5 @@
 summed over the two $T$-points $i+i_p-\half,k$ and $i+i_p+\half,k$, of
 \begin{equation}
-\label{eq:dvar_eiv_i}
+  \label{eq:dvar_eiv_i}
   _i^k{\mathbb{S}_u}_{i_p}^{k_p} (T)\,\delta_{i+ i_p}[T^k],
 \end{equation}
@@ -979,5 +1040,5 @@
 the $T$-points $i,k+k_p-\half$ (above) and $i,k+k_p+\half$ (below) of
 \begin{equation}
-\label{eq:dvar_eiv_k}
+  \label{eq:dvar_eiv_k}
   _i^k{\mathbb{S}_w}_{i_p}^{k_p} (T) \,\delta_{k+ k_p}[T^i].
 \end{equation}
@@ -987,4 +1048,5 @@
 
 \subsubsection{Reduction in gravitational PE}
+
 The vertical density flux associated with the vertical skew-flux always has the same sign as
 the vertical density gradient;
@@ -999,15 +1061,14 @@
     {\mathbb{S}_w}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k
     {\mathbb{S}_w}_{i_p}^{k_p} (S) \right]. \notag \\
-\intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from
-  \autoref{eq:skewfluxw}, gives}
-% and separating out
-% $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$,
-% gives two terms. The
-% first $\rtriad{R}$ term (the only term for $z$-coordinates) is:
- &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}
-\frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\
- &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
-     \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}}
-\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
+  \intertext{Substituting  ${\:}_i^k {\mathbb{S}_w}_{i_p}^{k_p}$ from \autoref{eq:skewfluxw}, gives}
+  % and separating out
+  % $\rtriadt{R}=\rtriad{R} + \delta_{i+i_p}[z_T^k]$,
+  % gives two terms. The
+  % first $\rtriad{R}$ term (the only term for $z$-coordinates) is:
+  &=-\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k {_i^k\tilde{\mathbb{R}}_{i_p}^{k_p}}
+    \frac{ -\alpha _i^k\delta_{i+ i_p}[T^k]+ \beta_i^k\delta_{i+ i_p}[S^k]} { {e_{1u}}_{\,i + i_p}^{\,k} } \notag \\
+  &=+\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+    \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right) {_i^k\mathbb{R}_{i_p}^{k_p}}
+    \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
 \end{align}
 using the definition of the triad slope $\rtriad{R}$, \autoref{eq:R} to
@@ -1018,27 +1079,29 @@
 \begin{multline}
   \label{eq:lat_densityPE}
- g \delta_{i+i_p}[z_T^k]
-\left[
--\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S)
-\right] \\
-= +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
-     \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
-\left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)
-\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
+  g \delta_{i+i_p}[z_T^k]
+  \left[
+    -\alpha _i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (T) + \beta_i^k {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (S)
+  \right] \\
+  = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+  \frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}
+  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)
+  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}},
 \end{multline}
 (using \autoref{eq:skewfluxu}) and so the total PE change \autoref{eq:vert_densityPE} +
 \autoref{eq:lat_densityPE} associated with the triad fluxes is
-\begin{multline}
-  \label{eq:tot_densityPE}
+\begin{multline*}
+  % \label{eq:tot_densityPE}
   g{e_{3w}}_{\,i}^{\,k+k_p}{\mathbb{S}_w}_{i_p}^{k_p} (\rho) +
-g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\
-= +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
-     \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2
-\frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}.
-\end{multline}
+  g\delta_{i+i_p}[z_T^k] {\:}_i^k {\mathbb{S}_u}_{i_p}^{k_p} (\rho) \\
+  = +\quarter g{A_e}_i^k{\: }{b_u}_{i+i_p}^k
+  \left({_i^k\mathbb{R}_{i_p}^{k_p}}+\frac{\delta_{i+i_p}[z_T^k]}{{e_{1u}}_{\,i + i_p}^{\,k}}\right)^2
+  \frac{-\alpha_i^k \delta_{k+ k_p}[T^i]+ \beta_i^k\delta_{k+ k_p}[S^i]} {{e_{3w}}_{\,i}^{\,k+k_p}}.
+\end{multline*}
 Where the fluid is stable, with $-\alpha_i^k \delta_{k+ k_p}[T^i]+
 \beta_i^k\delta_{k+ k_p}[S^i]<0$, this PE change is negative.
 
-\subsection{Treatment of the triads at the boundaries}\label{sec:skew_bdry}
+\subsection{Treatment of the triads at the boundaries}
+\label{sec:skew_bdry}
+
 Triad slopes \rtriadt{R} used for the calculation of the eddy-induced skew-fluxes are masked at the boundaries 
 in exactly the same way as are the triad slopes \rtriad{R} used for the iso-neutral diffusive fluxes, 
@@ -1049,10 +1112,14 @@
 The namelist parameter \np{ln\_botmix\_triad} has no effect on the eddy-induced skew-fluxes.
 
-\subsection{Limiting of the slopes within the interior}\label{sec:limitskew}
+\subsection{Limiting of the slopes within the interior}
+\label{sec:limitskew}
+
 Presently, the iso-neutral slopes $\tilde{r}_i$ relative to geopotentials are limited to be less than $1/100$, 
 exactly as in calculating the iso-neutral diffusion, \S \autoref{sec:limit}. 
 Each individual triad \rtriadt{R} is so limited.
 
-\subsection{Tapering within the surface mixed layer}\label{sec:taperskew}
+\subsection{Tapering within the surface mixed layer}
+\label{sec:taperskew}
+
 The slopes $\tilde{r}_i$ relative to geopotentials (and thus the individual triads \rtriadt{R}) 
 are always tapered linearly from their value immediately below the mixed layer to zero at the surface 
@@ -1072,5 +1139,7 @@
 (the horizontal flux convergence is relatively insignificant within the mixed-layer).
 
-\subsection{Streamfunction diagnostics}\label{sec:sfdiag}
+\subsection{Streamfunction diagnostics}
+\label{sec:sfdiag}
+
 Where the namelist parameter \np{ln\_traldf\_gdia}\forcode{ = .true.},
 diagnosed mean eddy-induced velocities are output.
@@ -1080,20 +1149,24 @@
 We follow \citep{Griffies_Bk04} and calculate the streamfunction at a given $uw$-point from
 the surrounding four triads according to:
-\begin{equation}
-  \label{eq:sfdiagi}
+\[
+  % \label{eq:sfdiagi}
   {\psi_1}_{i+1/2}^{k+1/2}={\quarter}\sum_{\substack{i_p,\,k_p}}
   {A_e}_{i+1/2-i_p}^{k+1/2-k_p}\:\triadd{i+1/2-i_p}{k+1/2-k_p}{R}{i_p}{k_p}.
-\end{equation}
+\]
 The streamfunction $\psi_1$ is calculated similarly at $vw$ points.
 The eddy-induced velocities are then calculated from the straightforward discretisation of \autoref{eq:eiv_v}:
-\begin{equation}\label{eq:eiv_v_discrete}
-\begin{split}
- {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\
- {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\
- {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{
- {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} -
- {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\
-\phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\},
-\end{split}
-\end{equation}
+\[
+  % \label{eq:eiv_v_discrete}
+  \begin{split}
+    {u^*}_{i+1/2}^{k} & = - \frac{1}{{e_{3u}}_{i}^{k}}\left({\psi_1}_{i+1/2}^{k+1/2}-{\psi_1}_{i+1/2}^{k+1/2}\right),   \\
+    {v^*}_{j+1/2}^{k} & = - \frac{1}{{e_{3v}}_{j}^{k}}\left({\psi_2}_{j+1/2}^{k+1/2}-{\psi_2}_{j+1/2}^{k+1/2}\right),   \\
+    {w^*}_{i,j}^{k+1/2} & =    \frac{1}{e_{1t}e_{2t}}\; \left\{
+      {e_{2u}}_{i+1/2}^{k+1/2} \,{\psi_1}_{i+1/2}^{k+1/2} -
+      {e_{2u}}_{i-1/2}^{k+1/2} \,{\psi_1}_{i-1/2}^{k+1/2} \right. + \\
+    \phantom{=} & \qquad\qquad\left. {e_{2v}}_{j+1/2}^{k+1/2} \,{\psi_2}_{j+1/2}^{k+1/2} - {e_{2v}}_{j-1/2}^{k+1/2} \,{\psi_2}_{j-1/2}^{k+1/2} \right\},
+  \end{split}
+\]
+
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ASM.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ASM.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ASM.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -11,7 +12,5 @@
 \minitoc
 
-
 \newpage
-$\ $\newline    % force a new line
 
 The ASM code adds the functionality to apply increments to the model variables: temperature, salinity,
@@ -45,8 +44,8 @@
 is corrected by adding the analysis increments for temperature, salinity, horizontal velocity and SSH as
 additional tendency terms to the prognostic equations:
-\begin{eqnarray}     \label{eq:wa_traj_iau}
-{\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] 
-\; + \; F_{i} \delta \tilde{\bf x}^{a} 
-\end{eqnarray}
+\begin{align*}
+  % \label{eq:wa_traj_iau}
+  {\bf x}^{a}(t_{i}) = M(t_{i}, t_{0})[{\bf x}^{b}(t_{0})] \; + \; F_{i} \delta \tilde{\bf x}^{a}
+\end{align*}
 where $F_{i}$ is a weighting function for applying the increments $\delta\tilde{\bf x}^{a}$ defined such that
 $\sum_{i=1}^{N} F_{i}=1$.
@@ -58,24 +57,30 @@
 In addition, two different weighting functions have been implemented.
 The first function employs constant weights, 
-\begin{eqnarray}    \label{eq:F1_i}
-F^{(1)}_{i}
-=\left\{ \begin{array}{ll}
-   0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\
-   1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\
-   0     &    {\rm if} \; \; \; t_{i} > t_{n}
-  \end{array} \right. 
-\end{eqnarray}
+\begin{align}
+  \label{eq:F1_i}
+  F^{(1)}_{i}
+  =\left\{
+  \begin{array}{ll}
+    0     &    {\rm if} \; \; \; t_{i} < t_{m}                \\
+    1/M &    {\rm if} \; \; \; t_{m} < t_{i} \leq t_{n} \\
+    0     &    {\rm if} \; \; \; t_{i} > t_{n}
+  \end{array}
+            \right. 
+\end{align}
 where $M = m-n$.
 The second function employs peaked hat-like weights in order to give maximum weight in the centre of the sub-window,
 with the weighting reduced linearly to a small value at the window end-points:
-\begin{eqnarray}     \label{eq:F2_i}
-F^{(2)}_{i}
-=\left\{ \begin{array}{ll}
-   0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\
-   \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\
-   \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\
-   0                            &   {\rm if} \; \; \; t_{i}        >    t_{n}
-  \end{array} \right.
-\end{eqnarray}
+\begin{align}
+  \label{eq:F2_i}
+  F^{(2)}_{i}
+  =\left\{
+  \begin{array}{ll}
+    0                           &    {\rm if} \; \; \; t_{i}       <     t_{m}                        \\
+    \alpha \, i               &    {\rm if} \; \; \; t_{m}    \leq t_{i}    \leq   t_{M/2}   \\
+    \alpha \, (M - i +1) &    {\rm if} \; \; \; t_{M/2}  <    t_{i}    \leq   t_{n}       \\
+    0                            &   {\rm if} \; \; \; t_{i}        >    t_{n}
+  \end{array}
+                                   \right.
+\end{align}
 where $\alpha^{-1} = \sum_{i=1}^{M/2} 2i$ and $M$ is assumed to be even. 
 The weights described by \autoref{eq:F2_i} provide a smoother transition of the analysis trajectory from
@@ -89,19 +94,22 @@
 The velocity increments may be initialized by the iterative application of a divergence damping operator.
 In iteration step $n$ new estimates of velocity increments $u^{n}_I$ and $v^{n}_I$ are updated by:
-\begin{equation} \label{eq:asm_dmp}
-\left\{ \begin{aligned}
- u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta _{i+1/2} \left( {A_D
-\;\chi^{n-1}_I } \right) \\
-\\
- v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta _{j+1/2} \left( {A_D
-\;\chi^{n-1}_I } \right) \\
-\end{aligned} \right.,
+\begin{equation}
+  \label{eq:asm_dmp}
+  \left\{
+    \begin{aligned}
+      u^{n}_I = u^{n-1}_I + \frac{1}{e_{1u} } \delta_{i+1/2} \left( {A_D
+          \;\chi^{n-1}_I } \right) \\ \\
+      v^{n}_I = v^{n-1}_I + \frac{1}{e_{2v} } \delta_{j+1/2} \left( {A_D
+          \;\chi^{n-1}_I } \right) \\
+    \end{aligned}
+  \right.,
 \end{equation}
 where
-\begin{equation} \label{eq:asm_div}
-\chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
-                \left( {\delta _i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right]
-                       +\delta _j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right).
-\end{equation}
+\[
+  % \label{eq:asm_div}
+  \chi^{n-1}_I = \frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u^{n-1}_I} \right]
+      +\delta_j \left[ {e_{1v}\,e_{3v}\,v^{n-1}_I} \right]} \right).
+\]
 By the application of \autoref{eq:asm_dmp} and \autoref{eq:asm_dmp} the divergence is filtered in each iteration,
 and the vorticity is left unchanged.
@@ -169,3 +177,5 @@
 \end{clines}
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_CONFIG.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_CONFIG.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_CONFIG.tex	(revision 10419)
@@ -1,13 +1,14 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter � Configurations
+% Chapter Configurations
 % ================================================================
 \chapter{Configurations}
 \label{chap:CFG}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -16,5 +17,4 @@
 \section{Introduction}
 \label{sec:CFG_intro}
-
 
 The purpose of this part of the manual is to introduce the \NEMO reference configurations.
@@ -36,7 +36,5 @@
 \label{sec:CFG_c1d}
 
-$\ $\newline
 BE careful: to be re-written according to suppression of jpizoom and jpjzoom !!!!
-$\ $\newline
 
 The 1D model option simulates a stand alone water column within the 3D \NEMO system.
@@ -102,5 +100,6 @@
   \begin{center}
     \includegraphics[width=0.98\textwidth]{Fig_ORCA_NH_mesh}
-    \caption{  \protect\label{fig:MISC_ORCA_msh}
+    \caption{
+      \protect\label{fig:MISC_ORCA_msh}
       ORCA mesh conception.
       The departure from an isotropic Mercator grid start poleward of 20\degN.
@@ -108,5 +107,6 @@
       are determined analytically and form the i-lines of the ORCA mesh (pseudo latitudes).
       Then, following \citet{Madec_Imbard_CD96}, the normal to the series of ellipses (red curves) is computed which
-      provides the j-lines of the mesh (pseudo longitudes).  }
+      provides the j-lines of the mesh (pseudo longitudes).
+    }
   \end{center}
 \end{figure}
@@ -133,5 +133,6 @@
     \includegraphics[width=1.0\textwidth]{Fig_ORCA_NH_msh05_e1_e2}
     \includegraphics[width=0.80\textwidth]{Fig_ORCA_aniso}
-    \caption {  \protect\label{fig:MISC_ORCA_e1e2}
+    \caption {
+      \protect\label{fig:MISC_ORCA_e1e2}
       \textit{Top}: Horizontal scale factors ($e_1$, $e_2$) and
       \textit{Bottom}: ratio of anisotropy ($e_1 / e_2$)
@@ -139,8 +140,9 @@
       South of 20\degN a Mercator grid is used ($e_1 = e_2$) so that the anisotropy ratio is 1.
       Poleward of 20\degN, the two "north pole" introduce a weak anisotropy over the ocean areas ($< 1.2$) except in
-      vicinity of Victoria Island (Canadian Arctic Archipelago). }
-\end{center}   \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-
+      vicinity of Victoria Island (Canadian Arctic Archipelago).
+    }
+  \end{center}
+\end{figure}
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 The method is applied to Mercator grid ($i.e.$ same zonal and meridional grid spacing) poleward of 20\degN,
@@ -161,5 +163,4 @@
 \label{subsec:CFG_orca_resolution}
 
-
 The NEMO system is provided with five built-in ORCA configurations which differ in the horizontal resolution.
 The value of the resolution is given by the resolution at the Equator expressed in degrees.
@@ -169,25 +170,27 @@
 (Tab. \autoref{tab:ORCA}).
 
-
-
-
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[!t]     \begin{center}
-\begin{tabular}{p{4cm} c c c c}
-Horizontal Grid         	             & \np{ORCA\_index} &  \np{jpiglo} & \np{jpjglo} &       \\  
-\hline  \hline
-\~4\deg	   &        4         &         92     &      76      &       \\
-\~2\deg        &        2         &       182     &    149      &        \\
-\~1\deg        &        1         &       362     &     292     &        \\
-\~0.5\deg     &        05       &       722     &     511     &        \\
-\~0.25\deg   &        025     &      1442    &   1021     &        \\
-%\key{orca\_r8}       &        8         &      2882    &   2042     &        \\
-%\key{orca\_r12}     &      12         &      4322    &   3062      &       \\
-\hline   \hline
-\end{tabular}
-\caption{ \protect\label{tab:ORCA}
-  Domain size of ORCA family configurations.
-  The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file. }
-\end{center}
+\begin{table}[!t]
+  \begin{center}
+    \begin{tabular}{p{4cm} c c c c}
+      Horizontal Grid         	             & \np{ORCA\_index} &  \np{jpiglo} & \np{jpjglo} &       \\
+      \hline
+      \hline
+      \~4\deg	   &        4         &         92     &      76      &       \\
+      \~2\deg        &        2         &       182     &    149      &        \\
+      \~1\deg        &        1         &       362     &     292     &        \\
+      \~0.5\deg     &        05       &       722     &     511     &        \\
+      \~0.25\deg   &        025     &      1442    &   1021     &        \\
+      % \key{orca\_r8}       &        8         &      2882    &   2042     &        \\
+      % \key{orca\_r12}     &      12         &      4322    &   3062      &       \\
+      \hline
+      \hline
+    \end{tabular}
+    \caption{
+      \protect\label{tab:ORCA}
+      Domain size of ORCA family configurations.
+      The flag for configurations of ORCA family need to be set in \textit{domain\_cfg} file.
+    }
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -284,7 +287,9 @@
   \begin{center}
     \includegraphics[width=1.0\textwidth]{Fig_GYRE}
-    \caption{  \protect\label{fig:GYRE}
+    \caption{
+      \protect\label{fig:GYRE}
       Snapshot of relative vorticity at the surface of the model domain in GYRE R9, R27 and R54.
-      From \citet{Levy_al_OM10}.}
+      From \citet{Levy_al_OM10}.
+    }
   \end{center}
 \end{figure}
@@ -315,3 +320,5 @@
 Unlike ordinary river points the Baltic inputs also include salinity and temperature data.
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIA.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIA.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIA.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,8 +7,8 @@
 \chapter{Output and Diagnostics (IOM, DIA, TRD, FLO)}
 \label{chap:DIA}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new line
 
 % ================================================================
@@ -168,32 +169,42 @@
 See the XML basics section below for more details on XML syntax and rules.
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|lXl|} \hline
-		variable name                                                           &
-		description                                                             &
-		example  \\	\hline	\hline
-		buffer\_size                                                            &
-		buffer size used by XIOS to send data from NEMO to XIOS. 
-		Larger is more efficient. 
-		Note that needed/used buffer sizes are summarized at the end of the job &
-		25000000 \\	\hline
-		buffer\_server\_factor\_size                                            &
-		ratio between NEMO and XIOS buffer size.
-		Should be 2.                                                            &
-		2        \\	\hline
-		info\_level                                                             &
-		verbosity level (0 to 100)                                              &
-		0        \\	\hline
-		using\_server                                                           &
-		activate attached(false) or detached(true) mode                         &
-		true     \\	\hline
-		using\_oasis                                                            &
-		XIOS is used with OASIS(true) or not (false)                            &
-		false    \\	\hline
-		oasis\_codes\_id                                                        &
-		when using oasis, define the identifier of NEMO in the namcouple. 
-		Note that the identifier of XIOS is xios.x                              &
-		oceanx   \\	\hline
-	\end{tabularx}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|lXl|}
+    \hline
+    variable name                                                           &
+    description                                                             &
+    example  \\
+    \hline
+    \hline
+    buffer\_size                                                            &
+    buffer size used by XIOS to send data from NEMO to XIOS.
+    Larger is more efficient.
+    Note that needed/used buffer sizes are summarized at the end of the job &
+    25000000 \\
+    \hline
+    buffer\_server\_factor\_size                                            &
+    ratio between NEMO and XIOS buffer size.
+    Should be 2.                                                            &
+    2        \\
+    \hline
+    info\_level                                                             &
+    verbosity level (0 to 100)                                              &
+    0        \\
+    \hline
+    using\_server                                                           &
+    activate attached(false) or detached(true) mode                         &
+    true     \\
+    \hline
+    using\_oasis                                                            &
+    XIOS is used with OASIS(true) or not (false)                            &
+    false    \\
+    \hline
+    oasis\_codes\_id                                                        &
+    when using oasis, define the identifier of NEMO in the namcouple.
+    Note that the identifier of XIOS is xios.x                              &
+    oceanx   \\
+    \hline
+  \end{tabularx}
 \end{table}
 
@@ -281,15 +292,22 @@
 Each tag family has hierarchy of three flavors (except for context):
 
-\begin{table} \scriptsize
-	\begin{tabular*}{\textwidth}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline
-		flavor  & description                                                                 &
-		example                          \\ \hline \hline
-		root    & declaration of the root element that can contain element groups or elements &
-		\xmlcode{<file_definition ... >} \\ \hline
-		group   & declaration of a group element that can contain element groups or elements  &
-		\xmlcode{<file_group      ... >} \\ \hline
-		element & declaration of an element that can contain elements                         &
-		\xmlcode{<file            ... >} \\ \hline
-	\end{tabular*}
+\begin{table}
+  \scriptsize
+  \begin{tabular*}{\textwidth}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|}
+    \hline
+    flavor  & description                                                                 &
+                                                                                            example                          \\
+    \hline
+    \hline
+    root    & declaration of the root element that can contain element groups or elements &
+                                                                                            \xmlcode{<file_definition ... >} \\
+    \hline
+    group   & declaration of a group element that can contain element groups or elements  &
+                                                                                            \xmlcode{<file_group      ... >} \\
+    \hline
+    element & declaration of an element that can contain elements                         &
+                                                                                            \xmlcode{<file            ... >} \\
+    \hline
+  \end{tabular*}
 \end{table}
 
@@ -306,51 +324,73 @@
 In \NEMO, we used the following contexts (that can be defined in any order):
 
-\begin{table} \scriptsize
-	\begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline
-		context         &	description                                                                & 
-		example                              \\	\hline \hline
-		context xios    &	context containing information for XIOS                                    & 
-		\xmlcode{<context id="xios" ... >}   \\	\hline
-		context nemo    &	context containing IO information for NEMO (mother grid when using AGRIF)  & 
-		\xmlcode{<context id="nemo" ... >}   \\	\hline
-		context 1\_nemo &	context containing IO information for NEMO child grid 1 (when using AGRIF) & 
-		\xmlcode{<context id="1_nemo" ... >} \\	\hline
-		context n\_nemo &	context containing IO information for NEMO child grid n (when using AGRIF) & 
-		\xmlcode{<context id="n_nemo" ... >} \\	\hline
+\begin{table}
+  \scriptsize
+  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|}
+    \hline
+    context         &	description                                                                &
+                                                                                                     example                              \\
+    \hline
+    \hline
+    context xios    &	context containing information for XIOS                                    &
+                                                                                                     \xmlcode{<context id="xios" ... >}   \\
+    \hline
+    context nemo    &	context containing IO information for NEMO (mother grid when using AGRIF)  & 
+                                                                                                     \xmlcode{<context id="nemo" ... >}   \\
+    \hline
+    context 1\_nemo &	context containing IO information for NEMO child grid 1 (when using AGRIF) & 
+                                                                                                     \xmlcode{<context id="1_nemo" ... >} \\
+    \hline
+    context n\_nemo &	context containing IO information for NEMO child grid n (when using AGRIF) & 
+                                                                                                     \xmlcode{<context id="n_nemo" ... >} \\
+    \hline
+  \end{tabular}
+\end{table}
+
+\noindent The xios context contains only 1 tag:
+
+\begin{table}
+  \scriptsize
+  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|}
+    \hline
+    context tag                                      &
+                                                       description                                      &
+                                                                                                          example                              \\
+    \hline
+    \hline
+    variable\_definition                             &
+                                                       define variables needed by XIOS.
+                                                       This can be seen as a kind of namelist for XIOS. &
+                                                                                                          \xmlcode{<variable_definition ... >} \\
+    \hline
 	\end{tabular}
 \end{table}
 
-\noindent The xios context contains only 1 tag:
-
-\begin{table} \scriptsize
-	\begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline
-		context tag                                      & 
-		description                                      & 
-		example                              \\ \hline	\hline
-		variable\_definition                             & 
-		define variables needed by XIOS. 
-		This can be seen as a kind of namelist for XIOS. & 
-		\xmlcode{<variable_definition ... >} \\ \hline
-	\end{tabular}
-\end{table}
-
 \noindent Each context tag related to NEMO (mother or child grids) is divided into 5 parts 
 (that can be defined in any order):
 
-\begin{table} \scriptsize
-	\begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|} \hline
-		context tag        &	description                                                               & 
-		example                            \\ \hline \hline
-		field\_definition  &	define all variables that can potentially be outputted                    & 
-		\xmlcode{<field_definition ... >}  \\ \hline
-		file\_definition   &	define the netcdf files to be created and the variables they will contain & 
-		\xmlcode{<file_definition ... >}   \\ \hline
-		axis\_definition   &	define vertical axis                                                      & 
-		\xmlcode{<axis_definition ... >}   \\ \hline
-		domain\_definition &	define the horizontal grids                                               & 
-		\xmlcode{<domain_definition ... >} \\ \hline
-		grid\_definition   &	define the 2D and 3D grids (association of an axis and a domain)          & 
-		\xmlcode{<grid_definition ... >}   \\ \hline
-	\end{tabular}
+\begin{table}
+  \scriptsize
+  \begin{tabular}{|p{0.15\textwidth}p{0.4\textwidth}p{0.35\textwidth}|}
+    \hline
+    context tag        &	description                                                               &
+                                                                                                       example                            \\
+    \hline
+    \hline
+    field\_definition  &	define all variables that can potentially be outputted                    &
+                                                                                                       \xmlcode{<field_definition ... >}  \\
+    \hline
+    file\_definition   &	define the netcdf files to be created and the variables they will contain &
+                                                                                                       \xmlcode{<file_definition ... >}   \\
+    \hline
+    axis\_definition   &	define vertical axis                                                      &
+                                                                                                       \xmlcode{<axis_definition ... >}   \\
+    \hline
+    domain\_definition &	define the horizontal grids                                               &
+                                                                                                       \xmlcode{<domain_definition ... >} \\
+    \hline
+    grid\_definition   &	define the 2D and 3D grids (association of an axis and a domain)          &
+                                                                                                       \xmlcode{<grid_definition ... >}   \\
+    \hline
+  \end{tabular}
 \end{table}
 
@@ -538,25 +578,39 @@
 the following part of the name and the name\_suffix (that can be inherited) will be automatically replaced by:
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|lX|} \hline
-		\centering placeholder string &
-		automatically replaced by                          \\ \hline \hline
-		\centering @expname@          &
-		the experiment name (from cn\_exp in the namelist) \\ \hline
-		\centering @freq@             &
-		output frequency (from attribute output\_freq)     \\ \hline
-		\centering @startdate@        &
-		starting date of the simulation (from nn\_date0 in the restart or the namelist). \newline
-		\verb?yyyymmdd?          format                   \\ \hline
-		\centering @startdatefull@    &
-		starting date of the simulation (from nn\_date0 in the restart or the namelist). \newline
-		\verb?yyyymmdd_hh:mm:ss? format                    \\ \hline
-		\centering @enddate@          &
-		ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). \newline
-		\verb?yyyymmdd?          format                    \\ \hline
-		\centering @enddatefull@      &
-		ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist). \newline
-		\verb?yyyymmdd_hh:mm:ss? format                    \\ \hline
-	\end{tabularx}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|lX|}
+    \hline
+    \centering placeholder string &
+    automatically replaced by                          \\
+    \hline
+    \hline
+    \centering @expname@          &
+    the experiment name (from cn\_exp in the namelist) \\
+    \hline
+    \centering @freq@             &
+    output frequency (from attribute output\_freq)     \\
+    \hline
+    \centering @startdate@        &
+    starting date of the simulation (from nn\_date0 in the restart or the namelist).
+    \newline
+    \verb?yyyymmdd?          format                   \\
+    \hline
+    \centering @startdatefull@    &
+    starting date of the simulation (from nn\_date0 in the restart or the namelist).
+    \newline
+    \verb?yyyymmdd_hh:mm:ss? format                    \\
+    \hline
+    \centering @enddate@          &
+    ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist).
+    \newline
+    \verb?yyyymmdd?          format                    \\
+    \hline
+    \centering @enddatefull@      &
+    ending date of the simulation   (from nn\_date0 and nn\_itend  in the namelist).
+    \newline
+    \verb?yyyymmdd_hh:mm:ss? format                    \\
+    \hline
+  \end{tabularx}
 \end{table}
 
@@ -585,34 +639,44 @@
 \\
 
-\begin{table} \scriptsize
-\begin{tabularx}{\textwidth}{|X|c|c|c|}   \hline
-	tag ids affected by automatic definition of some of their attributes & 
-	name attribute                                                       & 
-	attribute value                      \\ \hline \hline
-	field\_definition                                                    & 
-	freq\_op                                                             & 
-	\np{rn\_rdt}                         \\ \hline
-	SBC                                                                  & 
-	freq\_op                                                             & 
-	\np{rn\_rdt} $\times$ \np{nn\_fsbc}  \\ \hline
-	ptrc\_T                                                              & 
-	freq\_op                                                             & 
-	\np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ \hline
-	diad\_T                                                              & 
-	freq\_op                                                             & 
-	\np{rn\_rdt} $\times$ \np{nn\_dttrc} \\ \hline
-	EqT, EqU, EqW                                                        & 
-	jbegin, ni,                                                          & 
-	according to the grid                \\
-                                                                        & 
-	name\_suffix                                                         & 
-                                        \\ \hline
-	TAO, RAMA and PIRATA moorings                                        & 
-	zoom\_ibegin, zoom\_jbegin,                                          & 
-	according to the grid                \\
-                                                                        & 
-	name\_suffix                                                         & 
-                                        \\ \hline
-\end{tabularx}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|X|c|c|c|}
+    \hline
+    tag ids affected by automatic definition of some of their attributes &
+    name attribute                                                       &
+    attribute value                      \\
+    \hline
+    \hline
+    field\_definition                                                    &
+    freq\_op                                                             &
+    \np{rn\_rdt}                         \\
+    \hline
+    SBC                                                                  &
+    freq\_op                                                             &
+    \np{rn\_rdt} $\times$ \np{nn\_fsbc}  \\
+    \hline
+    ptrc\_T                                                              &
+    freq\_op                                                             &
+    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\
+    \hline
+    diad\_T                                                              &
+    freq\_op                                                             &
+    \np{rn\_rdt} $\times$ \np{nn\_dttrc} \\
+    \hline
+    EqT, EqU, EqW                                                        &
+    jbegin, ni,                                                          &
+    according to the grid                \\
+    &
+    name\_suffix                                                         &
+    \\
+    \hline
+    TAO, RAMA and PIRATA moorings                                        &
+    zoom\_ibegin, zoom\_jbegin,                                          &
+    according to the grid                \\
+    &
+    name\_suffix                                                         &
+    \\
+    \hline
+  \end{tabularx}
 \end{table}
 
@@ -774,346 +838,439 @@
 \subsubsection{Tag list per family}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|l|X|} \hline
-	tag name                                                                                     &
-	description                                                                                  &
-	accepted attribute                                                                           &
-	child of                                                                                     &
-	parent of                       \\ \hline \hline
-	simulation                                                                                   &
-	this tag is the root tag which encapsulates all the content of the XML file                  &
-	none                                                                                         &
-	none                                                                                         &
-	context                         \\ \hline
-	context                                                                                      &
-	encapsulates parts of the XML file dedicated to different codes or different parts of a code &
-	id (''xios'', ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom), src, time\_origin             &
-	simulation                                                                                   &
-	all root tags: ... \_definition \\ \hline
-	\end{tabularx}
-	\caption{Context tags}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|l|X|}
+    \hline
+    tag name                                                                                     &
+    description                                                                                  &
+    accepted attribute                                                                           &
+    child of                                                                                     &
+    parent of                       \\
+    \hline
+    \hline
+    simulation                                                                                   &
+    this tag is the root tag which encapsulates all the content of the XML file                  &
+    none                                                                                         &
+    none                                                                                         &
+    context                         \\
+    \hline
+    context                                                                                      &
+    encapsulates parts of the XML file dedicated to different codes or different parts of a code &
+    id (''xios'', ''nemo'' or ''n\_nemo'' for the nth AGRIF zoom), src, time\_origin             &
+    simulation                                                                                   &
+    all root tags: ... \_definition \\
+    \hline
+  \end{tabularx}
+  \caption{Context tags}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|l|} \hline
-	tag name                                                                                & 
-	description                                                                             & 
-	accepted attribute                                                                      & 
-	child of                                                                                & 
-	parent of             \\ \hline \hline
-	field\_definition                                                                       & 
-	encapsulates the definition of all the fields that can potentially be outputted         & 
-	axis\_ref, default\_value, domain\_ref, enabled, grid\_ref, level, operation, prec, src & 
-	context                                                                                 & 
-	field or field\_group \\ \hline
-	field\_group                                                                            & 
-	encapsulates a group of fields                                                          & 
-	axis\_ref, default\_value, domain\_ref, enabled, group\_ref, grid\_ref,
-	id, level, operation, prec, src                                                         & 
-	field\_definition, field\_group, file                                                   & 
-	field or field\_group \\ \hline
-	field                                                                                   & 
-	define a specific field                                                                 & 
-	axis\_ref, default\_value, domain\_ref, enabled, field\_ref, grid\_ref,
-	id, level, long\_name, name, operation, prec, standard\_name, unit                      & 
-	field\_definition, field\_group, file                                                   & 
-	none                  \\ \hline
-	\end{tabularx}
-	\caption{Field tags ("\tt{field\_*}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|l|}
+    \hline
+    tag name                                                                                &
+    description                                                                             &
+    accepted attribute                                                                      &
+    child of                                                                                &
+    parent of             \\
+    \hline
+    \hline
+    field\_definition                                                                       &
+    encapsulates the definition of all the fields that can potentially be outputted         &
+    axis\_ref, default\_value, domain\_ref, enabled, grid\_ref, level, operation, prec, src &
+    context                                                                                 &
+    field or field\_group \\
+    \hline
+    field\_group                                                                            &
+    encapsulates a group of fields                                                          &
+    axis\_ref, default\_value, domain\_ref, enabled, group\_ref, grid\_ref,
+    id, level, operation, prec, src                                                         &
+    field\_definition, field\_group, file                                                   &
+    field or field\_group \\
+    \hline
+    field                                                                                   &
+    define a specific field                                                                 &
+    axis\_ref, default\_value, domain\_ref, enabled, field\_ref, grid\_ref,
+    id, level, long\_name, name, operation, prec, standard\_name, unit                      &
+    field\_definition, field\_group, file                                                   &
+    none                  \\
+    \hline
+  \end{tabularx}
+  \caption{Field tags ("\tt{field\_*}")}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|l|} \hline
-	tag name                                                            & 
-	description                                                         & 
-	accepted attribute                                                  & 
-	child of                                                            & 
-	parent of           \\ \hline \hline
-	file\_definition                                                    & 
-	encapsulates the definition of all the files that will be outputted & 
-	enabled, min\_digits, name, name\_suffix, output\_level,
-	split\_freq\_format, split\_freq, sync\_freq, type, src             & 
-	context                                                             & 
-	file or file\_group \\ \hline
-	file\_group                                                         & 
-	encapsulates a group of files that will be outputted                & 
-	enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level,
-	split\_freq\_format, split\_freq, sync\_freq, type, src             & 
-	file\_definition, file\_group                                       & 
-	file or file\_group \\ \hline
-	file                                                                & 
-	define the contents of a file to be outputted                       & 
-	enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level,
-	split\_freq\_format, split\_freq, sync\_freq, type, src             & 
-	file\_definition, file\_group                                       & 
-	field               \\ \hline
-	\end{tabularx}
-	\caption{File tags ("\tt{file\_*}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|l|}
+    \hline
+    tag name                                                            &
+    description                                                         &
+    accepted attribute                                                  &
+    child of                                                            &
+    parent of           \\
+    \hline
+    \hline
+    file\_definition                                                    &
+    encapsulates the definition of all the files that will be outputted &
+    enabled, min\_digits, name, name\_suffix, output\_level,
+    split\_freq\_format, split\_freq, sync\_freq, type, src             &
+    context                                                             &
+    file or file\_group \\
+    \hline
+    file\_group                                                         &
+    encapsulates a group of files that will be outputted                &
+    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level,
+    split\_freq\_format, split\_freq, sync\_freq, type, src             &
+    file\_definition, file\_group                                       &
+    file or file\_group \\
+    \hline
+    file                                                                &
+    define the contents of a file to be outputted                       &
+    enabled, description, id, min\_digits, name, name\_suffix, output\_freq, output\_level,
+    split\_freq\_format, split\_freq, sync\_freq, type, src             &
+    file\_definition, file\_group                                       &
+    field               \\
+    \hline
+  \end{tabularx}
+  \caption{File tags ("\tt{file\_*}")}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline
-	tag name                                                                               & 
-	description                                                                            & 
-	accepted attribute                                                                     & 
-	child of                                                                               & 
-	parent of         \\ \hline \hline
-	axis\_definition                                                                       & 
-	define all the vertical axis potentially used by the variables                         & 
-	src                                                                                    & 
-	context                                                                                & 
-	axis\_group, axis \\ \hline
-	axis\_group                                                                            & 
-	encapsulates a group of vertical axis                                                  & 
-	id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size & 
-	axis\_definition, axis\_group                                                          & 
-	axis\_group, axis \\ \hline
-	axis                                                                                   & 
-	define a vertical axis                                                                 & 
-	id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size & 
-	axis\_definition, axis\_group                                                          & 
-	none					\\ \hline
-	\end{tabularx}
-	\caption{Axis tags ("\tt{axis\_*}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|X|}
+    \hline
+    tag name                                                                               &
+    description                                                                            &
+    accepted attribute                                                                     &
+    child of                                                                               &
+    parent of         \\
+    \hline
+    \hline
+    axis\_definition                                                                       &
+    define all the vertical axis potentially used by the variables                         &
+    src                                                                                    &
+    context                                                                                &
+    axis\_group, axis \\
+    \hline
+    axis\_group                                                                            &
+    encapsulates a group of vertical axis                                                  &
+    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &
+    axis\_definition, axis\_group                                                          &
+    axis\_group, axis \\
+    \hline
+    axis                                                                                   &
+    define a vertical axis                                                                 &
+    id, lon\_name, positive, src, standard\_name, unit, zoom\_begin, zoom\_end, zoom\_size &
+    axis\_definition, axis\_group                                                          &
+    none					\\
+    \hline
+  \end{tabularx}
+  \caption{Axis tags ("\tt{axis\_*}")}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline
-	tag name                                                            & 
-	description                                                         & 
-	accepted attribute                                                  & 
-	child of                                                            & 
-	parent of               \\ \hline \hline
-	domain\_\-definition                                                & 
-	define all the horizontal domains potentially used by the variables & 
-	src                                                                 & 
-	context                                                             & 
-	domain\_\-group, domain \\ \hline
-	domain\_group                                                       & 
-	encapsulates a group of horizontal domains                          & 
-	id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  & 
-	domain\_\-definition, domain\_group                                 & 
-	domain\_\-group, domain \\ \hline
-	domain                                                              & 
-	define an horizontal domain                                         & 
-	id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  & 
-	domain\_\-definition, domain\_group                                 & 
-	none                    \\ \hline
-	\end{tabularx}
-	\caption{Domain tags ("\tt{domain\_*)}"}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|X|}
+    \hline
+    tag name                                                            &
+    description                                                         &
+    accepted attribute                                                  &
+    child of                                                            &
+    parent of               \\
+    \hline
+    \hline
+    domain\_\-definition                                                &
+    define all the horizontal domains potentially used by the variables &
+    src                                                                 &
+    context                                                             &
+    domain\_\-group, domain \\
+    \hline
+    domain\_group                                                       &
+    encapsulates a group of horizontal domains                          &
+    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  &
+    domain\_\-definition, domain\_group                                 &
+    domain\_\-group, domain \\
+    \hline
+    domain                                                              &
+    define an horizontal domain                                         &
+    id, lon\_name, src, zoom\_ibegin, zoom\_jbegin, zoom\_ni, zoom\_nj  &
+    domain\_\-definition, domain\_group                                 &
+    none                    \\
+    \hline
+  \end{tabularx}
+  \caption{Domain tags ("\tt{domain\_*)}"}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|X|} \hline
-	tag name                                                                                     &
-	description                                                                                  &
-	accepted attribute                                                                           &
-	child of                                                                                     &
-	parent of                       \\ \hline \hline
-	grid\_definition                                                                               &
-	define all the grid (association of a domain and/or an axis) potentially used by the variables &
-	src                                                                                            &
-	context                                                                                        &
-	grid\_group, grid	\\ \hline
-	grid\_group                                                                                    &
-	encapsulates a group of grids                                                                  &
-	id, domain\_ref,axis\_ref                                                                      &
-	grid\_definition, grid\_group                                                                  &
-	grid\_group, grid	\\ \hline
-   grid                                                                                           &
-	define a grid                                                                                  &
-	id, domain\_ref,axis\_ref                                                                      &
-	grid\_definition, grid\_group                                                                  &
-	none 					\\ \hline
-	\end{tabularx}
-	\caption{Grid tags ("\tt{grid\_*}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|X|}
+    \hline
+    tag name                                                                                     &
+    description                                                                                  &
+    accepted attribute                                                                           &
+    child of                                                                                     &
+    parent of                       \\
+    \hline
+    \hline
+    grid\_definition                                                                               &
+    define all the grid (association of a domain and/or an axis) potentially used by the variables &
+    src                                                                                            &
+    context                                                                                        &
+    grid\_group, grid	\\
+    \hline
+    grid\_group                                                                                    &
+    encapsulates a group of grids                                                                  &
+    id, domain\_ref,axis\_ref                                                                      &
+    grid\_definition, grid\_group                                                                  &
+    grid\_group, grid	\\
+    \hline
+    grid                                                                                           &
+    define a grid                                                                                  &
+    id, domain\_ref,axis\_ref                                                                      &
+    grid\_definition, grid\_group                                                                  &
+    none 					\\
+    \hline
+  \end{tabularx}
+  \caption{Grid tags ("\tt{grid\_*}")}
 \end{table}
 
 \subsubsection{Attributes list per family}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|l|l|} \hline
-		attribute name                           & 
-		description                              & 
-		example                                  & 
-		accepted by            \\ \hline \hline
-		axis\_ref                                & 
-		refers to the id of a vertical axis      & 
-		axis\_ref="deptht"                       & 
-		field, grid families   \\	\hline
-		domain\_ref                              & 
-		refers to the id of a domain             & 
-		domain\_ref="grid\_T"                    & 
-		field or grid families \\ \hline
-		field\_ref                               & 
-		id of the field we want to add in a file & 
-		field\_ref="toce"                        & 
-		field                  \\ \hline
-		grid\_ref                                & 
-		refers to the id of a grid               & 
-		grid\_ref="grid\_T\_2D"                  & 
-		field family           \\ \hline
-		group\_ref                               & 
-		refer to a group of variables            & 
-		group\_ref="mooring"                     & 
-		field\_group           \\ \hline
-	\end{tabularx}
-	\caption{Reference attributes ("\tt{*\_ref}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|l|l|}
+    \hline
+    attribute name                           &
+    description                              &
+    example                                  &
+    accepted by            \\
+    \hline
+    \hline
+    axis\_ref                                &
+    refers to the id of a vertical axis      &
+    axis\_ref="deptht"                       &
+    field, grid families   \\
+    \hline
+    domain\_ref                              &
+    refers to the id of a domain             &
+    domain\_ref="grid\_T"                    &
+    field or grid families \\
+    \hline
+    field\_ref                               &
+    id of the field we want to add in a file &
+    field\_ref="toce"                        &
+    field                  \\
+    \hline
+    grid\_ref                                &
+    refers to the id of a grid               &
+    grid\_ref="grid\_T\_2D"                  &
+    field family           \\
+    \hline
+    group\_ref                               &
+    refer to a group of variables            &
+    group\_ref="mooring"                     &
+    field\_group           \\
+    \hline
+  \end{tabularx}
+  \caption{Reference attributes ("\tt{*\_ref}")}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|l|l|} \hline
-		attribute name                                     & 
-		description                                        & 
-		example                                            & 
-		accepted by   \\ \hline \hline
-		zoom\_ibegin                                       & 
-		starting point along x direction of the zoom.
-		Automatically defined for TAO/RAMA/PIRATA moorings & 
-		zoom\_ibegin="1"                                   & 
-		domain family \\ \hline
-		zoom\_jbegin                                       & 
-		starting point along y direction of the zoom.
-		Automatically defined for TAO/RAMA/PIRATA moorings & 
-		zoom\_jbegin="1"                                   & 
-		domain family \\ \hline
-		zoom\_ni                                           & 
-		zoom extent along x direction                      & 
-		zoom\_ni="1"                                       & 
-		domain family \\ \hline
-		zoom\_nj                                           & 
-		zoom extent along y direction                      & 
-		zoom\_nj="1"                                       & 
-		domain family \\ \hline
-	\end{tabularx}
-	\caption{Domain attributes ("\tt{zoom\_*}")}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|l|l|}
+    \hline
+    attribute name                                     &
+    description                                        &
+    example                                            &
+    accepted by   \\
+    \hline
+    \hline
+    zoom\_ibegin                                       &
+    starting point along x direction of the zoom.
+    Automatically defined for TAO/RAMA/PIRATA moorings &
+    zoom\_ibegin="1"                                   &
+    domain family \\
+    \hline
+    zoom\_jbegin                                       &
+    starting point along y direction of the zoom.
+    Automatically defined for TAO/RAMA/PIRATA moorings &
+    zoom\_jbegin="1"                                   &
+    domain family \\
+    \hline
+    zoom\_ni                                           &
+    zoom extent along x direction                      &
+    zoom\_ni="1"                                       &
+    domain family \\
+    \hline
+    zoom\_nj                                           &
+    zoom extent along y direction                      &
+    zoom\_nj="1"                                       &
+    domain family \\
+    \hline
+  \end{tabularx}
+  \caption{Domain attributes ("\tt{zoom\_*}")}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|l|l|} \hline
-		attribute name                                                                                       &
-		description                                                                                          &
-		example                                                                                              &
-		accepted by                            \\ \hline \hline
-		min\_digits                                                                                          &
-		specify the minimum of digits used in the core number in the name of the NetCDF file                 &
-		min\_digits="4"                                                                                      &
-		file family                            \\ \hline
-		name\_suffix                                                                                         &
-		suffix to be inserted after the name and before the cpu number and the ''.nc'' termination of a file &
-		name\_suffix="\_myzoom"                                                                              &
-		file family                            \\ \hline
-		output\_level                                                                                        &
-		output priority of variables in a file: 0 (high) to 10 (low).
-		All variables listed in the file with a level smaller or equal to output\_level will be output.
-		Other variables won't be output even if they are listed in the file.                                 &
-		output\_level="10"                                                                                   &
-		file family                            \\ \hline
-		split\_freq                                                                                          &
-		frequency at which to temporally split output files.
-		Units can be ts (timestep), y, mo, d, h, mi, s.
-		Useful for long runs to prevent over-sized output files.                                             &
-		split\_freq="1mo"                                                                                    &
-		file family                            \\ \hline
-		split\_freq\-\_format                                                                                &
-		date format used in the name of temporally split output files.
-		Can be specified using the following syntaxes: \%y, \%mo, \%d, \%h \%mi and \%s                      &
-		split\_freq\_format= "\%y\%mo\%d"                                                                    &
-		file family                            \\ \hline
-		sync\_freq                                                                                           &
-		NetCDF file synchronization frequency (update of the time\_counter).
-		Units can be ts (timestep), y, mo, d, h, mi, s.                                                      &
-		sync\_freq="10d"                                                                                     &
-		file family                            \\ \hline
-		type (1)                                                                                             &
-		specify if the output files are to be split spatially (multiple\_file) or not (one\_file)            &
-		type="multiple\_file"                                                                                &
-		file familly                           \\ \hline
-	\end{tabularx}
-	\caption{File attributes}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|l|l|}
+    \hline
+    attribute name                                                                                       &
+    description                                                                                          &
+    example                                                                                              &
+    accepted by                            \\
+    \hline
+    \hline
+    min\_digits                                                                                          &
+    specify the minimum of digits used in the core number in the name of the NetCDF file                 &
+    min\_digits="4"                                                                                      &
+    file family                            \\
+    \hline
+    name\_suffix                                                                                         &
+    suffix to be inserted after the name and before the cpu number and the ''.nc'' termination of a file &
+    name\_suffix="\_myzoom"                                                                              &
+    file family                            \\
+    \hline
+    output\_level                                                                                        &
+    output priority of variables in a file: 0 (high) to 10 (low).
+    All variables listed in the file with a level smaller or equal to output\_level will be output.
+    Other variables won't be output even if they are listed in the file.                                 &
+    output\_level="10"                                                                                   &
+    file family                            \\
+    \hline
+    split\_freq                                                                                          &
+    frequency at which to temporally split output files.
+    Units can be ts (timestep), y, mo, d, h, mi, s.
+    Useful for long runs to prevent over-sized output files.                                             &
+    split\_freq="1mo"                                                                                    &
+    file family                            \\
+    \hline
+    split\_freq\-\_format                                                                                &
+    date format used in the name of temporally split output files.
+    Can be specified using the following syntaxes: \%y, \%mo, \%d, \%h \%mi and \%s                      &
+    split\_freq\_format= "\%y\%mo\%d"                                                                    &
+    file family                            \\
+    \hline
+    sync\_freq                                                                                           &
+    NetCDF file synchronization frequency (update of the time\_counter).
+    Units can be ts (timestep), y, mo, d, h, mi, s.                                                      &
+    sync\_freq="10d"                                                                                     &
+    file family                            \\
+    \hline
+    type (1)                                                                                             &
+    specify if the output files are to be split spatially (multiple\_file) or not (one\_file)            &
+    type="multiple\_file"                                                                                &
+    file familly                           \\
+    \hline
+  \end{tabularx}
+  \caption{File attributes}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|l|l|} \hline
-		attribute name                                                                                       &
-		description                                                                                          &
-		example                                                                                              &
-		accepted by                            \\ \hline \hline
-		default\_value                                                                                       &
-		missing\_value definition                                                                            &
-		default\_value="1.e20"                                                                               &
-		field family                           \\ \hline
-		level                                                                                                &
-		output priority of a field: 0 (high) to 10 (low)                                                     &
-		level="1"                                                                                            &
-		field family                           \\ \hline
-		operation                                                                                            &
-		type of temporal operation: average, accumulate, instantaneous, min, max and once                    &
-		operation="average"                                                                                  &
-		field family                           \\ \hline
-		output\_freq                                                                                         &
-		operation frequency. units can be ts (timestep), y, mo, d, h, mi, s.                                 &
-		output\_freq="1d12h"                                                                                 &
-		field family                           \\ \hline
-		prec                                                                                                 &
-		output precision: real 4 or real 8                                                                   &
-		prec="4"                                                                                             &
-		field family                           \\ \hline
-		long\_name                                                                                           &
-		define the long\_name attribute in the NetCDF file                                                   &
-		long\_name="Vertical T levels"                                                                       &
-		field                                  \\ \hline
-		standard\_name                                                                                       &
-		define the standard\_name attribute in the NetCDF file                                               &
-		standard\_name= "Eastward\_Sea\_Ice\_Transport"                                                      &
-		field                                  \\ \hline
-	\end{tabularx}
-	\caption{Field attributes}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|l|l|}
+    \hline
+    attribute name                                                                                       &
+    description                                                                                          &
+    example                                                                                              &
+    accepted by                            \\
+    \hline
+    \hline
+    default\_value                                                                                       &
+    missing\_value definition                                                                            &
+    default\_value="1.e20"                                                                               &
+    field family                           \\
+    \hline
+    level                                                                                                &
+    output priority of a field: 0 (high) to 10 (low)                                                     &
+    level="1"                                                                                            &
+    field family                           \\
+    \hline
+    operation                                                                                            &
+    type of temporal operation: average, accumulate, instantaneous, min, max and once                    &
+    operation="average"                                                                                  &
+    field family                           \\
+    \hline
+    output\_freq                                                                                         &
+    operation frequency. units can be ts (timestep), y, mo, d, h, mi, s.                                 &
+    output\_freq="1d12h"                                                                                 &
+    field family                           \\
+    \hline
+    prec                                                                                                 &
+    output precision: real 4 or real 8                                                                   &
+    prec="4"                                                                                             &
+    field family                           \\
+    \hline
+    long\_name                                                                                           &
+    define the long\_name attribute in the NetCDF file                                                   &
+    long\_name="Vertical T levels"                                                                       &
+    field                                  \\
+    \hline
+    standard\_name                                                                                       &
+    define the standard\_name attribute in the NetCDF file                                               &
+    standard\_name= "Eastward\_Sea\_Ice\_Transport"                                                      &
+    field                                  \\
+    \hline
+  \end{tabularx}
+  \caption{Field attributes}
 \end{table}
 
-\begin{table} \scriptsize
-	\begin{tabularx}{\textwidth}{|l|X|X|X|} \hline
-		attribute name                                                                                       &
-		description                                                                                          &
-		example                                                                                              &
-		accepted by                            \\ \hline \hline
-		enabled                                                                                              &
-		switch on/off the output of a field or a file                                                        &
-		enabled=".true."                                                                                     &
-		field, file families                   \\ \hline
-		description                                                                                          &
-		just for information, not used                                                                       &
-		description="ocean T grid variables"                                                                 &
-		all tags                               \\ \hline
-		id                                                                                                   &
-		allow to identify a tag                                                                              &
-		id="nemo"                                                                                            &
-		accepted by all tags except simulation \\ \hline
-		name                                                                                                 &
-		name of a variable or a file. If the name of a file is undefined, its id is used as a name           &
-		name="tos"                                                                                           &
-		field or file families                 \\ \hline
-		positive                                                                                             &
-		convention used for the orientation of vertival axis (positive downward in \NEMO).                   &
-		positive="down"                                                                                      &
-		axis family                            \\ \hline
-		src                                                                                                  &
-		allow to include a file                                                                              &
-		src="./field\_def.xml"                                                                               &
-		accepted by all tags except simulation \\ \hline
-		time\_origin                                                                                         &
-		specify the origin of the time counter                                                               &
-		time\_origin="1900-01-01 00:00:00"                                                                   &
-		context                                \\ \hline
-		type (2)                                                                                             &
-		define the type of a variable tag                                                                    &
-		type="boolean"                                                                                       &
-		variable                               \\ \hline
-		unit                                                                                                 &
-		unit of a variable or the vertical axis                                                              &
-		unit="m"                                                                                             &
-		field and axis families                \\ \hline
-	\end{tabularx}
-	\caption{Miscellaneous attributes}
+\begin{table}
+  \scriptsize
+  \begin{tabularx}{\textwidth}{|l|X|X|X|}
+    \hline
+    attribute name                                                                                       &
+    description                                                                                          &
+    example                                                                                              &
+    accepted by                            \\
+    \hline
+    \hline
+    enabled                                                                                              &
+    switch on/off the output of a field or a file                                                        &
+    enabled=".true."                                                                                     &
+    field, file families                   \\
+    \hline
+    description                                                                                          &
+    just for information, not used                                                                       &
+    description="ocean T grid variables"                                                                 &
+    all tags                               \\
+    \hline
+    id                                                                                                   &
+    allow to identify a tag                                                                              &
+    id="nemo"                                                                                            &
+    accepted by all tags except simulation \\
+    \hline
+    name                                                                                                 &
+    name of a variable or a file. If the name of a file is undefined, its id is used as a name           &
+    name="tos"                                                                                           &
+    field or file families                 \\
+    \hline
+    positive                                                                                             &
+    convention used for the orientation of vertival axis (positive downward in \NEMO).                   &
+    positive="down"                                                                                      &
+    axis family                            \\
+    \hline
+    src                                                                                                  &
+    allow to include a file                                                                              &
+    src="./field\_def.xml"                                                                               &
+    accepted by all tags except simulation \\
+    \hline
+    time\_origin                                                                                         &
+    specify the origin of the time counter                                                               &
+    time\_origin="1900-01-01 00:00:00"                                                                   &
+    context                                \\
+    \hline
+    type (2)                                                                                             &
+    define the type of a variable tag                                                                    &
+    type="boolean"                                                                                       &
+    variable                               \\
+    \hline
+    unit                                                                                                 &
+    unit of a variable or the vertical axis                                                              &
+    unit="m"                                                                                             &
+    field and axis families                \\
+    \hline
+  \end{tabularx}
+  \caption{Miscellaneous attributes}
 \end{table}
 
@@ -1200,38 +1357,41 @@
 
 %------------------------------------------TABLE----------------------------------------------------
-\begin{table} \scriptsize \centering
-	\begin{tabular}{lrrr}
-		Filename                    & NetCDF3	& NetCDF4  & Reduction \\
-                                  & filesize	& filesize & \%        \\
-                                  & (KB)		& (KB)	  &           \\
-		ORCA2\_restart\_0000.nc     & 16420 	& 8860 	  & 47\%      \\
-		ORCA2\_restart\_0001.nc     & 16064 	& 11456    & 29\%      \\
-		ORCA2\_restart\_0002.nc     & 16064		& 9744	  & 40\%      \\
-		ORCA2\_restart\_0003.nc     & 16420		& 9404	  & 43\%      \\
-		ORCA2\_restart\_0004.nc     & 16200 	& 5844	  & 64\%      \\
-		ORCA2\_restart\_0005.nc     & 15848 	& 8172	  & 49\%      \\
-		ORCA2\_restart\_0006.nc     & 15848 	& 8012 	  & 50\%      \\
-		ORCA2\_restart\_0007.nc     & 16200 	& 5148 	  & 69\%      \\
-		ORCA2\_2d\_grid\_T\_0000.nc & 2200 		& 1504	  & 32\%      \\
-		ORCA2\_2d\_grid\_T\_0001.nc & 2200 		& 1748	  & 21\%      \\
-		ORCA2\_2d\_grid\_T\_0002.nc & 2200 		& 1592	  & 28\%      \\
-		ORCA2\_2d\_grid\_T\_0003.nc & 2200 		& 1540	  & 30\%      \\
-		ORCA2\_2d\_grid\_T\_0004.nc & 2200 		& 1204	  & 46\%      \\
-		ORCA2\_2d\_grid\_T\_0005.nc & 2200 		& 1444	  & 35\%      \\
-		ORCA2\_2d\_grid\_T\_0006.nc & 2200 		& 1428	  & 36\%      \\
-		ORCA2\_2d\_grid\_T\_0007.nc & 2200		& 1148	  & 48\%      \\
-		...                         & ...		& ...      & ...       \\
-		ORCA2\_2d\_grid\_W\_0000.nc & 4416		& 2240	  & 50\%      \\
-		ORCA2\_2d\_grid\_W\_0001.nc & 4416		& 2924	  & 34\%      \\
-		ORCA2\_2d\_grid\_W\_0002.nc & 4416		& 2512	  & 44\%      \\
-		ORCA2\_2d\_grid\_W\_0003.nc & 4416		& 2368	  & 47\%      \\
-		ORCA2\_2d\_grid\_W\_0004.nc & 4416		& 1432	  & 68\%      \\
-		ORCA2\_2d\_grid\_W\_0005.nc & 4416		& 1972	  & 56\%      \\
-		ORCA2\_2d\_grid\_W\_0006.nc & 4416		& 2028	  & 55\%      \\
-		ORCA2\_2d\_grid\_W\_0007.nc & 4416		& 1368	  & 70\%      \\
-	\end{tabular}
-	\caption{
-		\protect\label{tab:NC4}
-		Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression}
+\begin{table}
+  \scriptsize
+  \centering
+  \begin{tabular}{lrrr}
+    Filename                    & NetCDF3	& NetCDF4  & Reduction \\
+                                & filesize	& filesize & \%        \\
+                                & (KB)		& (KB)	  &           \\
+    ORCA2\_restart\_0000.nc     & 16420 	& 8860 	  & 47\%      \\
+    ORCA2\_restart\_0001.nc     & 16064 	& 11456    & 29\%      \\
+    ORCA2\_restart\_0002.nc     & 16064		& 9744	  & 40\%      \\
+    ORCA2\_restart\_0003.nc     & 16420		& 9404	  & 43\%      \\
+    ORCA2\_restart\_0004.nc     & 16200 	& 5844	  & 64\%      \\
+    ORCA2\_restart\_0005.nc     & 15848 	& 8172	  & 49\%      \\
+    ORCA2\_restart\_0006.nc     & 15848 	& 8012 	  & 50\%      \\
+    ORCA2\_restart\_0007.nc     & 16200 	& 5148 	  & 69\%      \\
+    ORCA2\_2d\_grid\_T\_0000.nc & 2200 		& 1504	  & 32\%      \\
+    ORCA2\_2d\_grid\_T\_0001.nc & 2200 		& 1748	  & 21\%      \\
+    ORCA2\_2d\_grid\_T\_0002.nc & 2200 		& 1592	  & 28\%      \\
+    ORCA2\_2d\_grid\_T\_0003.nc & 2200 		& 1540	  & 30\%      \\
+    ORCA2\_2d\_grid\_T\_0004.nc & 2200 		& 1204	  & 46\%      \\
+    ORCA2\_2d\_grid\_T\_0005.nc & 2200 		& 1444	  & 35\%      \\
+    ORCA2\_2d\_grid\_T\_0006.nc & 2200 		& 1428	  & 36\%      \\
+    ORCA2\_2d\_grid\_T\_0007.nc & 2200		& 1148	  & 48\%      \\
+    ...                         & ...		& ...      & ...       \\
+    ORCA2\_2d\_grid\_W\_0000.nc & 4416		& 2240	  & 50\%      \\
+    ORCA2\_2d\_grid\_W\_0001.nc & 4416		& 2924	  & 34\%      \\
+    ORCA2\_2d\_grid\_W\_0002.nc & 4416		& 2512	  & 44\%      \\
+    ORCA2\_2d\_grid\_W\_0003.nc & 4416		& 2368	  & 47\%      \\
+    ORCA2\_2d\_grid\_W\_0004.nc & 4416		& 1432	  & 68\%      \\
+    ORCA2\_2d\_grid\_W\_0005.nc & 4416		& 1972	  & 56\%      \\
+    ORCA2\_2d\_grid\_W\_0006.nc & 4416		& 2028	  & 55\%      \\
+    ORCA2\_2d\_grid\_W\_0007.nc & 4416		& 1368	  & 70\%      \\
+  \end{tabular}
+  \caption{
+    \protect\label{tab:NC4}
+    Filesize comparison between NetCDF3 and NetCDF4 with chunking and compression
+  }
 \end{table}
 %----------------------------------------------------------------------------------------------------
@@ -1329,10 +1489,13 @@
 
 \noindent Example: \\
-\noindent {\scriptsize \texttt{
-100.00000  90.00000  -1.50000 1.00000   0.00000   \\
-102.00000  90.00000  -1.50000 1.00000   0.00000   \\
-104.00000  90.00000  -1.50000 1.00000   0.00000   \\
-106.00000  90.00000  -1.50000 1.00000   0.00000   \\
-108.00000  90.00000  -1.50000 1.00000   0.00000}} \\
+\noindent
+{\scriptsize
+  \texttt{
+    100.00000  90.00000  -1.50000 1.00000   0.00000   \\
+    102.00000  90.00000  -1.50000 1.00000   0.00000   \\
+    104.00000  90.00000  -1.50000 1.00000   0.00000   \\
+    106.00000  90.00000  -1.50000 1.00000   0.00000   \\
+    108.00000  90.00000  -1.50000 1.00000   0.00000}
+} \\
 
 In the other case (longitude and latitude), input filename is init\_float.
@@ -1351,10 +1514,13 @@
 
 \noindent Example: \\
-\noindent {\scriptsize \texttt{
- 20.0 0.0 0.0 0 1 1    \\
--21.0 0.0 0.0 0 1 1    \\
--22.0 0.0 0.0 0 1 1    \\
--23.0 0.0 0.0 0 1 1    \\
--24.0 0.0 0.0 0 1 1 }} \\
+\noindent
+{\scriptsize
+  \texttt{
+    20.0 0.0 0.0 0 1 1    \\
+    -21.0 0.0 0.0 0 1 1    \\
+    -22.0 0.0 0.0 0 1 1    \\
+    -23.0 0.0 0.0 0 1 1    \\
+    -24.0 0.0 0.0 0 1 1 }
+} \\
 
 \np{jpnfl} is the total number of floats during the run.
@@ -1426,5 +1592,7 @@
  The Harmonic analysis solve the following equation:
 
-\[h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i}\]
+ \[
+   h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[A_{j}cos(\nu_{j}t_{j}-\phi_{j})] = e_{i}
+ \]
 
 With $A_{j}$, $\nu_{j}$, $\phi_{j}$, the amplitude, frequency and phase for each wave and $e_{i}$ the error.
@@ -1432,5 +1600,7 @@
 We can rewrite this equation:
 
-\[h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[C_{j}cos(\nu_{j}t_{j})+S_{j}sin(\nu_{j}t_{j})] = e_{i}\]
+\[
+  h_{i} - A_{0} + \sum^{nb\_ana}_{j=1}[C_{j}cos(\nu_{j}t_{j})+S_{j}sin(\nu_{j}t_{j})] = e_{i}
+\]
 
 with $A_{j}=\sqrt{C^{2}_{j}+S^{2}_{j}}$ and $\phi_{j}=arctan(S_{j}/C_{j})$.
@@ -1497,13 +1667,15 @@
 
 \noindent If nclass $\neq$ 0, the next lines contain the class type and the nclass bounds: \\
-{\scriptsize \texttt{
-long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name \\
-classtype                                                         \\
-zbound1                                                           \\
-zbound2                                                           \\
-.                                                                 \\
-.                                                                 \\
-nclass-1                                                          \\
-nclass}}
+{\scriptsize
+  \texttt{
+    long1 lat1 long2 lat2 nclass (ok/no)strpond (no)ice section\_name \\
+    classtype                                                         \\
+    zbound1                                                           \\
+    zbound2                                                           \\
+    .                                                                 \\
+    .                                                                 \\
+    nclass-1                                                          \\
+    nclass}
+}
 
 \noindent where \texttt{classtype} can be:
@@ -1527,23 +1699,28 @@
  Examples of two sections, the ACC\_Drake\_Passage with no classes,
  and the ATL\_Cuba\_Florida with 4 temperature clases (5 class bounds), are shown: \\
-\noindent {\scriptsize \texttt{
--68.    -54.5   -60.    -64.7  00 okstrpond noice ACC\_Drake\_Passage \\
--80.5    22.5   -80.5    25.5  05 nostrpond noice ATL\_Cuba\_Florida  \\
-ztem                                                                  \\
--2.0                                                                  \\
- 4.5                                                                  \\
- 7.0                                                                  \\
-12.0                                                                  \\
-40.0}}
+ \noindent
+ {\scriptsize
+   \texttt{
+     -68.    -54.5   -60.    -64.7  00 okstrpond noice ACC\_Drake\_Passage \\
+     -80.5    22.5   -80.5    25.5  05 nostrpond noice ATL\_Cuba\_Florida  \\
+     ztem                                                                  \\
+     -2.0                                                                  \\
+     4.5                                                                  \\
+     7.0                                                                  \\
+     12.0                                                                  \\
+     40.0}
+ }
 
 \subsubsection{To read the output files}
 
 The output format is: \\
-{\scriptsize \texttt{
-date, time-step number, section number,                \\
-section name, section slope coefficient, class number, \\
-class name, class bound 1 , classe bound2,             \\
-transport\_direction1, transport\_direction2,          \\
-transport\_total}}                                     \\
+{\scriptsize
+  \texttt{
+    date, time-step number, section number,                \\
+    section name, section slope coefficient, class number, \\
+    class name, class bound 1 , classe bound2,             \\
+    transport\_direction1, transport\_direction2,          \\
+    transport\_total}
+}                                     \\
 
 For sections with classes, the first \texttt{nclass-1} lines correspond to the transport for each class and
@@ -1559,15 +1736,17 @@
 direction: \\
 
-\begin{table} \scriptsize
-	\begin{tabular}{|l|l|l|l|l|} \hline
-		section slope coefficient      & section type & direction 1 & direction 2 & total transport
-		\\ \hline
-		0.                             & horizontal	 & northward	& southward   & postive: northward
-		\\ \hline
-		1000.                          & vertical     & eastward    & westward    & postive: eastward
-		\\ \hline
-		\texttt{$\neq$ 0, $\neq$ 1000.} & diagonal     & eastward    & westward	  & postive: eastward
-		\\ \hline
-	\end{tabular}
+\begin{table}
+  \scriptsize
+  \begin{tabular}{|l|l|l|l|l|}
+    \hline
+    section slope coefficient      & section type & direction 1 & direction 2 & total transport    \\
+    \hline
+    0.                             & horizontal	 & northward	& southward   & postive: northward    \\
+    \hline
+    1000.                          & vertical     & eastward    & westward    & postive: eastward		\\
+    \hline
+    \texttt{$\neq$ 0, $\neq$ 1000.} & diagonal     & eastward    & westward	  & postive: eastward		\\
+    \hline
+  \end{tabular}
 \end{table}
 
@@ -1609,15 +1788,19 @@
 A non-Boussinesq fluid conserves mass. It satisfies the following relations:
 
-\[	\begin{split}
-		\mathcal{M} &=  \mathcal{V}  \;\bar{\rho} \\
-		\mathcal{V} &=  \mathcal{A}  \;\bar{\eta}
-	\end{split}  \label{eq:MV_nBq}
-\]
+\begin{equation}
+  \begin{split}
+    \mathcal{M} &=  \mathcal{V}  \;\bar{\rho} \\
+    \mathcal{V} &=  \mathcal{A}  \;\bar{\eta}
+  \end{split}
+  \label{eq:MV_nBq}
+\end{equation}
 
 Temporal changes in total mass is obtained from the density conservation equation:
 
-\[ \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} ) 
-	= \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface}
- \label{eq:Co_nBq} \]
+\begin{equation}
+  \frac{1}{e_3} \partial_t ( e_3\,\rho) + \nabla( \rho \, \textbf{U} )
+  = \left. \frac{\textit{emp}}{e_3}\right|_\textit{surface}
+  \label{eq:Co_nBq}
+\end{equation}
 
 where $\rho$ is the \textit{in situ} density, and \textit{emp} the surface mass exchanges with the other media of
@@ -1625,6 +1808,8 @@
 Its global averaged leads to the total mass change 
 
-\[ \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}}
- \label{eq:Mass_nBq} \]
+\begin{equation}
+  \partial_t \mathcal{M} = \mathcal{A} \;\overline{\textit{emp}}
+  \label{eq:Mass_nBq}
+\end{equation}
 
 where $\overline{\textit{emp}} = \int_S \textit{emp}\,ds$ is the net mass flux through the ocean surface.
@@ -1632,7 +1817,9 @@
 the evolution equation of the mean sea level
 
-\[ \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}} 
-                         - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}}
- \label{eq:ssh_nBq} \]
+\begin{equation}
+  \partial_t \bar{\eta} = \frac{\overline{\textit{emp}}}{ \bar{\rho}}
+  - \frac{\mathcal{V}}{\mathcal{A}} \;\frac{\partial_t \bar{\rho} }{\bar{\rho}}
+  \label{eq:ssh_nBq}
+\end{equation}
 
 The first term in equation \autoref{eq:ssh_nBq} alters sea level by adding or subtracting mass from the ocean. 
@@ -1643,12 +1830,15 @@
 In particular, the mass conservation equation, \autoref{eq:Co_nBq}, degenerates into the incompressibility equation:
 
-\[ \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} )
-   = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface}
- \label{eq:Co_Bq} \]
+\[
+  \frac{1}{e_3} \partial_t ( e_3 ) + \nabla( \textbf{U} ) = \left. \frac{\textit{emp}}{\rho_o \,e_3}\right|_ \textit{surface}
+  % \label{eq:Co_Bq}
+\]
 
 and the global average of this equation now gives the temporal change of the total volume,
 
-\[ \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o}
- \label{eq:V_Bq} \]
+\[
+  \partial_t \mathcal{V} = \mathcal{A} \;\frac{\overline{\textit{emp}}}{\rho_o}
+  % \label{eq:V_Bq}
+\]
 
 Only the volume is conserved, not mass, or, more precisely, the mass which is conserved is the Boussinesq mass,
@@ -1666,6 +1856,8 @@
 $\eta_s$, a spatially uniform variable, as follows:
 
-\[ \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A}
- \label{eq:M_Bq} \]
+\begin{equation}
+  \mathcal{M}_o = \mathcal{M} + \rho_o \,\eta_s \,\mathcal{A}
+  \label{eq:M_Bq}
+\end{equation}
 
 Any change in $\mathcal{M}$ which cannot be explained by the net mass flux through the ocean surface
@@ -1675,6 +1867,8 @@
 in \autoref{eq:M_Bq} leads to a very simple form for the steric height:
 
-\[	\eta_s = - \frac{1}{\mathcal{A}} \mathcal{D}
- \label{eq:steric_Bq} \]
+\begin{equation}
+  \eta_s = - \frac{1}{\mathcal{A}} \mathcal{D}
+  \label{eq:steric_Bq}
+\end{equation}
 
 The above formulation of the steric height of a Boussinesq ocean requires four remarks.
@@ -1698,7 +1892,8 @@
 In the non linear free surface case, $i.e.$ \key{vvl} defined, it is given by
 
-\[ \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }
-                   { \sum_{i,\,j,\,k}       e_{1t} e_{2t} e_{3t} }
- \label{eq:discrete_steric_Bq_nfs} \]
+\[
+  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t} e_{2t} e_{3t} }{ \sum_{i,\,j,\,k}       e_{1t} e_{2t} e_{3t} }
+  % \label{eq:discrete_steric_Bq_nfs}
+\]
 
 whereas in the linear free surface,
@@ -1706,7 +1901,9 @@
 better approximate the total ocean mass and thus the steric sea level:
 
-\[	\eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
-                   { \sum_{i,\,j,\,k}       e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}      e_{1t}e_{2t} \eta }
- \label{eq:discrete_steric_Bq_fs} \]
+\[
+  \eta_s = - \frac{ \sum_{i,\,j,\,k} d_a\; e_{1t}e_{2t}e_{3t} + \sum_{i,\,j} d_a\; e_{1t}e_{2t} \eta }
+                  { \sum_{i,\,j,\,k}       e_{1t}e_{2t}e_{3t} + \sum_{i,\,j}       e_{1t}e_{2t} \eta }
+  % \label{eq:discrete_steric_Bq_fs}
+\]
 
 The fourth and last remark concerns the effective sea level and the presence of sea-ice.
@@ -1725,6 +1922,8 @@
 It is given by:
 
-\[	\eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv
- \label{eq:thermosteric_Bq} \]
+\[
+  \eta_s = - \frac{1}{\mathcal{A}} \int_D d_a(T,S_o,p_o) \,dv
+  % \label{eq:thermosteric_Bq}
+\]
 
 where $S_o$ and $p_o$ are the initial salinity and pressure, respectively.
@@ -1777,11 +1976,14 @@
   \begin{center}
     \includegraphics[width=1.0\textwidth]{Fig_mask_subasins}
-    \caption{	\protect\label{fig:mask_subasins}
+    \caption{
+      \protect\label{fig:mask_subasins}
 		Decomposition of the World Ocean (here ORCA2) into sub-basin used in to
       compute the heat and salt transports as well as the meridional stream-function:
       Atlantic basin (red), Pacific basin (green), Indian basin (bleue), Indo-Pacific basin (bleue+green).
 		Note that semi-enclosed seas (Red, Med and Baltic seas) as well as Hudson Bay are removed from the sub-basins.
-      Note also that the Arctic Ocean has been split into Atlantic and Pacific basins along the North fold line.}
-\end{center} \end{figure}  
+      Note also that the Arctic Ocean has been split into Atlantic and Pacific basins along the North fold line.
+    }
+  \end{center}
+\end{figure}  
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1835,6 +2037,8 @@
 The advective Courant numbers can be calculated according to
 
-\[	C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}}
- \label{eq:CFL} \]
+\[
+  C_u = |u|\frac{\rdt}{e_{1u}}, \quad C_v = |v|\frac{\rdt}{e_{2v}}, \quad C_w = |w|\frac{\rdt}{e_{3w}}
+  % \label{eq:CFL}
+\]
 
 in the zonal, meridional and vertical directions respectively.
@@ -1854,3 +2058,5 @@
 % ================================================================
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DIU.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -28,5 +29,5 @@
 \end{itemize}
 
-Models are provided for both the warm layer, \mdfl{diurnal_bulk}, and the cool skin, \mdl{cool_skin}.
+Models are provided for both the warm layer, \mdl{diurnal\_bulk}, and the cool skin, \mdl{cool\_skin}.
 Foundation SST is not considered as it can be obtained either from the main NEMO model
 ($i.e.$ from the temperature of the top few model levels) or from some other source.  
@@ -61,10 +62,10 @@
 The warm layer is calculated using the model of \citet{Takaya_al_JGR10} (TAKAYA10 model hereafter).
 This is a simple flux based model that is defined by the equations
-\begin{eqnarray}
+\begin{align}
 \frac{\partial{\Delta T_{\rm{wl}}}}{\partial{t}}&=&\frac{Q(\nu+1)}{D_T\rho_w c_p
 \nu}-\frac{(\nu+1)ku^*_{w}f(L_a)\Delta T}{D_T\Phi\!\left(\frac{D_T}{L}\right)} \mbox{,}
 \label{eq:ecmwf1} \\
 L&=&\frac{\rho_w c_p u^{*^3}_{w}}{\kappa g \alpha_w Q }\mbox{,}\label{eq:ecmwf2}
-\end{eqnarray}
+\end{align}
 where $\Delta T_{\rm{wl}}$ is the temperature difference between the top of the warm layer and the depth $D_T=3$\,m at which there is assumed to be no diurnal signal.
 In equation (\autoref{eq:ecmwf1}) $\alpha_w=2\times10^{-4}$ is the thermal expansion coefficient of water,
@@ -72,5 +73,5 @@
 $\rho_w$ is the water density, and $L$ is the Monin-Obukhov length.
 The tunable variable $\nu$ is a shape parameter that defines the expected subskin temperature profile via
-$T(z)=T(0)-\left(\frac{z}{D_T}\right)^\nu\DeltaT_{\rm{wl}}$,
+$T(z) = T(0) - \left( \frac{z}{D_T} \right)^\nu \Delta T_{\rm{wl}}$,
 where $T$ is the absolute temperature and $z\le D_T$ is the depth below the top of the warm layer.
 The influence of wind on TAKAYA10 comes through the magnitude of the friction velocity of the water $u^*_{w}$,
@@ -80,7 +81,8 @@
 The symbol $Q$ in equation (\autoref{eq:ecmwf1}) is the instantaneous total thermal energy flux into
 the diurnal layer, $i.e.$
-\begin{equation}
-Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,} \label{eq:e_flux_eqn}
-\end{equation}
+\[
+  Q = Q_{\rm{sol}} + Q_{\rm{lw}} + Q_{\rm{h}}\mbox{,}
+  % \label{eq:e_flux_eqn}
+\]
 where $Q_{\rm{h}}$ is the sensible and latent heat flux, $Q_{\rm{lw}}$ is the long wave flux,
 and $Q_{\rm{sol}}$ is the solar flux absorbed within the diurnal warm layer.
@@ -118,8 +120,8 @@
 The cool skin is modelled using the framework of \citet{Saunders_JAS82} who used a formulation of the near surface temperature difference based upon the heat flux and the friction velocity $u^*_{w}$.
 As the cool skin is so thin (~1\,mm) we ignore the solar flux component to the heat flux and the Saunders equation for the cool skin temperature difference $\Delta T_{\rm{cs}}$ becomes
-\begin{equation}
-\label{eq:sunders_eqn}
-\Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,}
-\end{equation}
+\[
+  % \label{eq:sunders_eqn}
+  \Delta T_{\rm{cs}}=\frac{Q_{\rm{ns}}\delta}{k_t} \mbox{,}
+\]
 where $Q_{\rm{ns}}$ is the, usually negative, non-solar heat flux into the ocean and
 $k_t$ is the thermal conductivity of sea water.
@@ -136,19 +138,21 @@
 both low and high wind speeds.
 Specifically,
-\begin{equation}
-\label{eq:artale_lambda_eqn}
-\lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
-\end{equation}
+\[
+  % \label{eq:artale_lambda_eqn}
+  \lambda = \frac{ 8.64\times10^4 u^*_{w} k_t }{ \rho c_p h \mu \gamma }\mbox{,}
+\]
 where $h=10$\,m is a reference depth and
 $\gamma$ is a dimensionless function of wind speed $u$:
-\begin{equation}
-\label{eq:artale_gamma_eqn}
-\gamma = \left\{ \begin{matrix}
-                     0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
-                     1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
-                     6\mbox{,} & \ge 10\,\mbox{ms}^{-1} \\
-                 \end{matrix}
-          \right.
-\end{equation}
+\[
+  % \label{eq:artale_gamma_eqn}
+  \gamma =
+  \begin{cases}
+    0.2u+0.5\mbox{,} & u \le 7.5\,\mbox{ms}^{-1} \\
+    1.6u-10\mbox{,} & 7.5 < u < 10\,\mbox{ms}^{-1} \\
+    6\mbox{,} & u \ge 10\,\mbox{ms}^{-1} \\
+  \end{cases}
+\]
+
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DOM.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DOM.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DOM.tex	(revision 10419)
@@ -1,9 +1,11 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter 2 ——— Space and Time Domain (DOM)
+% Chapter 2 Space and Time Domain (DOM)
 % ================================================================
 \chapter{Space Domain (DOM)}
 \label{chap:DOM}
+
 \minitoc
 
@@ -16,7 +18,5 @@
 %     - domclo:  closed sea and lakes.... management of closea sea area : specific to global configuration, both forced and coupled
 
-
 \newpage
-$\ $\newline    % force a new line
 
 Having defined the continuous equations in \autoref{chap:PE} and chosen a time discretization \autoref{chap:STP},
@@ -25,6 +25,4 @@
 and other information relevant to the main directory routines as well as the DOM (DOMain) directory.
 
-$\ $\newline    % force a new line
-
 % ================================================================
 % Fundamentals of the Discretisation
@@ -40,12 +38,17 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]    \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_cell}
-\caption{ \protect\label{fig:cell}
-  Arrangement of variables.
-  $t$ indicates scalar points where temperature, salinity, density, pressure and horizontal divergence are defined.
-  ($u$,$v$,$w$) indicates vector points,
-  and $f$ indicates vorticity points where both relative and planetary vorticities are defined}
-\end{center}   \end{figure}
+\begin{figure}[!tb]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_cell}
+    \caption{
+      \protect\label{fig:cell}
+      Arrangement of variables.
+      $t$ indicates scalar points where temperature, salinity, density, pressure and
+      horizontal divergence are defined.
+      ($u$,$v$,$w$) indicates vector points,
+      and $f$ indicates vorticity points where both relative and planetary vorticities are defined
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -83,21 +86,24 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{table}[!tb]
-\begin{center} \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
-\hline
-T	&$i$ 		& $j$		& $k$		 \\ \hline
-u	& $i+1/2$	& $j$		& $k$	 	\\ \hline
-v	& $i$	 	& $j+1/2$	& $k$	 	\\ \hline
-w	& $i$	 	& $j$		& $k+1/2$ 	\\ \hline
-f	& $i+1/2$ 	& $j+1/2$	& $k$	 	\\ \hline
-uw	& $i+1/2$ 	& $j$		& $k+1/2$ 	\\ \hline
-vw	& $i$	 	& $j+1/2$	& $k+1/2$ 	\\ \hline
-fw	& $i+1/2$ 	& $j+1/2$	& $k+1/2$ 	\\ \hline
-\end{tabular}
-\caption{ \protect\label{tab:cell}
-  Location of grid-points as a function of integer or integer and a half value of the column, line or level.
-  This indexing is only used for the writing of the semi-discrete equation.
-  In the code, the indexing uses integer values only and has a reverse direction in the vertical
-  (see \autoref{subsec:DOM_Num_Index})}
-\end{center}
+  \begin{center}
+    \begin{tabular}{|p{46pt}|p{56pt}|p{56pt}|p{56pt}|}
+      \hline
+      T	&$i$ 		& $j$		& $k$		 \\ \hline
+      u	& $i+1/2$	& $j$		& $k$	 	\\ \hline
+      v	& $i$	 	& $j+1/2$	& $k$	 	\\ \hline
+      w	& $i$	 	& $j$		& $k+1/2$ 	\\ \hline
+      f	& $i+1/2$ 	& $j+1/2$	& $k$	 	\\ \hline
+      uw	& $i+1/2$ 	& $j$		& $k+1/2$ 	\\ \hline
+      vw	& $i$	 	& $j+1/2$	& $k+1/2$ 	\\ \hline
+      fw	& $i+1/2$ 	& $j+1/2$	& $k+1/2$ 	\\ \hline
+    \end{tabular}
+    \caption{
+      \protect\label{tab:cell}
+      Location of grid-points as a function of integer or integer and a half value of the column, line or level.
+      This indexing is only used for the writing of the semi-discrete equation.
+      In the code, the indexing uses integer values only and has a reverse direction in the vertical
+      (see \autoref{subsec:DOM_Num_Index})
+    }
+  \end{center}
 \end{table}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -111,10 +117,11 @@
 Given the values of a variable $q$ at adjacent points,
 the differencing and averaging operators at the midpoint between them are:
-\begin{subequations} \label{eq:di_mi}
-\begin{align}
- \delta _i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)		\\
- \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
-\end{align}
-\end{subequations}
+\[
+  % \label{eq:di_mi}
+  \begin{split}
+    \delta_i [q]       &=  \  \    q(i+1/2)  - q(i-1/2)		\\
+    \overline q^{\,i} &= \left\{ q(i+1/2) + q(i-1/2) \right\} \; / \; 2
+  \end{split}
+\]
 
 Similar operators are defined with respect to $i+1/2$, $j$, $j+1/2$, $k$, and $k+1/2$.
@@ -123,36 +130,41 @@
 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 } [q] \;\,\mathbf{i}
-		+	\frac{1}{e_{2v} } \delta _{j+1/2 } [q] \;\,\mathbf{j}
-		+	\frac{1}{e_{3w}} \delta _{k+1/2} [q] \;\,\mathbf{k}
-\end{equation}
-\begin{multline} \label{eq:DOM_lap}
-\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}
+\[
+  % \label{eq:DOM_grad}
+  \nabla q\equiv 	\frac{1}{e_{1u} } \delta_{i+1/2 } [q] \;\,\mathbf{i}
+  +	\frac{1}{e_{2v} } \delta_{j+1/2 } [q] \;\,\mathbf{j}
+  +	\frac{1}{e_{3w}} \delta_{k+1/2} [q] \;\,\mathbf{k}
+\]
+\begin{multline*}
+  % \label{eq:DOM_lap}
+  \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 \autoref{eq:PE_curl} and \autoref{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)  &\ \mathbf{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)  &\ \mathbf{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)  &\ \mathbf{k}
- \end{eqnarray}
-\begin{eqnarray} \label{eq:DOM_div}
-\nabla \cdot \rm{\bf A} \equiv 
-    \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{eqnarray}
+\begin{align*}
+  % \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)  &\ \mathbf{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)  &\ \mathbf{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)  &\ \mathbf{k}
+\end{align*}
+\begin{align*}
+  % \label{eq:DOM_div}
+  \nabla \cdot \rm{\bf A} \equiv
+  \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{align*}
 
 The vertical average over the whole water column denoted by an overbar becomes for a quantity $q$ which
 is a masked field (i.e. equal to zero inside solid area):
-\begin{equation} \label{eq:DOM_bar}
-\bar q 	=         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk} 
-		\equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
+\begin{equation}
+  \label{eq:DOM_bar}
+  \bar q 	=         \frac{1}{H}    \int_{k^b}^{k^o} {q\;e_{3q} \,dk}
+  \equiv \frac{1}{H_q }\sum\limits_k {q\;e_{3q} }
 \end{equation}
 where $H_q$  is the ocean depth, which is the masked sum of the vertical scale factors at $q$ points,
@@ -162,9 +174,11 @@
 
 In continuous form, the following properties are satisfied:
-\begin{equation} \label{eq:DOM_curl_grad}
-\nabla \times \nabla q ={\rm {\bf {0}}}
+\begin{equation}
+  \label{eq:DOM_curl_grad}
+  \nabla \times \nabla q ={\rm {\bf {0}}}
 \end{equation}
-\begin{equation} \label{eq:DOM_div_curl}
-\nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
+\begin{equation}
+  \label{eq:DOM_div_curl}
+  \nabla \cdot \left( {\nabla \times {\rm {\bf A}}} \right)=0
 \end{equation}
 
@@ -179,11 +193,11 @@
 $\overline{\,\cdot\,}^{\,k}$) are symmetric linear operators,
 $i.e.$
-\begin{align} 
-\label{eq:DOM_di_adj}
-\sum\limits_i { a_i \;\delta _i \left[ b \right]} 
-	&\equiv -\sum\limits_i {\delta _{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
-\label{eq:DOM_mi_adj}
-\sum\limits_i { a_i \;\overline b^{\,i}} 
-	& \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} } 
+\begin{align}
+  \label{eq:DOM_di_adj}
+  \sum\limits_i { a_i \;\delta_i \left[ b \right]}
+  &\equiv -\sum\limits_i {\delta_{i+1/2} \left[ a \right]\;b_{i+1/2} }      \\
+  \label{eq:DOM_mi_adj}
+  \sum\limits_i { a_i \;\overline b^{\,i}}
+  & \equiv \quad \sum\limits_i {\overline a ^{\,i+1/2}\;b_{i+1/2} }
 \end{align}
 
@@ -200,10 +214,14 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]  \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_index_hor}
-\caption{   \protect\label{fig:index_hor}
-  Horizontal integer indexing used in the \textsc{Fortran} code.
-  The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices}
-\end{center}   \end{figure}
+\begin{figure}[!tb]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_index_hor}
+    \caption{
+      \protect\label{fig:index_hor}
+      Horizontal integer indexing used in the \textsc{Fortran} code.
+      The dashed area indicates the cell in which variables contained in arrays have the same $i$- and $j$-indices
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -249,11 +267,15 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!pt]    \begin{center}
-\includegraphics[width=.90\textwidth]{Fig_index_vert}
-\caption{ \protect\label{fig:index_vert}
-  Vertical integer indexing used in the \textsc{Fortran } code.
-  Note that the $k$-axis is orientated downward.
-  The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.}
-\end{center}   \end{figure}
+\begin{figure}[!pt]
+  \begin{center}
+    \includegraphics[width=.90\textwidth]{Fig_index_vert}
+    \caption{
+      \protect\label{fig:index_vert}
+      Vertical integer indexing used in the \textsc{Fortran } code.
+      Note that the $k$-axis is orientated downward.
+      The dashed area indicates the cell in which variables contained in arrays have the same $k$-index.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -272,7 +294,4 @@
 the code is run in parallel using domain decomposition (\key{mpp\_mpi} defined,
 see \autoref{sec:LBC_mpp}).
-
-
-$\ $\newline    % force a new line
 
 % ================================================================
@@ -351,18 +370,18 @@
 The model computes the grid-point positions and scale factors in the horizontal plane as follows:
 \begin{flalign*}
-\lambda_t &\equiv \text{glamt}= \lambda(i)	  & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
-\lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
-\lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
-\lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2) 
+  \lambda_t &\equiv \text{glamt}= \lambda(i)	  & \varphi_t &\equiv \text{gphit} = \varphi(j)\\
+  \lambda_u &\equiv \text{glamu}= \lambda(i+1/2)& \varphi_u &\equiv \text{gphiu}= \varphi(j)\\
+  \lambda_v &\equiv \text{glamv}= \lambda(i)       & \varphi_v &\equiv \text{gphiv} = \varphi(j+1/2)\\
+  \lambda_f &\equiv \text{glamf }= \lambda(i+1/2)& \varphi_f &\equiv \text{gphif }= \varphi(j+1/2)
 \end{flalign*}
 \begin{flalign*}
-e_{1t} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j)  |&
-e_{2t} &\equiv \text{e2t} = r_a |\varphi'(j)|  \\
-e_{1u} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)	\; \cos\varphi(j)  |&
-e_{2u} &\equiv \text{e2t} = r_a |\varphi'(j)|\\
-e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j+1/2)  |&
-e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
-e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
-e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
+  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)|\\
+  e_{1v} &\equiv \text{e1t} = r_a |\lambda'(i)		\; \cos\varphi(j+1/2)  |&
+  e_{2v} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|\\
+  e_{1f} &\equiv \text{e1t} = r_a |\lambda'(i+1/2)\; \cos\varphi(j+1/2)  |&
+  e_{2f} &\equiv \text{e2t} = r_a |\varphi'(j+1/2)|
 \end{flalign*}
 where the last letter of each computational name indicates the grid point considered and
@@ -385,14 +404,19 @@
 An example of the effect of such a choice is shown in \autoref{fig:zgr_e3}.
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]     \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_zgr_e3}
-\caption{ \protect\label{fig:zgr_e3}
-  Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
-  and (b) analytically derived grid-point position and scale factors.
-  For both grids here,
-  the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while
-  in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
-  Note the resulting difference between the value of the grid-size $\Delta_k$ and those of the scale factor $e_k$. }
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_zgr_e3}
+    \caption{
+      \protect\label{fig:zgr_e3}
+      Comparison of (a) traditional definitions of grid-point position and grid-size in the vertical,
+      and (b) analytically derived grid-point position and scale factors.
+      For both grids here,
+      the same $w$-point depth has been chosen but in (a) the $t$-points are set half way between $w$-points while
+      in (b) they are defined from an analytical function: $z(k)=5\,(k-1/2)^3 - 45\,(k-1/2)^2 + 140\,(k-1/2) - 150$.
+      Note the resulting difference between the value of the grid-size $\Delta_k$ and
+      those of the scale factor $e_k$.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -420,6 +444,4 @@
 the output grid written when \np{nn\_msh} $\not= 0$ is no more equal to the input grid.
 
-$\ $\newline    % force a new line
-
 % ================================================================
 % Domain: Vertical Grid (domzgr)
@@ -443,16 +465,20 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]    \begin{center}
-\includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps}
-\caption{  \protect\label{fig:z_zps_s_sps}
-  The ocean bottom as seen by the model:
-  (a) $z$-coordinate with full step,
-  (b) $z$-coordinate with partial step,
-  (c) $s$-coordinate: terrain following representation,
-  (d) hybrid $s-z$ coordinate,
-  (e) hybrid $s-z$ coordinate with partial step, and
-  (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}).
-  Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).}
-\end{center}   \end{figure}
+\begin{figure}[!tb]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_z_zps_s_sps}
+    \caption{
+      \protect\label{fig:z_zps_s_sps}
+      The ocean bottom as seen by the model:
+      (a) $z$-coordinate with full step,
+      (b) $z$-coordinate with partial step,
+      (c) $s$-coordinate: terrain following representation,
+      (d) hybrid $s-z$ coordinate,
+      (e) hybrid $s-z$ coordinate with partial step, and
+      (f) same as (e) but in the non-linear free surface (\protect\np{ln\_linssh}\forcode{ = .false.}).
+      Note that the non-linear free surface can be used with any of the 5 coordinates (a) to (e).
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -482,5 +508,6 @@
 \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}. 
+  so that the computation of the number of wet ocean point in each water column is by-passed
+}. 
 If \np{ln\_isfcav}\forcode{ = .true.},
 an extra file input file describing the ice shelf draft (in meters) (\ifile{isf\_draft\_meter}) is needed.
@@ -563,11 +590,15 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!tb]    \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_zgr}
-\caption{ \protect\label{fig:zgr}
-  Default vertical mesh for ORCA2: 30 ocean levels (L30).
-  Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from
-  \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.}
-\end{center}   \end{figure}
+\begin{figure}[!tb]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_zgr}
+    \caption{
+      \protect\label{fig:zgr}
+      Default vertical mesh for ORCA2: 30 ocean levels (L30).
+      Vertical level functions for (a) T-point depth and (b) the associated scale factor as computed from
+      \autoref{eq:DOM_zgr_ana_1} using \autoref{eq:DOM_zgr_coef} in $z$-coordinate.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -589,9 +620,10 @@
 For climate-related studies it is often desirable to concentrate the vertical resolution near the ocean surface.
 The following function is proposed as a standard for a $z$-coordinate (with either full or partial steps): 
-\begin{equation} \label{eq:DOM_zgr_ana_1}
-\begin{split}
- z_0 (k) 	&= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\ 
- e_3^0 (k) 	&= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right| 
-\end{split}
+\begin{equation}
+  \label{eq:DOM_zgr_ana_1}
+  \begin{split}
+    z_0 (k) 	&= h_{sur} -h_0 \;k-\;h_1 \;\log \left[ {\,\cosh \left( {{(k-h_{th} )} / {h_{cr} }} \right)\,} \right] \\
+    e_3^0 (k) 	&= \left| -h_0 -h_1 \;\tanh \left( {{(k-h_{th} )} / {h_{cr} }} \right) \right|
+  \end{split}
 \end{equation}
 where $k=1$ to \jp{jpk} for $w$-levels and $k=1$ to $k=1$ for $T-$levels.
@@ -601,9 +633,10 @@
 If the ice shelf cavities are opened (\np{ln\_isfcav}\forcode{ = .true.}), the definition of $z_0$ is the same.
 However, definition of $e_3^0$ at $t$- and $w$-points is respectively changed to:
-\begin{equation} \label{eq:DOM_zgr_ana_2}
-\begin{split}
- e_3^T(k) &= z_W (k+1) - z_W (k)   \\
- e_3^W(k) &= z_T (k)   - z_T (k-1) \\
-\end{split}
+\begin{equation}
+  \label{eq:DOM_zgr_ana_2}
+  \begin{split}
+    e_3^T(k) &= z_W (k+1) - z_W (k)   \\
+    e_3^W(k) &= z_T (k)   - z_T (k-1) \\
+  \end{split}
 \end{equation}
 This formulation decrease the self-generated circulation into the ice shelf cavity 
@@ -614,11 +647,12 @@
 a depth which varies from 0 at the sea surface to a minimum of $-5000~m$.
 This leads to the following conditions:
-\begin{equation} \label{eq:DOM_zgr_coef}
-\begin{split}
- e_3 (1+1/2)		&=10. \\ 
- e_3 (jpk-1/2)	&=500. \\ 
- z(1)			&=0. \\ 
- z(jpk)			&=-5000. \\ 
-\end{split}
+\begin{equation}
+  \label{eq:DOM_zgr_coef}
+  \begin{split}
+    e_3 (1+1/2)		&=10. \\
+    e_3 (jpk-1/2)	&=500. \\
+    z(1)			&=0. \\
+    z(jpk)			&=-5000. \\
+  \end{split}
 \end{equation}
 
@@ -654,42 +688,47 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{table}     \begin{center} \begin{tabular}{c||r|r|r|r}
-\hline
-\textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \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
-3	&	\textbf{25.00}	&	  20.00 &	\textbf{ 10.00} & 	 10.00 \\	\hline
-4	&	\textbf{35.01}	&	  30.00 & 	\textbf{ 10.01} & 	 10.00 \\	\hline
-5	&	\textbf{45.01}	&	  40.01 &	\textbf{ 10.01} &	 10.01 \\	\hline
-6	&	\textbf{55.03}	&	  50.02 &	\textbf{ 10.02} & 	 10.02 \\	\hline
-7	&	\textbf{65.06}	&	  60.04 &	\textbf{ 10.04} &	 10.03 \\	\hline
-8	&	\textbf{75.13}	&	  70.09 &	\textbf{ 10.09} &	 10.06 \\	\hline
-9	&	\textbf{85.25}	&	  80.18 &	\textbf{ 10.17} &	 10.12 \\	\hline
-10	& 	\textbf{95.49}	& 	  90.35 &	\textbf{ 10.33} &	 10.24 \\	\hline
-11	& 	\textbf{105.97}	& 	 100.69 &	\textbf{ 10.65} &	 10.47 \\	\hline
-12	& 	\textbf{116.90}	& 	 111.36 &	\textbf{ 11.27} &	 10.91 \\	\hline
-13	& 	\textbf{128.70}	& 	 122.65 &	\textbf{ 12.47} &	 11.77 \\	\hline
-14	& 	\textbf{142.20}	& 	 135.16 &	\textbf{ 14.78} &	 13.43 \\	\hline
-15	& 	\textbf{158.96}	& 	 150.03 &	\textbf{ 19.23} &	 16.65 \\	\hline
-16	& 	\textbf{181.96}	& 	 169.42 &	\textbf{ 27.66} &	 22.78 \\	\hline
-17	& 	\textbf{216.65}	& 	 197.37 & 	\textbf{ 43.26} &	 34.30 \\ \hline
-18	& 	\textbf{272.48}	& 	 241.13 & 	\textbf{ 70.88} &	 55.21 \\ \hline
-19	& 	\textbf{364.30}	& 	 312.74 & 	\textbf{116.11} &	 90.99 \\ \hline
-20	& 	\textbf{511.53}	& 	 429.72 & 	\textbf{181.55} & 	146.43 \\ \hline
-21	& 	\textbf{732.20}	& 	 611.89 & 	\textbf{261.03} & 	220.35 \\ \hline
-22	& 	\textbf{1033.22}&	 872.87 & 	\textbf{339.39} & 	301.42 \\ \hline
-23	& 	\textbf{1405.70}&	1211.59 & \textbf{402.26} & 	373.31 \\ \hline
-24	& 	\textbf{1830.89}&	1612.98 & \textbf{444.87} & 	426.00 \\ \hline
-25	& 	\textbf{2289.77}&	2057.13 & \textbf{470.55} & 	459.47 \\ \hline
-26	& 	\textbf{2768.24}&	2527.22 & \textbf{484.95} & 	478.83 \\ \hline
-27	& 	\textbf{3257.48}&	3011.90 & \textbf{492.70} & 	489.44 \\ \hline
-28	& 	\textbf{3752.44}&	3504.46 & \textbf{496.78} & 	495.07 \\ \hline
-29	& 	\textbf{4250.40}&	4001.16 & \textbf{498.90} & 	498.02 \\ \hline
-30	& 	\textbf{4749.91}&	4500.02 & \textbf{500.00} &	499.54 \\ \hline
-31	& 	\textbf{5250.23}&	5000.00 &	\textbf{500.56} &	500.33 \\ \hline
-\end{tabular} \end{center} 
-\caption{ \protect\label{tab:orca_zgr}
-  Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from
-  \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}}
+\begin{table}
+  \begin{center}
+    \begin{tabular}{c||r|r|r|r}
+      \hline
+      \textbf{LEVEL}& \textbf{gdept\_1d}& \textbf{gdepw\_1d}& \textbf{e3t\_1d }& \textbf{e3w\_1d  } \\ \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
+      3	&	\textbf{25.00}	&	  20.00 &	\textbf{ 10.00} & 	 10.00 \\	\hline
+      4	&	\textbf{35.01}	&	  30.00 & 	\textbf{ 10.01} & 	 10.00 \\	\hline
+      5	&	\textbf{45.01}	&	  40.01 &	\textbf{ 10.01} &	 10.01 \\	\hline
+      6	&	\textbf{55.03}	&	  50.02 &	\textbf{ 10.02} & 	 10.02 \\	\hline
+      7	&	\textbf{65.06}	&	  60.04 &	\textbf{ 10.04} &	 10.03 \\	\hline
+      8	&	\textbf{75.13}	&	  70.09 &	\textbf{ 10.09} &	 10.06 \\	\hline
+      9	&	\textbf{85.25}	&	  80.18 &	\textbf{ 10.17} &	 10.12 \\	\hline
+      10	& 	\textbf{95.49}	& 	  90.35 &	\textbf{ 10.33} &	 10.24 \\	\hline
+      11	& 	\textbf{105.97}	& 	 100.69 &	\textbf{ 10.65} &	 10.47 \\	\hline
+      12	& 	\textbf{116.90}	& 	 111.36 &	\textbf{ 11.27} &	 10.91 \\	\hline
+      13	& 	\textbf{128.70}	& 	 122.65 &	\textbf{ 12.47} &	 11.77 \\	\hline
+      14	& 	\textbf{142.20}	& 	 135.16 &	\textbf{ 14.78} &	 13.43 \\	\hline
+      15	& 	\textbf{158.96}	& 	 150.03 &	\textbf{ 19.23} &	 16.65 \\	\hline
+      16	& 	\textbf{181.96}	& 	 169.42 &	\textbf{ 27.66} &	 22.78 \\	\hline
+      17	& 	\textbf{216.65}	& 	 197.37 & 	\textbf{ 43.26} &	 34.30 \\ \hline
+      18	& 	\textbf{272.48}	& 	 241.13 & 	\textbf{ 70.88} &	 55.21 \\ \hline
+      19	& 	\textbf{364.30}	& 	 312.74 & 	\textbf{116.11} &	 90.99 \\ \hline
+      20	& 	\textbf{511.53}	& 	 429.72 & 	\textbf{181.55} & 	146.43 \\ \hline
+      21	& 	\textbf{732.20}	& 	 611.89 & 	\textbf{261.03} & 	220.35 \\ \hline
+      22	& 	\textbf{1033.22}&	 872.87 & 	\textbf{339.39} & 	301.42 \\ \hline
+      23	& 	\textbf{1405.70}&	1211.59 & \textbf{402.26} & 	373.31 \\ \hline
+      24	& 	\textbf{1830.89}&	1612.98 & \textbf{444.87} & 	426.00 \\ \hline
+      25	& 	\textbf{2289.77}&	2057.13 & \textbf{470.55} & 	459.47 \\ \hline
+      26	& 	\textbf{2768.24}&	2527.22 & \textbf{484.95} & 	478.83 \\ \hline
+      27	& 	\textbf{3257.48}&	3011.90 & \textbf{492.70} & 	489.44 \\ \hline
+      28	& 	\textbf{3752.44}&	3504.46 & \textbf{496.78} & 	495.07 \\ \hline
+      29	& 	\textbf{4250.40}&	4001.16 & \textbf{498.90} & 	498.02 \\ \hline
+      30	& 	\textbf{4749.91}&	4500.02 & \textbf{500.00} &	499.54 \\ \hline
+      31	& 	\textbf{5250.23}&	5000.00 &	\textbf{500.56} &	500.33 \\ \hline
+    \end{tabular}
+  \end{center}
+  \caption{
+    \protect\label{tab:orca_zgr}
+    Default vertical mesh in $z$-coordinate for 30 layers ORCA2 configuration as computed from
+    \autoref{eq:DOM_zgr_ana_2} using the coefficients given in \autoref{eq:DOM_zgr_coef}
+  }
 \end{table}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -739,10 +778,11 @@
 the product of a depth field and either a stretching function or its derivative, respectively:
 
-\begin{equation} \label{eq:DOM_sco_ana}
-\begin{split}
- z(k) 		&= h(i,j) \; z_0(k)	\\
- e_3(k)	&= h(i,j) \; z_0'(k)
-\end{split}
-\end{equation}
+\[
+  % \label{eq:DOM_sco_ana}
+  \begin{split}
+    z(k) 		&= h(i,j) \; z_0(k)	\\
+    e_3(k)	&= h(i,j) \; z_0'(k)
+  \end{split}
+\]
 
 where $h$ is the depth of the last $w$-level ($z_0(k)$) defined at the $t$-point location in the horizontal and
@@ -763,8 +803,8 @@
 This uses a depth independent $\tanh$ function for the stretching \citep{Madec_al_JPO96}:
 
-\begin{equation}
+\[
   z = s_{min}+C\left(s\right)\left(H-s_{min}\right)
-  \label{eq:SH94_1}
-\end{equation}
+  % \label{eq:SH94_1}
+\]
 
 where $s_{min}$ is the depth at which the $s$-coordinate stretching starts and
@@ -772,16 +812,17 @@
 and $z$ is the depth (negative down from the asea surface).
 
-\begin{equation}
+\[
   s = -\frac{k}{n-1} \quad \text{ and } \quad 0 \leq k \leq n-1
-  \label{eq:DOM_s}
-\end{equation}
-
-\begin{equation} \label{eq:DOM_sco_function}
-\begin{split}
-C(s)	&=  \frac{ \left[	  \tanh{ \left( \theta \, (s+b) \right)} 
-	  	 			- \tanh{ \left(  \theta \, b      \right)}  \right]}
-		      {2\;\sinh \left( \theta \right)}
-\end{split}
-\end{equation}
+  % \label{eq:DOM_s}
+\]
+
+\[
+  % \label{eq:DOM_sco_function}
+  \begin{split}
+    C(s)	&=  \frac{ \left[	  \tanh{ \left( \theta \, (s+b) \right)}
+        - \tanh{ \left(  \theta \, b      \right)}  \right]}
+    {2\;\sinh \left( \theta \right)}
+  \end{split}
+\]
 
 A stretching function,
@@ -789,17 +830,21 @@
 is also available and is more commonly used for shelf seas modelling:
 
-\begin{equation}
+\[
   C\left(s\right) =   \left(1 - b \right)\frac{ \sinh\left( \theta s\right)}{\sinh\left(\theta\right)} +      \\
   b\frac{ \tanh \left[ \theta \left(s + \frac{1}{2} \right)\right] - \tanh\left(\frac{\theta}{2}\right)}{ 2\tanh\left (\frac{\theta}{2}\right)}
-  \label{eq:SH94_2}
-\end{equation}
-
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]    \begin{center}
-\includegraphics[width=1.0\textwidth]{Fig_sco_function}
-\caption{  \protect\label{fig:sco_function}
-  Examples of the stretching function applied to a seamount;
-  from left to right: surface, surface and bottom, and bottom intensified resolutions}
-\end{center}   \end{figure}
+  % \label{eq:SH94_2}
+\]
+
+%>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_sco_function}
+    \caption{
+      \protect\label{fig:sco_function}
+      Examples of the stretching function applied to a seamount;
+      from left to right: surface, surface and bottom, and bottom intensified resolutions
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -815,19 +860,21 @@
 In this case the a stretching function $\gamma$ is defined such that:
 
-\begin{equation}
-z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
-\label{eq:z}
-\end{equation}
+\[
+  z = -\gamma h \quad \text{ with } \quad 0 \leq \gamma \leq 1
+  % \label{eq:z}
+\]
 
 The function is defined with respect to $\sigma$, the unstretched terrain-following coordinate:
 
-\begin{equation} \label{eq:DOM_gamma_deriv}
-\gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right)
-\end{equation}
+\[
+  % \label{eq:DOM_gamma_deriv}
+  \gamma= A\left(\sigma-\frac{1}{2}\left(\sigma^{2}+f\left(\sigma\right)\right)\right)+B\left(\sigma^{3}-f\left(\sigma\right)\right)+f\left(\sigma\right)
+\]
 
 Where:
-\begin{equation} \label{eq:DOM_gamma}
-f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1} 
-\end{equation}
+\[
+  % \label{eq:DOM_gamma}
+  f\left(\sigma\right)=\left(\alpha+2\right)\sigma^{\alpha+1}-\left(\alpha+1\right)\sigma^{\alpha+2} \quad \text{ and } \quad \sigma = \frac{k}{n-1}
+\]
 
 This gives an analytical stretching of $\sigma$ that is solvable in $A$ and $B$ as a function of
@@ -837,7 +884,8 @@
 The bottom cell depth in this example is given as a function of water depth:
 
-\begin{equation} \label{eq:DOM_zb}
-Z_b= h a + b
-\end{equation}
+\[
+  % \label{eq:DOM_zb}
+  Z_b = h a + b
+\]
 
 where the namelist parameters \np{rn\_zb\_a} and \np{rn\_zb\_b} are $a$ and $b$ respectively.
@@ -851,6 +899,7 @@
      the \citet{Siddorn_Furner_OM12} $S$-coordinate (dashed lines) in
      the surface 100m for a idealised bathymetry that goes from 50m to 5500m depth.
-     For clarity every third coordinate surface is shown.}
-    \label{fig:fig_compare_coordinates_surface}
+     For clarity every third coordinate surface is shown.
+   }
+   \label{fig:fig_compare_coordinates_surface}
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -919,12 +968,12 @@
 From the \textit{mbathy} and \textit{misfdep} array, the mask fields are defined as follows:
 \begin{align*}
-tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\
-                                \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\
-                                \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\
-umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
-vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)	\\
-fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
-             &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
-wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1) 
+  tmask(i,j,k) &= \begin{cases}   \; 0&   \text{ if $k < misfdep(i,j) $ } \\
+    \; 1&   \text{ if $misfdep(i,j) \leq k\leq mbathy(i,j)$  }    \\
+    \; 0&   \text{ if $k > mbathy(i,j)$  }    \end{cases}     \\
+  umask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+  vmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j+1,k)	\\
+  fmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i+1,j,k)	\\
+               &    \ \ \, * tmask(i,j,k) \ * \ tmask(i+1,j,k) \\
+  wmask(i,j,k) &=         \; tmask(i,j,k) \ * \ tmask(i,j,k-1) \text{ with } wmask(i,j,1) = tmask(i,j,1)
 \end{align*}
 
@@ -965,3 +1014,6 @@
   see \rou{istate\_t\_s} subroutine called from \mdl{istate} module.
 \end{description}
+
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DYN.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DYN.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_DYN.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,8 +7,6 @@
 \chapter{Ocean Dynamics (DYN)}
 \label{chap:DYN}
+
 \minitoc
-
-%\vspace{2.cm}
-$\ $\newline      %force an empty line
 
 Using the representation described in \autoref{chap:DOM},
@@ -19,9 +18,9 @@
 
 The prognostic ocean dynamics equation can be summarized as follows:
-\begin{equation*}
-\text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
-						{\text{COR} + \text{ADV}                       }
-			+ \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
-\end{equation*}
+\[
+  \text{NXT} = \dbinom	{\text{VOR} + \text{KEG} + \text {ZAD} }
+  {\text{COR} + \text{ADV}                       }
+  + \text{HPG} + \text{SPG} + \text{LDF} + \text{ZDF}
+\]
 NXT stands for next, referring to the time-stepping.
 The first group of terms on the rhs of this equation corresponds to the Coriolis and advection terms that
@@ -57,6 +56,4 @@
 MISC correspond to "extracting tendency terms" or "vorticity balance"?}
 
-$\ $\newline    % force a new ligne
-
 % ================================================================
 % Sea Surface Height evolution & Diagnostics variables
@@ -72,16 +69,18 @@
 
 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)
+\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} 
+\[
+  % \label{eq:divcur_div}
+  \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+  \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
+      +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
+\]
 
 Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
@@ -106,16 +105,17 @@
 
 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}
+\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 $\rho _w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
+and $\rho_w$=1,035~Kg/m$^3$ is the reference density of sea water (Boussinesq approximation).
 If river runoff is expressed as a surface freshwater flux (see \autoref{chap:SBC}) then
 \textit{emp} can be written as the evaporation minus precipitation, minus the river runoff. 
@@ -131,10 +131,13 @@
 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|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }  	\\
-&\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k  
-                                         - \frac{1} {2 \rdt} \left(  \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
-\end{aligned}   \right.
+\begin{equation}
+  \label{eq:wzv}
+  \left\{
+    \begin{aligned}
+      &\left. w \right|_{k_b-1/2} \quad= 0    \qquad \text{where } k_b \text{ is the level just above the sea floor }  	\\
+      &\left. w \right|_{k+1/2}     = \left. w \right|_{k-1/2}  +  \left. e_{3t} \right|_{k}\;  \left. \chi \right|_k
+      - \frac{1} {2 \rdt} \left(  \left. e_{3t}^{t+1}\right|_{k} - \left. e_{3t}^{t-1}\right|_{k}\right)
+    \end{aligned}
+  \right.
 \end{equation}
 
@@ -208,13 +211,14 @@
 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}    \\
-{- \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}  
-\end{aligned} 
- \right.
+\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}    \\
+      {- \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}
+    \end{aligned}
+  \right.
 \end{equation} 
 
@@ -227,11 +231,14 @@
 The kinetic energy conserving scheme (ENE scheme) conserves the global kinetic energy but not the global enstrophy.
 It is given by:
-\begin{equation} \label{eq:dynvor_ene}
-\left\{   \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} }    \\
-{- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
-                            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
-\end{aligned}    \right.
+\begin{equation}
+  \label{eq:dynvor_ene}
+  \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} }    \\
+      {- \frac{1}{e_{2v}}\; {\overline {\left( {\frac{\zeta +f}{e_{3f} }} \right)
+            \;  \overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} }
+    \end{aligned}
+  \right.
 \end{equation} 
 
@@ -245,16 +252,19 @@
 It consists of the ENS scheme (\autoref{eq:dynvor_ens}) for the relative vorticity term,
 and of the ENE scheme (\autoref{eq:dynvor_ene}) applied to the planetary vorticity term.
-\begin{equation} \label{eq:dynvor_mix}
-\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 {\left( {\frac{f}{e_{3f} }} \right) 
- \;\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 {\left( {\frac{f}{e_{3f} }} \right)
- \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
-\end{aligned}     } \right.
-\end{equation} 
+\[
+  % \label{eq:dynvor_mix}
+  \left\{ {
+      \begin{aligned}
+        {+\frac{1}{e_{1u} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^{\,i}
+          \; {\overline{\overline {\left( {e_{1v}\,e_{3v}\;v} \right)}} }^{\,i,j+1/2} -\frac{1}{e_{1u} }
+          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
+              \;\overline {\left( {e_{1v}\,e_{3v}\;v} \right)} ^{\,i+1/2}} }^{\,j} } \\
+        {-\frac{1}{e_{2v} }\; {\overline {\left( {\frac{\zeta }{e_{3f} }} \right)} }^j
+          \; {\overline{\overline {\left( {e_{2u}\,e_{3u}\;u} \right)}} }^{\,i+1/2,j} +\frac{1}{e_{2v} }
+          \; {\overline {\left( {\frac{f}{e_{3f} }} \right)
+              \;\overline {\left( {e_{2u}\,e_{3u}\;u} \right)} ^{\,j+1/2}} }^{\,i} } \hfill
+      \end{aligned}
+    } \right.
+\]
 
 %-------------------------------------------------------------
@@ -285,21 +295,27 @@
 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  = \frac{\zeta +f} {e_{3f} }
-\end{equation}
+\[
+  % \label{eq:pot_vor}
+  q  = \frac{\zeta +f} {e_{3f} }
+\]
 where the relative vorticity is defined by (\autoref{eq:divcur_cur}),
 the Coriolis parameter is given by $f=2 \,\Omega \;\sin \varphi _f $ and the layer thickness at $f$-points is: 
-\begin{equation} \label{eq:een_e3f}
-e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
+\begin{equation}
+  \label{eq:een_e3f}
+  e_{3f} = \overline{\overline {e_{3t} }} ^{\,i+1/2,j+1/2}
 \end{equation}
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]    \begin{center}
-\includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
-\caption{ \protect\label{fig:DYN_een_triad}
-  Triads used in the energy and enstrophy conserving scheme (een) for
-  $u$-component (upper panel) and $v$-component (lower panel).}
-\end{center}   \end{figure}
-%>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.70\textwidth]{Fig_DYN_een_triad}
+    \caption{
+      \protect\label{fig:DYN_een_triad}
+      Triads used in the energy and enstrophy conserving scheme (een) for
+      $u$-component (upper panel) and $v$-component (lower panel).
+    }
+  \end{center}
+\end{figure}
+% >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 A key point in \autoref{eq:een_e3f} is how the averaging in the \textbf{i}- and \textbf{j}- directions is made. 
@@ -316,20 +332,22 @@
 the following triad combinations of the neighbouring potential vorticities defined at f-points
 (\autoref{fig:DYN_een_triad}): 
-\begin{equation} \label{eq: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)
+\begin{equation}
+  \label{eq: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} }   \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.
+\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+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} 
 
@@ -353,10 +371,13 @@
 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]    
-\end{aligned} \right.
-\end{equation} 
+\[
+  % \label{eq:dynkeg}
+  \left\{
+    \begin{aligned}
+      -\frac{1}{2 \; e_{1u} }  & \ \delta_{i+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]   \\
+      -\frac{1}{2 \; e_{2v} }  & \ \delta_{j+1/2} \left[ {\overline {u^2}^{\,i} + \overline{v^2}^{\,j}} \right]
+    \end{aligned}
+  \right.
+\]
 
 %--------------------------------------------------------------------------------------------------------------
@@ -371,10 +392,13 @@
 Indeed, the change of KE due to the vertical advection is exactly balanced by
 the change of KE due to the gradient of KE (see \autoref{apdx:C}).
-\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.
-\end{equation} 
+\[
+  % \label{eq:dynzad}
+  \left\{
+    \begin{aligned}
+      -\frac{1} {e_{1u}\,e_{2u}\,e_{3u}} &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,i+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}  \\
+      -\frac{1} {e_{1v}\,e_{2v}\,e_{3v}}  &\ \overline{\ \overline{ e_{1t}\,e_{2t}\;w } ^{\,j+1/2}  \;\delta_{k+1/2} \left[ u \right]\  }^{\,k}
+    \end{aligned}
+  \right.
+\]
 When \np{ln\_dynzad\_zts}\forcode{ = .true.},
 a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
@@ -412,9 +436,10 @@
 This altered Coriolis parameter is thus discretised at $f$-points.
 It is given by: 
-\begin{multline} \label{eq:dyncor_metric}
-f+\frac{1}{e_1 e_2 }\left( {v\frac{\partial e_2 }{\partial i}  -  u\frac{\partial e_1 }{\partial j}} \right)  \\
-   \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} 
+\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)
+\end{multline*} 
 
 Any of the (\autoref{eq:dynvor_ens}), (\autoref{eq:dynvor_ene}) and (\autoref{eq:dynvor_een}) schemes can be used to
@@ -430,19 +455,20 @@
 
 The discrete expression of the advection term is given by:
-\begin{equation} \label{eq:dynadv}
-\left\{ 
-\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)   \\
-\\
-\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) \\
-\end{aligned}
-\right.
-\end{equation}
+\[
+  % \label{eq:dynadv}
+  \left\{
+    \begin{aligned}
+      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}}
+      \left(      \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i       }  \ u_t      \right]
+        + \delta_{j       } \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2}  \ u_f      \right] \right.  \ \;   \\
+      \left.   + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2}  \ u_{uw} \right] \right)   \\
+      \\
+      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
+      \left(     \delta_{i       } \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f       \right]
+        + \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i       } \ v_t       \right] \right.  \ \, \, \\
+      \left.  + \delta_{k      } \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw}  \right] \right) \\
+    \end{aligned}
+  \right.
+\]
 
 Two advection schemes are available:
@@ -462,9 +488,12 @@
 
 In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
-\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.
+\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.
 \end{equation} 
 
@@ -484,11 +513,13 @@
 an upstream-biased parabolic interpolation.
 For example, the evaluation of $u_T^{ubs} $ is done as follows:
-\begin{equation} \label{eq:dynadv_ubs}
-u_T^{ubs} =\overline u ^i-\;\frac{1}{6} 	\begin{cases}
-		u"_{i-1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ } 	\\
-		u"_{i+1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
-\end{cases}
+\begin{equation}
+  \label{eq:dynadv_ubs}
+  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
+  \begin{cases}
+    u"_{i-1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  \geqslant 0$ } 	\\
+    u"_{i+1/2}& 	\text{if $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
+  \end{cases}
 \end{equation}
-where $u"_{i+1/2} =\delta _{i+1/2} \left[ {\delta _i \left[ u \right]} \right]$.
+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{Shchepetkin_McWilliams_OM05}.
@@ -560,23 +591,29 @@
 
 for $k=km$ (surface layer, $jk=1$ in the code)
-\begin{equation} \label{eq:dynhpg_zco_surf}
-\left\{ \begin{aligned}
-					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k=km} 
-&= \frac{1}{2} g \ 	\left. \delta _{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
-     					\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k=km} 
-&= \frac{1}{2} g \ 	\left. \delta _{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
-\end{aligned} \right.
+\begin{equation}
+  \label{eq:dynhpg_zco_surf}
+  \left\{
+    \begin{aligned}
+      \left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k=km}
+      &= \frac{1}{2} g \ 	\left. \delta_{i+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
+      \left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k=km}
+      &= \frac{1}{2} g \ 	\left. \delta_{j+1/2} \left[  e_{3w} \ \rho \right] \right|_{k=km}   \\
+    \end{aligned}
+  \right.
 \end{equation} 
 
 for $1<k<km$ (interior layer)
-\begin{equation} \label{eq:dynhpg_zco}
-\left\{ \begin{aligned}
-					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k} 
-&=					\left. \delta _{i+1/2} \left[  p^h 			 \right] \right|_{k-1} 
-+    \frac{1}{2}\;g\;	\left. \delta _{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
-     					\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k} 
-&=     				\left. \delta _{j+1/2} \left[  p^h  			 \right] \right|_{k-1} 
-+    \frac{1}{2}\;g\;	\left. \delta _{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
-\end{aligned} \right.
+\begin{equation}
+  \label{eq:dynhpg_zco}
+  \left\{
+    \begin{aligned}
+      \left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k}
+      &=					\left. \delta_{i+1/2} \left[  p^h 			 \right] \right|_{k-1}
+      +    \frac{1}{2}\;g\;	\left. \delta_{i+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
+      \left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k}
+      &=     				\left. \delta_{j+1/2} \left[  p^h  			 \right] \right|_{k-1}
+      +    \frac{1}{2}\;g\;	\left. \delta_{j+1/2} \left[  e_{3w} \ \overline {\rho}^{k+1/2} \right] \right|_{k}   \\
+    \end{aligned}
+  \right.
 \end{equation} 
 
@@ -620,11 +657,14 @@
 
 $\bullet$ Traditional coding (see for example \citet{Madec_al_JPO96}: (\np{ln\_dynhpg\_sco}\forcode{ = .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{1}    					{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  p^h  \right]  
-+ \frac{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta _{j+1/2} \left[  z_t   \right]    \\
-\end{aligned} \right.
+\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{1}    					{\rho_o \, e_{2v}} \;	\delta_{j+1/2} \left[  p^h  \right]
+      + \frac{g\; \overline {\rho}^{j+1/2}}	{\rho_o \, e_{2v}} \;	\delta_{j+1/2} \left[  z_t   \right]    \\
+    \end{aligned}
+  \right.
 \end{equation} 
 
@@ -693,13 +733,15 @@
 $\bullet$ leapfrog scheme (\np{ln\_dynhpg\_imp}\forcode{ = .true.}):
 
-\begin{equation} \label{eq:dynhpg_lf}
-\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]
+\begin{equation}
+  \label{eq:dynhpg_lf}
+  \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}\forcode{ = .true.}):
-\begin{equation} \label{eq:dynhpg_imp}
-\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]
+\begin{equation}
+  \label{eq:dynhpg_imp}
+  \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}
 
@@ -720,9 +762,10 @@
 so that no additional storage array has to be defined.
 The density used to compute the hydrostatic pressure gradient (whatever the formulation) is evaluated as follows:
-\begin{equation} \label{eq:rho_flt}
-	\rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
- \quad	  \text{with}	\quad 
-	\widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
-\end{equation}
+\[
+  % \label{eq:rho_flt}
+  \rho^t = \rho( \widetilde{T},\widetilde {S},z_t)
+  \quad	  \text{with}	\quad
+  \widetilde{X} = 1 / 4 \left(  X^{t+\rdt} +2 \,X^t + X^{t-\rdt}  \right)
+\]
 
 Note that in the semi-implicit case, it is necessary to save the filtered density,
@@ -739,6 +782,4 @@
 \nlst{namdyn_spg} 
 %------------------------------------------------------------------------------------------------------------
-
-$\ $\newline      %force an empty line
 
 Options are defined through the \ngn{namdyn\_spg} namelist variables.
@@ -781,9 +822,12 @@
 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}\,\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.
+\begin{equation}
+  \label{eq:dynspg_exp}
+  \left\{
+    \begin{aligned}
+      - \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} 
 
@@ -817,15 +861,17 @@
 %%%
 The barotropic mode solves the following equations:
-\begin{subequations} \label{eq:BT}
-  \begin{equation}     \label{eq:BT_dyn}
-\frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
- -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h} 
--g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
-  \end{equation}
-
-  \begin{equation} \label{eq:BT_ssh}
-\frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
-  \end{equation}
-\end{subequations}
+% \begin{subequations}
+%  \label{eq:BT}
+\begin{equation}
+  \label{eq:BT_dyn}
+  \frac{\partial {\rm \overline{{\bf U}}_h} }{\partial t}=
+  -f\;{\rm {\bf k}}\times {\rm \overline{{\bf U}}_h}
+  -g\nabla _h \eta -\frac{c_b^{\textbf U}}{H+\eta} \rm {\overline{{\bf U}}_h} + \rm {\overline{\bf G}}
+\end{equation}
+\[
+  % \label{eq:BT_ssh}
+  \frac{\partial \eta }{\partial t}=-\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]+P-E
+\]
+% \end{subequations}
 where $\rm {\overline{\bf G}}$ is a forcing term held constant, containing coupling term between modes,
 surface atmospheric forcing as well as slowly varying barotropic terms not explicitly computed to gain efficiency.
@@ -839,23 +885,27 @@
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
-\begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
-\caption{  \protect\label{fig:DYN_dynspg_ts}
-  Schematic of the split-explicit time stepping scheme for the external and internal modes.
-  Time increases to the right. In this particular exemple,
-  a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
-  Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$.
-  Variables with $k$ superscript refer to instantaneous barotropic variables,
-  $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and
-  secondary weights (blue vertical bars).
-  The former are used to obtain time filtered quantities at $t+\rdt$ while
-  the latter are used to obtain time averaged transports to advect tracers.
-  a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.},
-  \protect\np{ln\_bt\_av}\forcode{ = .true.}.
-  b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.},
-  \protect\np{ln\_bt\_av}\forcode{ = .true.}.
-  c) Forward time integration with no time filtering (POM-like scheme):
-  \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}. }
-\end{center}    \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.7\textwidth]{Fig_DYN_dynspg_ts}
+    \caption{
+      \protect\label{fig:DYN_dynspg_ts}
+      Schematic of the split-explicit time stepping scheme for the external and internal modes.
+      Time increases to the right. In this particular exemple,
+      a boxcar averaging window over $nn\_baro$ barotropic time steps is used ($nn\_bt\_flt=1$) and $nn\_baro=5$.
+      Internal mode time steps (which are also the model time steps) are denoted by $t-\rdt$, $t$ and $t+\rdt$.
+      Variables with $k$ superscript refer to instantaneous barotropic variables,
+      $< >$ and $<< >>$ operator refer to time filtered variables using respectively primary (red vertical bars) and
+      secondary weights (blue vertical bars).
+      The former are used to obtain time filtered quantities at $t+\rdt$ while
+      the latter are used to obtain time averaged transports to advect tracers.
+      a) Forward time integration: \protect\np{ln\_bt\_fw}\forcode{ = .true.},
+      \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+      b) Centred time integration: \protect\np{ln\_bt\_fw}\forcode{ = .false.},
+      \protect\np{ln\_bt\_av}\forcode{ = .true.}.
+      c) Forward time integration with no time filtering (POM-like scheme):
+      \protect\np{ln\_bt\_fw}\forcode{ = .true.}, \protect\np{ln\_bt\_av}\forcode{ = .false.}.
+    }
+  \end{center}
+\end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
 
@@ -918,13 +968,15 @@
 We have 
 
-\begin{equation} \label{eq: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{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+  = 2 \rdt \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right]
+\]
+\begin{multline*}
+  % \label{eq: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*}
 \
 
@@ -938,39 +990,43 @@
 a single cycle.
 This is also the time that sets the barotropic time steps via 
-\begin{equation} \label{eq:DYN_spg_ts_t}
-t_n=\tau+n\rdt   
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_t}
+  t_n=\tau+n\rdt
+\]
 with $n$ an integer.
 The density scaled surface pressure is evaluated via 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_ps}
+  p_s^{(b)}(\tau,t_{n}) =
+  \begin{cases}
+    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
+    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
+  \end{cases}
+\]
 To get started, we assume the following initial conditions 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \begin{split}
+    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)}    \\
+    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
+  \end{split}
+\]
 with 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_etaF}
+  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\rdt,t_{n})
+\]
 the time averaged surface height taken from the previous barotropic cycle.
 Likewise, 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}	\\ \\
+  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \rdt \ \text{RHS}_{n=0}
+\]
 with 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\rdt,t_{n})
+\]
 the time averaged vertically integrated transport.
 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
@@ -979,13 +1035,14 @@
 the vertically integrated velocity is time averaged to produce the updated vertically integrated velocity at
 baroclinic time $\tau + \rdt \tau$ 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}(\tau+\rdt) = \overline{\textbf{U}^{(b)}(\tau+\rdt)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\]
 The surface height on the new baroclinic time step is then determined via a baroclinic leap-frog using
 the following form 
 
-\begin{equation} \label{eq:DYN_spg_ts_ssh}
-\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\rdt \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]  
+\begin{equation}
+  \label{eq: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}
 
@@ -1000,14 +1057,16 @@
 its stability and reasonably good maintenance of tracer conservation properties (see ??).
 
-\begin{equation} \label{eq:DYN_spg_ts_sshf}
-\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
+\begin{equation}
+  \label{eq:DYN_spg_ts_sshf}
+  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
 \end{equation}
 Another approach tried was 
 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_sshf2}
+  \eta^{F}(\tau-\Delta) = \eta(\tau)
+  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\rdt)
+    + \overline{\eta^{(b)}}(\tau-\rdt) -2 \;\eta(\tau) \right]
+\]
 
 which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
@@ -1034,13 +1093,14 @@
 %% gm %%======>>>>   given here the discrete eqs provided to the solver
 \gmcomment{               %%% copy from chap-model basics 
-\begin{equation} \label{eq:spg_flt}
-\frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
-- g \nabla \left( \tilde{\rho} \ \eta \right) 
-- g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right) 
-\end{equation}
-where $T_c$, is a parameter with dimensions of time which characterizes the force, 
-$\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
-and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient, 
-non-linear and viscous terms in \autoref{eq:PE_dyn}.
+  \[
+    % \label{eq:spg_flt}
+    \frac{\partial {\rm {\bf U}}_h }{\partial t}= {\rm {\bf M}}
+    - g \nabla \left( \tilde{\rho} \ \eta \right)
+    - g \ T_c \nabla \left( \widetilde{\rho} \ \partial_t \eta \right)
+  \]
+  where $T_c$, is a parameter with dimensions of time which characterizes the force,
+  $\widetilde{\rho} = \rho / \rho_o$ is the dimensionless density,
+  and $\rm {\bf M}$ represents the collected contributions of the Coriolis, hydrostatic pressure gradient,
+  non-linear and viscous terms in \autoref{eq:PE_dyn}.
 }   %end gmcomment
 
@@ -1091,14 +1151,16 @@
 
 For lateral iso-level diffusion, the discrete operator is: 
-\begin{equation} \label{eq:dynldf_lap}
-\left\{ \begin{aligned}
- D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta _{i+1/2} \left[ {A_T^{lm} 
-\;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta _j \left[ 
-{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
-\\
- D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta _{j+1/2} \left[ {A_T^{lm} 
-\;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta _i \left[ 
-{A_f^{lm} \;e_{3f} \zeta } \right] \\ 
-\end{aligned} \right.
+\begin{equation}
+  \label{eq:dynldf_lap}
+  \left\{
+    \begin{aligned}
+      D_u^{l{\rm {\bf U}}} =\frac{1}{e_{1u} }\delta_{i+1/2} \left[ {A_T^{lm}
+          \;\chi } \right]-\frac{1}{e_{2u} {\kern 1pt}e_{3u} }\delta_j \left[ 
+        {A_f^{lm} \;e_{3f} \zeta } \right] \\ \\
+      D_v^{l{\rm {\bf U}}} =\frac{1}{e_{2v} }\delta_{j+1/2} \left[ {A_T^{lm}
+          \;\chi } \right]+\frac{1}{e_{1v} {\kern 1pt}e_{3v} }\delta_i \left[ 
+        {A_f^{lm} \;e_{3f} \zeta } \right]
+    \end{aligned}
+  \right.
 \end{equation} 
 
@@ -1124,50 +1186,41 @@
 It must be emphasized that this formulation ignores constraints on the stress tensor such as symmetry.
 The resulting discrete representation is:
-\begin{equation} \label{eq:dyn_ldf_iso}
-\begin{split}
- 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}}
- \right)} \right]} 	\right.
-\\ 
-& \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f} 
-}\,\delta _{j+1/2} [u] - e_{1f}\, r_{2f} 
-\,\overline{\overline {\delta _{k+1/2} [u]}} ^{\,j+1/2,\,k}} 
-\right)} \right] 
-\\ 
-&\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline 
-{\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} } 
-\right.} \right. 
-\\ 
-&  \ \qquad \qquad \qquad \quad\ 
-- e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2}
-\\ 
-& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
-+\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2} 
-\right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} 
-\\
-\\
- D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} }    \\
-&  \left\{\quad  {\delta _{i+1/2} \left[ {A_f^{lm}  \left( 
-	 {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta _{i+1/2}[v]
-	-e_{2f} \; r_{1f} \,\overline{\overline {\delta _{k+1/2}[v]}}^{\,i+1/2,\,k}}
- \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} 
-\,\overline{\overline {\delta _{k+1/2} [v]}} ^{\,j,\,k}} 
-\right)} \right] 
-\\ 
-& \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
-{\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. 
-\\
-&  \ \qquad \qquad \qquad \quad\ 
-- e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2}
-\\ 
-& \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
-+\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2} 
-\right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
- \end{split}
+\begin{equation}
+  \label{eq:dyn_ldf_iso}
+  \begin{split}
+    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}}
+            \right)} \right]} 	\right. \\
+    & \qquad +\ \delta_j \left[ {A_f^{lm} \left( {\frac{e_{1f}\,e_{3f} }{e_{2f}
+            }\,\delta_{j+1/2} [u] - e_{1f}\, r_{2f}
+            \,\overline{\overline {\delta_{k+1/2} [u]}} ^{\,j+1/2,\,k}}
+        \right)} \right] \\
+    &\qquad +\ \delta_k \left[ {A_{uw}^{lm} \left( {-e_{2u} \, r_{1uw} \,\overline{\overline
+              {\delta_{i+1/2} [u]}}^{\,i+1/2,\,k+1/2} }
+        \right.} \right. \\
+    &  \ \qquad \qquad \qquad \quad\
+    - e_{1u} \, r_{2uw} \,\overline{\overline {\delta_{j+1/2} [u]}} ^{\,j,\,k+1/2} \\
+    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\
+                +\frac{e_{1u}\, e_{2u} }{e_{3uw} }\,\left( {r_{1uw}^2+r_{2uw}^2}
+                \right)\,\delta_{k+1/2} [u]} \right)} \right]\;\;\;} \right\} \\ \\
+    D_v^{l\textbf{V}} &= \frac{1}{e_{1v} \, e_{2v} \, e_{3v} } \\
+    &  \left\{\quad  {\delta_{i+1/2} \left[ {A_f^{lm}  \left(
+              {\frac{e_{2f} \; e_{3f} }{e_{1f} } \,\delta_{i+1/2}[v]
+                -e_{2f} \; r_{1f} \,\overline{\overline {\delta_{k+1/2}[v]}}^{\,i+1/2,\,k}}
+            \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}
+            \,\overline{\overline {\delta_{k+1/2} [v]}} ^{\,j,\,k}}
+        \right)} \right] \\
+    & \qquad +\ \delta_k \left[ {A_{vw}^{lm} \left( {-e_{2v} \, r_{1vw} \,\overline{\overline 
+              {\delta_{i+1/2} [v]}}^{\,i+1/2,\,k+1/2} }\right.} \right. \\
+    &  \ \qquad \qquad \qquad \quad\
+    - e_{1v} \, r_{2vw} \,\overline{\overline {\delta_{j+1/2} [v]}} ^{\,j+1/2,\,k+1/2} \\
+    & \left. {\left. { \ \qquad \qquad \qquad \ \ \ \left. {\ 
+                +\frac{e_{1v}\, e_{2v} }{e_{3vw} }\,\left( {r_{1vw}^2+r_{2vw}^2}
+                \right)\,\delta_{k+1/2} [v]} \right)} \right]\;\;\;} \right\} 
+  \end{split}
 \end{equation}
 where $r_1$ and $r_2$ are the slopes between the surface along which the diffusion operator acts and
@@ -1211,13 +1264,16 @@
 The formulation of the vertical subgrid scale physics is the same whatever the vertical coordinate is.
 The vertical diffusion operators given by \autoref{eq:PE_zdf} take the following semi-discrete space form:
-\begin{equation} \label{eq:dynzdf}
-\left\{   \begin{aligned}
-D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta _k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
-                              \ \delta _{k+1/2} [\,u\,]         \right]     \\
-\\
-D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta _k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
-                              \ \delta _{k+1/2} [\,v\,]         \right]
-\end{aligned}   \right.
-\end{equation} 
+\[
+  % \label{eq:dynzdf}
+  \left\{
+    \begin{aligned}
+      D_u^{vm} &\equiv \frac{1}{e_{3u}} \ \delta_k \left[ \frac{A_{uw}^{vm} }{e_{3uw} }
+        \ \delta_{k+1/2} [\,u\,]         \right]     \\
+      \\
+      D_v^{vm} &\equiv \frac{1}{e_{3v}} \ \delta_k \left[ \frac{A_{vw}^{vm} }{e_{3vw} }
+        \ \delta_{k+1/2} [\,v\,]         \right]
+    \end{aligned}
+  \right.
+\]
 where $A_{uw}^{vm} $ and $A_{vw}^{vm} $ are the vertical eddy viscosity and diffusivity coefficients.
 The way these coefficients are evaluated depends on the vertical physics used (see \autoref{chap:ZDF}).
@@ -1226,9 +1282,10 @@
 At the surface, the momentum fluxes are prescribed as the boundary condition on
 the vertical turbulent momentum fluxes,
-\begin{equation} \label{eq:dynzdf_sbc}
-\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
-	 = \frac{1}{\rho _o} \binom{\tau _u}{\tau _v }
+\begin{equation}
+  \label{eq:dynzdf_sbc}
+  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{z=1}
+  = \frac{1}{\rho_o} \binom{\tau_u}{\tau_v }
 \end{equation}
-where $\left( \tau _u ,\tau _v \right)$ are the two components of the wind stress vector in
+where $\left( \tau_u ,\tau_v \right)$ are the two components of the wind stress vector in
 the (\textbf{i},\textbf{j}) coordinate system.
 The high mixing coefficients in the surface mixed layer ensure that the surface wind stress is distributed in 
@@ -1286,20 +1343,26 @@
 $\bullet$ vector invariant form or linear free surface
 (\np{ln\_dynhpg\_vec}\forcode{ = .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} 
+\[
+  % \label{eq:dynnxt_vec}
+  \left\{
+    \begin{aligned}
+      &u^{t+\rdt} = u_f^{t-\rdt} + 2\rdt  \ \text{RHS}_u^t  	\\
+      &u_f^t \;\quad = u^t+\gamma \,\left[ {u_f^{t-\rdt} -2u^t+u^{t+\rdt}} \right]
+    \end{aligned}
+  \right.
+\]
 
 $\bullet$ flux form and nonlinear free surface
 (\np{ln\_dynhpg\_vec}\forcode{ = .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} 
+\[
+  % \label{eq:dynnxt_flux}
+  \left\{
+    \begin{aligned}
+      &\left(e_{3u}\,u\right)^{t+\rdt} = \left(e_{3u}\,u\right)_f^{t-\rdt} + 2\rdt \; e_{3u} \;\text{RHS}_u^t  	\\
+      &\left(e_{3u}\,u\right)_f^t \;\quad = \left(e_{3u}\,u\right)^t
+      +\gamma \,\left[ {\left(e_{3u}\,u\right)_f^{t-\rdt} -2\left(e_{3u}\,u\right)^t+\left(e_{3u}\,u\right)^{t+\rdt}} \right]
+    \end{aligned}
+  \right.
+\]
 where RHS is the right hand side of the momentum equation,
 the subscript $f$ denotes filtered values and $\gamma$ is the Asselin coefficient.
@@ -1314,3 +1377,5 @@
 
 % ================================================================
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LBC.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LBC.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LBC.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,9 +7,8 @@
 \chapter{Lateral Boundary Condition (LBC)}
 \label{chap:LBC}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
-
 
 %gm% add here introduction to this chapter
@@ -44,8 +44,9 @@
 Evaluating this quantity as,
 
-\begin{equation} \label{eq:lbc_aaaa}
-\frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT} 
-}{e_{1u} } \; \delta _{i+1 / 2} \left[ T \right]\;\;mask_u 
-\end{equation}
+\[
+  % \label{eq:lbc_aaaa}
+  \frac{A^{lT} }{e_1 }\frac{\partial T}{\partial i}\equiv \frac{A_u^{lT}
+  }{e_{1u} } \; \delta_{i+1 / 2} \left[ T \right]\;\;mask_u
+\]
 (where mask$_{u}$ is the mask array at a $u$-point) ensures that the heat flux is zero inside land and
 at the boundaries, since mask$_{u}$ is zero at solid boundaries which in this case are defined at $u$-points
@@ -53,10 +54,14 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]     \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_LBC_uv}
-\caption{  \protect\label{fig:LBC_uv}
-  Lateral boundary (thick line) at T-level.
-  The velocity normal to the boundary is set to zero.}
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_LBC_uv}
+    \caption{
+      \protect\label{fig:LBC_uv}
+      Lateral boundary (thick line) at T-level.
+      The velocity normal to the boundary is set to zero.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -78,14 +83,18 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!p] \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_LBC_shlat}
-\caption{     \protect\label{fig:LBC_shlat} 
-  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}
+\begin{figure}[!p]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_LBC_shlat}
+    \caption{
+      \protect\label{fig:LBC_shlat}
+      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}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -106,15 +115,16 @@
   Therefore, the vorticity along the coastlines is given by: 
 
-\begin{equation*}
-\zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
-\end{equation*}
-where $u$ and $v$ are masked fields.
-Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
-the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
-\begin{equation} \label{eq:lbc_bbbb}
-\zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta _{i+1/2} 
-\left[ {e_{2v} \,v} \right]-\delta _{j+1/2} \left[ {e_{1u} \,u} \right]} 
-\right)\;\mbox{mask}_f 
-\end{equation}
+  \[
+    \zeta \equiv 2 \left(\delta_{i+1/2} \left[e_{2v} v \right] - \delta_{j+1/2} \left[e_{1u} u \right] \right) / \left(e_{1f} e_{2f} \right) \ ,
+  \]
+  where $u$ and $v$ are masked fields.
+  Setting the mask$_{f}$ array to $2$ along the coastline provides a vorticity field computed with
+  the no-slip boundary condition, simply by multiplying it by the mask$_{f}$ :
+  \[
+    % \label{eq:lbc_bbbb}
+    \zeta \equiv \frac{1}{e_{1f} {\kern 1pt}e_{2f} }\left( {\delta_{i+1/2}
+        \left[ {e_{2v} \,v} \right]-\delta_{j+1/2} \left[ {e_{1u} \,u} \right]}
+    \right)\;\mbox{mask}_f
+  \]
 
 \item["partial" free-slip boundary condition (0$<$\np{rn\_shlat}$<$2):] the tangential velocity at
@@ -182,9 +192,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]     \begin{center}
-\includegraphics[width=1.0\textwidth]{Fig_LBC_jperio}
-\caption{    \protect\label{fig:LBC_jperio}
-  setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.}
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_LBC_jperio}
+    \caption{
+      \protect\label{fig:LBC_jperio}
+      setting of (a) east-west cyclic  (b) symmetric across the equator boundary conditions.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -202,11 +216,15 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_North_Fold_T}
-\caption{    \protect\label{fig:North_Fold_T}
-  North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
-  as used in ORCA 2, 1/4, and 1/12.
-  Pink shaded area corresponds to the inner domain mask (see text). }
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_North_Fold_T}
+    \caption{
+      \protect\label{fig:North_Fold_T}
+      North fold boundary with a $T$-point pivot and cyclic east-west boundary condition ($jperio=4$),
+      as used in ORCA 2, 1/4, and 1/12.
+      Pink shaded area corresponds to the inner domain mask (see text).
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -260,9 +278,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_mpp}
-\caption{   \protect\label{fig:mpp}
-  Positioning of a sub-domain when massively parallel processing is used. }
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_mpp}
+    \caption{
+      \protect\label{fig:mpp}
+      Positioning of a sub-domain when massively parallel processing is used.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -279,8 +301,8 @@
 The whole domain dimensions are named \np{jpiglo}, \np{jpjglo} and \jp{jpk}.
 The relationship between the whole domain and a sub-domain is:
-\begin{eqnarray} 
-      jpi & = & ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci  \nonumber \\
-      jpj & = & ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj  \label{eq:lbc_jpi}
-\end{eqnarray}
+\[
+  jpi = ( jpiglo-2*jpreci + (jpni-1) ) / jpni + 2*jpreci
+  jpj = ( jpjglo-2*jprecj + (jpnj-1) ) / jpnj + 2*jprecj
+\]
 where \jp{jpni}, \jp{jpnj} are the number of processors following the i- and j-axis.
 
@@ -289,7 +311,8 @@
 An element of $T_{l}$, a local array (subdomain) corresponds to an element of $T_{g}$,
 a global array (whole domain) by the relationship: 
-\begin{equation} \label{eq:lbc_nimpp}
-T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
-\end{equation}
+\[
+  % \label{eq:lbc_nimpp}
+  T_{g} (i+nimpp-1,j+njmpp-1,k) = T_{l} (i,j,k),
+\]
 with  $1 \leq i \leq jpi$, $1  \leq j \leq jpj $ , and  $1  \leq k \leq jpk$.
 
@@ -335,14 +358,18 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]     \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_mppini2}
-\caption {    \protect\label{fig:mppini2}
-  Example of Atlantic domain defined for the CLIPPER projet.
-  Initial grid is composed of 773 x 1236 horizontal points.
-  (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
-  52 subdomains are land areas.
-  (b) 52 subdomains are eliminated (white rectangles) and
-  the resulting number of processors really used during the computation is jpnij=128.}
-\end{center}   \end{figure}
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_mppini2}
+    \caption {
+      \protect\label{fig:mppini2}
+      Example of Atlantic domain defined for the CLIPPER projet.
+      Initial grid is composed of 773 x 1236 horizontal points.
+      (a) the domain is split onto 9 \time 20 subdomains (jpni=9, jpnj=20).
+      52 subdomains are land areas.
+      (b) 52 subdomains are eliminated (white rectangles) and
+      the resulting number of processors really used during the computation is jpnij=128.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -400,6 +427,4 @@
 The choice of algorithm is currently as follows:
 
-\mbox{}
-
 \begin{itemize}
 \item[0.] No boundary condition applied.
@@ -410,6 +435,4 @@
   ({\it dynspg\_ts}). 
 \end{itemize}
-
-\mbox{}
 
 The main choice for the boundary data is to use initial conditions as boundary data
@@ -445,7 +468,8 @@
 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}
+\[
+  % \label{eq:bdy_frs1}
+  \Phi(d) = \alpha(d)\Phi_{e}(d) + (1-\alpha(d))\Phi_{m}(d)\;\;\;\;\; d=1,N
+\]
 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
@@ -453,11 +477,13 @@
 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}
+\[
+  % \label{eq:bdy_frs2}
+  -\frac{1}{\tau}\left(\Phi - \Phi_{e}\right)
+\]
 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}
+\[
+  % \label{eq:bdy_frs3}
+  \tau = \frac{1-\alpha}{\alpha}  \,\rdt
+\]
 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.
@@ -466,7 +492,8 @@
 
 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}
+\[
+  % \label{eq:bdy_frs4}
+  \alpha(d) = 1 - \tanh\left(\frac{d-1}{2}\right),       \quad d=1,N
+\]
 The width of the FRS zone is specified in the namelist as \np{nn\_rimwidth}.
 This is typically set to a value between 8 and 10.
@@ -532,9 +559,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]      \begin{center}
-\includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom}
-\caption {      \protect\label{fig:LBC_bdy_geom}
-  Example of geometry of unstructured open boundary}
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_LBC_bdy_geom}
+    \caption {
+      \protect\label{fig:LBC_bdy_geom}
+      Example of geometry of unstructured open boundary
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -553,6 +584,4 @@
 (and therefore restrictions on the order of the data in the file).
 In particular:
-
-\mbox{}
 
 \begin{enumerate}
@@ -564,6 +593,4 @@
 \end{enumerate}
 
-\mbox{}
-
 These restrictions mean that data files used with previous versions of the model may not work with version 3.4.
 A fortran utility {\it bdy\_reorder} exists in the TOOLS directory which
@@ -571,9 +598,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]     \begin{center}
-\includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header}
-\caption {     \protect\label{fig:LBC_nc_header}
-  Example of the header for a \protect\ifile{coordinates.bdy} file}
-\end{center}   \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.0\textwidth]{Fig_LBC_nc_header}
+    \caption {
+      \protect\label{fig:LBC_nc_header}
+      Example of the header for a \protect\ifile{coordinates.bdy} file
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -608,6 +639,5 @@
  To be written....
 
-
-
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LDF.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LDF.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_LDF.tex	(revision 10419)
@@ -1,16 +1,15 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
 % ================================================================
-% Chapter ———  Lateral Ocean Physics (LDF)
+% Chapter Lateral Ocean Physics (LDF)
 % ================================================================
 \chapter{Lateral Ocean Physics (LDF)}
 \label{chap:LDF}
+
 \minitoc
 
-
 \newpage
-$\ $\newline    % force a new ligne
-
 
 The lateral physics terms in the momentum and tracer equations have been described in \autoref{eq:PE_zdf} and
@@ -43,6 +42,8 @@
 
 %%%
-\gmcomment{  we should emphasize here that the implementation is a rather old one. 
-Better work can be achieved by using \citet{Griffies_al_JPO98, Griffies_Bk04} iso-neutral scheme. }
+\gmcomment{
+  we should emphasize here that the implementation is a rather old one.
+  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 not act along the model levels.
@@ -68,21 +69,18 @@
 %gm { Steven : My version is obviously wrong since I'm left with an arbitrary constant which is the local vertical temperature gradient}
 
-\begin{equation} \label{eq:ldfslp_geo}
-\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] \ \ \
-\\
- 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}
- 		&\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}
-		&\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}] 
- \\
-\end{aligned}
+\begin{equation}
+  \label{eq:ldfslp_geo}
+  \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] \ \ \ \\
+    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}
+    &\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}
+    &\approx \frac{1}{e_{2w}}\; \delta_{j+1/2}[z_{vw}]
+  \end{aligned}
 \end{equation}
 
@@ -94,4 +92,5 @@
 \subsection{Slopes for tracer iso-neutral mixing}
 \label{subsec:LDF_slp_iso}
+
 In iso-neutral mixing  $r_1$ and $r_2$ are the slopes between the iso-neutral and computational surfaces.
 Their formulation does not depend on the vertical coordinate used.
@@ -101,21 +100,18 @@
 the three directions to zero leads to the following definition for the neutral slopes:
 
-\begin{equation} \label{eq:ldfslp_iso}
-\begin{split}
- r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
- 								{\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}}
-\\
- r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
- 								{\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}}
-\\
- r_{1w} &= \frac{e_{3w}}{e_{1w}}\; 
- 			\frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
-				 {\delta_{k+1/2}[\rho]}
-\\
- r_{2w} &= \frac{e_{3w}}{e_{2w}}\; 
- 			\frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
-				 {\delta_{k+1/2}[\rho]}
-\\
-\end{split}
+\begin{equation}
+  \label{eq:ldfslp_iso}
+  \begin{split}
+    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac{\delta_{i+1/2}[\rho]}
+    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,i+1/2,\,k}} \\
+    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac{\delta_{j+1/2}\left[\rho \right]}
+    {\overline{\overline{\delta_{k+1/2}[\rho]}}^{\,j+1/2,\,k}} \\
+    r_{1w} &= \frac{e_{3w}}{e_{1w}}\;
+    \frac{\overline{\overline{\delta_{i+1/2}[\rho]}}^{\,i,\,k+1/2}}
+    {\delta_{k+1/2}[\rho]} \\
+    r_{2w} &= \frac{e_{3w}}{e_{2w}}\;
+    \frac{\overline{\overline{\delta_{j+1/2}[\rho]}}^{\,j,\,k+1/2}}
+    {\delta_{k+1/2}[\rho]}
+  \end{split}
 \end{equation}
 
@@ -161,35 +157,32 @@
   locally referenced potential density, we stay in the $T$-$S$ plane and consider the balance between
   the neutral direction diffusive fluxes of potential temperature and salinity:
-\begin{equation}
-\alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
-\end{equation}
-%gm{  where vector F is ....}
+  \[
+    \alpha \ \textbf{F}(T) = \beta \ \textbf{F}(S)
+  \]
+  % gm{  where vector F is ....}
 
 This constraint leads to the following definition for the slopes:
 
-\begin{equation} \label{eq:ldfslp_iso2}
-\begin{split}
- r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
- 		{\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
-		{\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
-		 -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} }
-\\
- r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
- 		{\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
-		{\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
-		 -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }
-\\
- r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
-		{\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
-		 -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
- 		{\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
-\\
- r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
-		{\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
-		 -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
- 		{\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]}
-\\
-\end{split}
-\end{equation}
+\[
+  % \label{eq:ldfslp_iso2}
+  \begin{split}
+    r_{1u} &= \frac{e_{3u}}{e_{1u}}\; \frac
+    {\alpha_u \;\delta_{i+1/2}[T] - \beta_u \;\delta_{i+1/2}[S]}
+    {\alpha_u \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,i+1/2,\,k}
+      -\beta_u  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,i+1/2,\,k} } \\
+    r_{2v} &= \frac{e_{3v}}{e_{2v}}\; \frac
+    {\alpha_v \;\delta_{j+1/2}[T] - \beta_v \;\delta_{j+1/2}[S]}
+    {\alpha_v \;\overline{\overline{\delta_{k+1/2}[T]}}^{\,j+1/2,\,k}
+      -\beta_v  \;\overline{\overline{\delta_{k+1/2}[S]}}^{\,j+1/2,\,k} }    \\
+    r_{1w} &= \frac{e_{3w}}{e_{1w}}\; \frac
+    {\alpha_w \;\overline{\overline{\delta_{i+1/2}[T]}}^{\,i,\,k+1/2}
+      -\beta_w  \;\overline{\overline{\delta_{i+1/2}[S]}}^{\,i,\,k+1/2} }
+    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
+    r_{2w} &= \frac{e_{3w}}{e_{2w}}\; \frac
+    {\alpha_w \;\overline{\overline{\delta_{j+1/2}[T]}}^{\,j,\,k+1/2}
+      -\beta_w  \;\overline{\overline{\delta_{j+1/2}[S]}}^{\,j,\,k+1/2} }
+    {\alpha_w \;\delta_{k+1/2}[T] - \beta_w \;\delta_{k+1/2}[S]} \\
+  \end{split}
+\]
 where $\alpha$ and $\beta$, the thermal expansion and saline contraction coefficients introduced in
 \autoref{subsec:TRA_bn2}, have to be evaluated at the three velocity points.
@@ -222,9 +215,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!ht]      \begin{center}
-\includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1}
-\caption {    \protect\label{fig:LDF_ZDF1}
-  averaging procedure for isopycnal slope computation.}
-\end{center}    \end{figure}
+\begin{figure}[!ht]
+  \begin{center}
+    \includegraphics[width=0.70\textwidth]{Fig_LDF_ZDF1}
+    \caption {
+      \protect\label{fig:LDF_ZDF1}
+      averaging procedure for isopycnal slope computation.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -253,5 +250,6 @@
   \begin{center}
     \includegraphics[width=0.70\textwidth]{Fig_eiv_slp}
-    \caption {     \protect\label{fig:eiv_slp}
+    \caption{
+      \protect\label{fig:eiv_slp}
       Vertical profile of the slope used for lateral mixing in the mixed layer:
       \textit{(a)} in the real ocean the slope is the iso-neutral slope in the ocean interior,
@@ -265,5 +263,6 @@
       \textit{(c)} profile of slope actually used in \NEMO: a linear decrease of the slope from
       zero at the surface to its ocean interior value computed just below the mixed layer.
-      Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.}
+      Note the huge change in the slope at the base of the mixed layer between \textit{(b)} and \textit{(c)}.
+    }
   \end{center}
 \end{figure}
@@ -283,12 +282,13 @@
 $i.e.$ \autoref{eq:ldfslp_geo} and \autoref{eq:ldfslp_iso} :
 
-\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_{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}\\
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:ldfslp_dyn}
+  \begin{aligned}
+    &r_{1t}\ \ = \overline{r_{1u}}^{\,i}       &&&    r_{1f}\ \ &= \overline{r_{1u}}^{\,i+1/2} \\
+    &r_{2f} \ \ = \overline{r_{2v}}^{\,j+1/2} &&& 	r_{2t}\ &= \overline{r_{2v}}^{\,j} \\
+    &r_{1uw}  = \overline{r_{1w}}^{\,i+1/2} &&\ \ \text{and} \ \ &   r_{1vw}&= \overline{r_{1w}}^{\,j+1/2} \\
+    &r_{2uw}= \overline{r_{2w}}^{\,j+1/2} &&&         r_{2vw}&= \overline{r_{2w}}^{\,j+1/2}\\
+  \end{aligned}
+\]
 
 The major issue remaining is in the specification of the boundary conditions.
@@ -353,10 +353,12 @@
 By default the horizontal variation of the eddy coefficient depends on the local mesh size and
 the type of operator used:
-\begin{equation} \label{eq:title}
-  A_l = \left\{     
-   \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 } 
-   \end{aligned}    \right.
+\begin{equation}
+  \label{eq:title}
+  A_l = \left\{
+    \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 }
+    \end{aligned}
+  \right.
 \end{equation}
 where $e_{max}$ is the maximum of $e_1$ and $e_2$ taken over the whole masked ocean domain,
@@ -393,6 +395,4 @@
 This specification is actually used when an ORCA key and both \key{traldf\_eiv} and \key{traldf\_c2d} are defined.
 
-$\ $\newline    % force a new ligne
-
 The following points are relevant when the eddy coefficient varies spatially:
 
@@ -439,5 +439,6 @@
   GM diffusivity $A_e$ are directly set by \np{rn\_aeih\_0} and \np{rn\_aeiv\_0}.
   If 2D-varying coefficients are set with \key{traldf\_c2d} then $A_l$ is reduced in proportion with horizontal
-  scale factor according to \autoref{eq:title} \footnote{
+  scale factor according to \autoref{eq:title}
+  \footnote{
     Except in global ORCA  $0.5^{\circ}$ runs with \key{traldf\_eiv},
     where $A_l$ is set like $A_e$ but with a minimum vale of $100\;\mathrm{m}^2\;\mathrm{s}^{-1}$
@@ -445,5 +446,6 @@
   In idealised setups with \key{traldf\_c2d}, $A_e$ is reduced similarly, but if \key{traldf\_eiv} is set in
   the global configurations with \key{traldf\_c2d}, a horizontally varying $A_e$ is instead set from
-  the Held-Larichev parameterisation \footnote{
+  the Held-Larichev parameterisation
+  \footnote{
     In this case, $A_e$ at low latitudes $|\theta|<20^{\circ}$ is further reduced by a factor $|f/f_{20}|$,
     where $f_{20}$ is the value of $f$ at $20^{\circ}$~N
@@ -458,10 +460,11 @@
 and the sum \autoref{eq:ldfslp_geo} + \autoref{eq:ldfslp_iso} in $s$-coordinates.
 The eddy induced velocity is given by: 
-\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}
+\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}
 where $A^{eiv}$ is the eddy induced velocity coefficient whose value is set through \np{rn\_aeiv},
@@ -479,6 +482,5 @@
 and thus the advective eddy fluxes of heat and salt, are set to zero. 
 
-
-
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_OBS.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_OBS.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_OBS.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -11,7 +12,5 @@
 \minitoc
 
-
 \newpage
-$\ $\newline    % force a new line
 
 The observation and model comparison code (OBS) reads in observation files
@@ -573,4 +572,5 @@
 
 \subsubsection{Horizontal interpolation}
+
 Consider an observation point ${\rm P}$ with with longitude and latitude $({\lambda_{}}_{\rm P}, \phi_{\rm P})$ and
 the four nearest neighbouring model grid points ${\rm A}$, ${\rm B}$, ${\rm C}$ and ${\rm D}$ with
@@ -578,11 +578,11 @@
 All horizontal interpolation methods implemented in NEMO estimate the value of a model variable $x$ at point $P$ as
 a weighted linear combination of the values of the model variables at the grid points ${\rm A}$, ${\rm B}$ etc.:
-\begin{eqnarray}
-{x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & 
-\frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} + 
-                   {w_{}}_{\rm B} {x_{}}_{\rm B} + 
-                   {w_{}}_{\rm C} {x_{}}_{\rm C} + 
-                   {w_{}}_{\rm D} {x_{}}_{\rm D} \right)
-\end{eqnarray}
+\begin{align*}
+  {x_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} &
+                                                   \frac{1}{w} \left( {w_{}}_{\rm A} {x_{}}_{\rm A} +
+                                                   {w_{}}_{\rm B} {x_{}}_{\rm B} +
+                                                   {w_{}}_{\rm C} {x_{}}_{\rm C} +
+                                                   {w_{}}_{\rm D} {x_{}}_{\rm D} \right)
+\end{align*}
 where ${w_{}}_{\rm A}$, ${w_{}}_{\rm B}$ etc. are the respective weights for the model field at
 points ${\rm A}$, ${\rm B}$ etc., and $w = {w_{}}_{\rm A} + {w_{}}_{\rm B} + {w_{}}_{\rm C} + {w_{}}_{\rm D}$.
@@ -597,11 +597,10 @@
   For example, the weight given to the field ${x_{}}_{\rm A}$ is specified as the product of the distances
   from ${\rm P}$ to the other points:
-  \begin{eqnarray}
-  {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D})
-  \nonumber
-  \end{eqnarray}
+  \begin{align*}
+    {w_{}}_{\rm A} = s({\rm P}, {\rm B}) \, s({\rm P}, {\rm C}) \, s({\rm P}, {\rm D})
+  \end{align*}
   where 
-  \begin{eqnarray}
-   s\left ({\rm P}, {\rm M} \right ) 
+  \begin{align*}
+    s\left ({\rm P}, {\rm M} \right ) 
      & \hspace{-2mm} = \hspace{-2mm} & 
       \cos^{-1} \! \left\{ 
@@ -610,46 +609,35 @@
                \cos ({\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P}) 
                    \right\}
-   \end{eqnarray}
+   \end{align*}
    and $M$ corresponds to $B$, $C$ or $D$.
    A more stable form of the great-circle distance formula for small distances ($x$ near 1)
    involves the arcsine function ($e.g.$ see p.~101 of \citet{Daley_Barker_Bk01}:
-   \begin{eqnarray}
-   s\left( {\rm P}, {\rm M} \right) 
-     & \hspace{-2mm} = \hspace{-2mm} & 
-      \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\}
-   \nonumber
-   \end{eqnarray}
+   \begin{align*}
+     s\left( {\rm P}, {\rm M} \right) & \hspace{-2mm} = \hspace{-2mm} & \sin^{-1} \! \left\{ \sqrt{ 1 - x^2 } \right\}
+   \end{align*}
    where
-   \begin{eqnarray}
-    x & \hspace{-2mm} = \hspace{-2mm} & 
-      {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P}
-   \nonumber
-   \end{eqnarray}
+   \begin{align*}
+     x & \hspace{-2mm} = \hspace{-2mm} &
+                                         {a_{}}_{\rm M} {a_{}}_{\rm P} + {b_{}}_{\rm M} {b_{}}_{\rm P} + {c_{}}_{\rm M} {c_{}}_{\rm P}
+   \end{align*}
    and 
-   \begin{eqnarray}
-      {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M}, 
-      \nonumber \\
-      {a_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm P}, 
-      \nonumber \\
-      {b_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \cos {\phi_{}}_{\rm M}, 
-      \nonumber \\
-      {b_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm P}, 
-      \nonumber \\
-      {c_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \sin {\phi_{}}_{\rm M}, 
-      \nonumber \\
+   \begin{align*}
+      {a_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm M}, \\
+      {a_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \sin {\phi_{}}_{\rm P}, \\
+      {b_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \cos {\phi_{}}_{\rm M}, \\
+      {b_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \cos {\phi_{}}_{\rm P}, \\
+      {c_{}}_{\rm M} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm M} \sin {\phi_{}}_{\rm M}, \\
       {c_{}}_{\rm P} & \hspace{-2mm} = \hspace{-2mm} & \cos {\phi_{}}_{\rm P} \sin {\phi_{}}_{\rm P}.
-      \nonumber
-   \nonumber
-  \end{eqnarray}
+  \end{align*}
 
 \item[2.] {\bf Great-Circle distance-weighted interpolation with small angle approximation.}
   Similar to the previous interpolation but with the distance $s$ computed as
-  \begin{eqnarray}
-    s\left( {\rm P}, {\rm M} \right) 
-     & \hspace{-2mm} = \hspace{-2mm} & 
-      \sqrt{ \left( {\phi_{}}_{\rm M} - {\phi_{}}_{\rm P} \right)^{2} 
-      + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2}
-        \cos^{2} {\phi_{}}_{\rm M} }
-  \end{eqnarray}
+  \begin{align*}
+    s\left( {\rm P}, {\rm M} \right)
+    & \hspace{-2mm} = \hspace{-2mm} &
+                                      \sqrt{ \left( {\phi_{}}_{\rm M} - {\phi_{}}_{\rm P} \right)^{2}
+                                      + \left( {\lambda_{}}_{\rm M} - {\lambda_{}}_{\rm P} \right)^{2}
+                                      \cos^{2} {\phi_{}}_{\rm M} }
+  \end{align*}
   where $M$ corresponds to $A$, $B$, $C$ or $D$.
 
@@ -688,19 +676,27 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}      \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rec}
-\caption{      \protect\label{fig:obsavgrec}
-  Weights associated with each model grid box (blue lines and numbers)
-  for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg.}
-\end{center}      \end{figure}
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rec}
+    \caption{
+      \protect\label{fig:obsavgrec}
+      Weights associated with each model grid box (blue lines and numbers)
+      for an observation at -170.5E, 56.0N with a rectangular footprint of 1\deg x 1\deg.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}      \begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rad}
-\caption{      \protect\label{fig:obsavgrad}
-  Weights associated with each model grid box (blue lines and numbers)
-  for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg.} 
-\end{center}      \end{figure}
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_OBS_avg_rad}
+    \caption{
+      \protect\label{fig:obsavgrad}
+      Weights associated with each model grid box (blue lines and numbers)
+      for an observation at -170.5E, 56.0N with a radial footprint with diameter 1\deg.
+    } 
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -719,29 +715,29 @@
 denote the bottom left, bottom right, top left and top right corner points of the cell, respectively. 
 To determine if P is inside the cell, we verify that the cross-products 
-\begin{eqnarray}
-\begin{array}{lllll}
-{{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC}
-& = & [({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )
-    - ({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
-{{\bf r}_{}}_{\rm PB} \times {{\bf r}_{}}_{\rm PA}
-& = & [({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )
-    - ({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
-{{\bf r}_{}}_{\rm PC} \times {{\bf r}_{}}_{\rm PD}
-& = & [({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm D}   \; -\; {\phi_{}}_{\rm P} )
-    - ({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
-{{\bf r}_{}}_{\rm PD} \times {{\bf r}_{}}_{\rm PB}
-& = & [({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )
-    - ({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
-      ({\phi_{}}_{\rm D}  \;  - \; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
-\end{array}
-\label{eq:cross}
-\end{eqnarray}
+\begin{align*}
+  \begin{array}{lllll}
+    {{\bf r}_{}}_{\rm PA} \times {{\bf r}_{}}_{\rm PC}
+    & = & [({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )
+          - ({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+    {{\bf r}_{}}_{\rm PB} \times {{\bf r}_{}}_{\rm PA}
+    & = & [({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm A}   \; -\; {\phi_{}}_{\rm P} )
+          - ({\lambda_{}}_{\rm A}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+    {{\bf r}_{}}_{\rm PC} \times {{\bf r}_{}}_{\rm PD}
+    & = & [({\lambda_{}}_{\rm C}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm D}   \; -\; {\phi_{}}_{\rm P} )
+          - ({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm C}   \; -\; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+    {{\bf r}_{}}_{\rm PD} \times {{\bf r}_{}}_{\rm PB}
+    & = & [({\lambda_{}}_{\rm D}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm B}   \; -\; {\phi_{}}_{\rm P} )
+          - ({\lambda_{}}_{\rm B}\; -\; {\lambda_{}}_{\rm P} )
+          ({\phi_{}}_{\rm D}  \;  - \; {\phi_{}}_{\rm P} )] \; \widehat{\bf k} \\
+  \end{array}
+  % \label{eq:cross}
+\end{align*}
 point in the opposite direction to the unit normal $\widehat{\bf k}$
 (i.e., that the coefficients of $\widehat{\bf k}$ are negative),
@@ -774,9 +770,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}      \begin{center}
-\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_local}
-\caption{      \protect\label{fig:obslocal}
-  Example of the distribution of observations with the geographical distribution of observational data.} 
-\end{center}      \end{figure}
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_local}
+    \caption{
+      \protect\label{fig:obslocal}
+      Example of the distribution of observations with the geographical distribution of observational data.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -799,9 +799,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}     \begin{center}
-\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_global}
-\caption{      \protect\label{fig:obsglobal}
-  Example of the distribution of observations with the round-robin distribution of observational data.}
-\end{center}     \end{figure}
+\begin{figure}
+  \begin{center}
+    \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_ASM_obsdist_global}
+    \caption{
+      \protect\label{fig:obsglobal}
+      Example of the distribution of observations with the round-robin distribution of observational data.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1153,7 +1157,4 @@
 This technique has not been used before so experimentation is needed before results can be trusted.
 
-
-
-
 \newpage
 
@@ -1367,10 +1368,14 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}     \begin{center}
-%\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_main}
-\includegraphics[width=9cm,angle=-90.]{Fig_OBS_dataplot_main}
-\caption{      \protect\label{fig:obsdataplotmain}
-  Main window of dataplot.}
-\end{center}     \end{figure}
+\begin{figure}
+  \begin{center}
+    % \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_main}
+    \includegraphics[width=9cm,angle=-90.]{Fig_OBS_dataplot_main}
+    \caption{
+      \protect\label{fig:obsdataplotmain}
+      Main window of dataplot.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1379,14 +1384,17 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}     \begin{center}
-%\includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_prof}
-\includegraphics[width=7cm,angle=-90.]{Fig_OBS_dataplot_prof}
-\caption{      \protect\label{fig:obsdataplotprofile}
-  Profile plot from dataplot produced by right clicking on a point in the main window.}
-\end{center}     \end{figure}
+\begin{figure}
+  \begin{center}
+    % \includegraphics[width=10cm,height=12cm,angle=-90.]{Fig_OBS_dataplot_prof}
+    \includegraphics[width=7cm,angle=-90.]{Fig_OBS_dataplot_prof}
+    \caption{
+      \protect\label{fig:obsdataplotprofile}
+      Profile plot from dataplot produced by right clicking on a point in the main window.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
-
-
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_SBC.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -9,15 +10,14 @@
 
 \newpage
-$\ $\newline    % force a new ligne
+
 %---------------------------------------namsbc--------------------------------------------------
 
 \nlst{namsbc}
 %--------------------------------------------------------------------------------------------------------------
-$\ $\newline    % force a new ligne
 
 The ocean needs six fields as surface boundary condition:
 \begin{itemize}
 \item
-  the two components of the surface ocean stress $\left( {\tau _u \;,\;\tau _v} \right)$
+  the two components of the surface ocean stress $\left( {\tau_u \;,\;\tau_v} \right)$
 \item
   the incoming solar and non solar heat fluxes $\left( {Q_{ns} \;,\;Q_{sr} } \right)$
@@ -161,17 +161,22 @@
 
 %-------------------------------------------------TABLE---------------------------------------------------
-\begin{table}[tb]   \begin{center}   \begin{tabular}{|l|l|l|l|}
-\hline
-Variable description					& Model variable	& Units	& point \\	\hline
-i-component of the surface current	& ssu\_m	& $m.s^{-1}$	& U \\	\hline
-j-component of the surface current	& ssv\_m	& $m.s^{-1}$	& V \\	\hline
-Sea surface temperature				& sst\_m	& \r{}$K$		& T \\	\hline
-Sea surface salinty					& sss\_m	& $psu$			& T \\	\hline
-\end{tabular}
-\caption{  \protect\label{tab:ssm}
-  Ocean variables provided by the ocean to the surface module (SBC).
-  The variable are averaged over nn{\_}fsbc time step,
-  $i.e.$ the frequency of computation of surface fluxes.}
-\end{center}   \end{table}
+\begin{table}[tb]
+  \begin{center}
+    \begin{tabular}{|l|l|l|l|}
+      \hline
+      Variable description					& Model variable	& Units	& point \\	\hline
+      i-component of the surface current	& ssu\_m	& $m.s^{-1}$	& U \\	\hline
+      j-component of the surface current	& ssv\_m	& $m.s^{-1}$	& V \\	\hline
+      Sea surface temperature				& sst\_m	& \r{}$K$		& T \\	\hline
+      Sea surface salinty					& sss\_m	& $psu$			& T \\	\hline
+    \end{tabular}
+    \caption{
+      \protect\label{tab:ssm}
+      Ocean variables provided by the ocean to the surface module (SBC).
+      The variable are averaged over nn{\_}fsbc time step,
+      $i.e.$ the frequency of computation of surface fluxes.
+    }
+  \end{center}
+\end{table}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -239,23 +244,25 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp] 
-\begin{center}
-\begin{tabular}{|l|c|c|c|}
-\hline
-                         & daily or weekLLL	        & monthly                   &   yearly          \\   \hline
-\np{clim}\forcode{ = .false.}	& fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline
-\np{clim}\forcode{ = .true.}	 	   & not possible 	              &  fn\_m??.nc             &   fn                \\   \hline
-\end{tabular}
-\end{center}
-\caption{ \protect\label{tab:fldread}
-  naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency.
-  The stem name is assumed to be 'fn'.
-  For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week
-  ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon').
-  The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits.
-  Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc',
-  where 'PPPP' is the process number coded with 4 digits;
-  (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.}
-\end{table}
+  \begin{table}[htbp]
+    \begin{center}
+      \begin{tabular}{|l|c|c|c|}
+        \hline
+        & daily or weekLLL	        & monthly                   &   yearly          \\   \hline
+        \np{clim}\forcode{ = .false.}	& fn\_yYYYYmMMdDD.nc  &   fn\_yYYYYmMM.nc   &   fn\_yYYYY.nc  \\   \hline
+        \np{clim}\forcode{ = .true.}	 	   & not possible 	              &  fn\_m??.nc             &   fn                \\   \hline
+      \end{tabular}
+    \end{center}
+    \caption{
+      \protect\label{tab:fldread}
+      naming nomenclature for climatological or interannual input file, as a function of the Open/close frequency.
+      The stem name is assumed to be 'fn'.
+      For weekly files, the 'LLL' corresponds to the first three letters of the first day of the week
+      ($i.e.$ 'sun','sat','fri','thu','wed','tue','mon').
+      The 'YYYY', 'MM' and 'DD' should be replaced by the actual year/month/day, always coded with 4 or 2 digits.
+      Note that (1) in mpp, if the file is split over each subdomain, the suffix '.nc' is replaced by '\_PPPP.nc',
+      where 'PPPP' is the process number coded with 4 digits;
+      (2) when using AGRIF, the prefix '\_N' is added to files, where 'N'  is the child grid number.
+    }
+  \end{table}
 %--------------------------------------------------------------------------------------------------------------
   
@@ -378,7 +385,7 @@
 
 Symbolically, the algorithm used is:
-\begin{equation}
-f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
-\end{equation}
+\[
+  f_{m}(i,j) = f_{m}(i,j) + \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
+\]
 where function idx() transforms a one dimensional index src(k) into a two dimensional index,
 and wgt(1) corresponds to variable "wgt01" for example.
@@ -391,11 +398,12 @@
 The symbolic algorithm used to calculate values on the model grid is now:
 
-\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*}
+\[
+  \begin{split}
+    f_{m}(i,j) =  f_{m}(i,j) +& \sum_{k=1}^{4} {wgt(k)f(idx(src(k)))}
+    +   \sum_{k=5}^{8} {wgt(k)\left.\frac{\partial f}{\partial i}\right| _{idx(src(k))} }    \\
+    +& \sum_{k=9}^{12} {wgt(k)\left.\frac{\partial f}{\partial j}\right| _{idx(src(k))} }
+    +   \sum_{k=13}^{16} {wgt(k)\left.\frac{\partial ^2 f}{\partial i \partial j}\right| _{idx(src(k))} }
+  \end{split}
+\]
 The gradients here are taken with respect to the horizontal indices and not distances since
 the spatial dependency has been absorbed into the weights.
@@ -638,19 +646,20 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]   \label{tab:CORE}
-\begin{center}
-\begin{tabular}{|l|c|c|c|}
-\hline
-Variable desciption					& Model variable	& Units	 & point \\		\hline
-i-component of the 10m air velocity	& utau		& $m.s^{-1}$			& T  \\ 	\hline
-j-component of the 10m air velocity	& vtau		& $m.s^{-1}$			& T  \\	\hline
-10m air temperature					& tair		& \r{}$K$				& T 	\\	\hline
-Specific humidity					& humi		& \%					& T \\		\hline
-Incoming long wave radiation		& qlw		& $W.m^{-2}$			& T \\		\hline
-Incoming short wave radiation		& qsr		& $W.m^{-2}$			& T \\		\hline
-Total precipitation (liquid + solid)	& precip	& $Kg.m^{-2}.s^{-1}$	& T \\ 	\hline
-Solid precipitation					& snow		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
-\end{tabular}
-\end{center}
+\begin{table}[htbp]
+  \label{tab:CORE}
+  \begin{center}
+    \begin{tabular}{|l|c|c|c|}
+      \hline
+      Variable desciption					& Model variable	& Units	 & point \\		\hline
+      i-component of the 10m air velocity	& utau		& $m.s^{-1}$			& T  \\ 	\hline
+      j-component of the 10m air velocity	& vtau		& $m.s^{-1}$			& T  \\	\hline
+      10m air temperature					& tair		& \r{}$K$				& T 	\\	\hline
+      Specific humidity					& humi		& \%					& T \\		\hline
+      Incoming long wave radiation		& qlw		& $W.m^{-2}$			& T \\		\hline
+      Incoming short wave radiation		& qsr		& $W.m^{-2}$			& T \\		\hline
+      Total precipitation (liquid + solid)	& precip	& $Kg.m^{-2}.s^{-1}$	& T \\ 	\hline
+      Solid precipitation					& snow		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+    \end{tabular}
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -695,19 +704,20 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]   \label{tab:CLIO}
-\begin{center}
-\begin{tabular}{|l|l|l|l|}
-\hline
-Variable desciption				& Model variable	& Units				& point \\	\hline
-i-component of the ocean stress		& utau			& $N.m^{-2}$			& U \\	\hline
-j-component of the ocean stress		& vtau			& $N.m^{-2}$			& V \\	\hline
-Wind speed module					& vatm			& $m.s^{-1}$			& T \\	\hline
-10m air temperature					& tair			& \r{}$K$				& T \\	\hline
-Specific humidity						& humi			& \%					& T \\	\hline
-Cloud cover							& 				& \%					& T \\	\hline
-Total precipitation (liquid + solid)	& precip		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
-Solid precipitation 					& snow			& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
-\end{tabular}
-\end{center}
+\begin{table}[htbp]
+  \label{tab:CLIO}
+  \begin{center}
+    \begin{tabular}{|l|l|l|l|}
+      \hline
+      Variable desciption				& Model variable	& Units				& point \\	\hline
+      i-component of the ocean stress		& utau			& $N.m^{-2}$			& U \\	\hline
+      j-component of the ocean stress		& vtau			& $N.m^{-2}$			& V \\	\hline
+      Wind speed module					& vatm			& $m.s^{-1}$			& T \\	\hline
+      10m air temperature					& tair			& \r{}$K$				& T \\	\hline
+      Specific humidity						& humi			& \%					& T \\	\hline
+      Cloud cover							& 				& \%					& T \\	\hline
+      Total precipitation (liquid + solid)	& precip		& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+      Solid precipitation 					& snow			& $Kg.m^{-2}.s^{-1}$	& T \\	\hline
+    \end{tabular}
+  \end{center}
 \end{table}
 %--------------------------------------------------------------------------------------------------------------
@@ -773,7 +783,8 @@
 When used to force the dynamics, the atmospheric pressure is further transformed into
 an equivalent inverse barometer sea surface height, $\eta_{ib}$, using:
-\begin{equation} \label{eq:SBC_ssh_ib}
-	\eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right) 
-\end{equation}
+\[
+  % \label{eq:SBC_ssh_ib}
+  \eta_{ib} = -  \frac{1}{g\,\rho_o}  \left( P_{atm} - P_o \right)
+\]
 where $P_{atm}$ is the atmospheric pressure and $P_o$ a reference atmospheric pressure.
 A value of $101,000~N/m^2$ is used unless \np{ln\_ref\_apr} is set to true.
@@ -806,8 +817,9 @@
 is activated if \np{ln\_tide} and \np{ln\_tide\_pot} are both set to \np{.true.} in \ngn{nam\_tide}.
 This translates as an additional barotropic force in the momentum equations \ref{eq:PE_dyn} such that:
-\begin{equation}     \label{eq:PE_dyn_tides}
-\frac{\partial {\rm {\bf U}}_h }{\partial t}= ...
-+g\nabla (\Pi_{eq} + \Pi_{sal}) 
-\end{equation} 
+\[
+  % \label{eq:PE_dyn_tides}
+  \frac{\partial {\rm {\bf U}}_h }{\partial t}= ...
+  +g\nabla (\Pi_{eq} + \Pi_{sal})
+\]
 where $\Pi_{eq}$ stands for the equilibrium tidal forcing and $\Pi_{sal}$ a self-attraction and loading term (SAL). 
  
@@ -816,15 +828,15 @@
 For the three types of tidal frequencies it reads: \\
 Long period tides :
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(\frac{1}{2}-\frac{3}{2}sin^{2}\phi)cos(\omega_{l}t+V_{l})
+\]
 diurnal tides :
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(sin 2\phi)cos(\omega_{l}t+\lambda+V_{l})
+\]
 Semi-diurnal tides:
-\begin{equation}
-\Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l})
-\end{equation}
+\[
+  \Pi_{eq}(l)=A_{l}(1+k-h)(cos^{2}\phi)cos(\omega_{l}t+2\lambda+V_{l})
+\]
 Here $A_{l}$ is the amplitude, $\omega_{l}$ is the frequency, $\phi$ the latitude, $\lambda$ the longitude,
 $V_{0l}$ a phase shift with respect to Greenwich meridian and $t$ the time.
@@ -837,7 +849,7 @@
 (\np{ln\_read\_load=.true.}) or use a ``scalar approximation'' (\np{ln\_scal\_load=.true.}).
 In the latter case, it reads:\\
-\begin{equation}
-\Pi_{sal} = \beta \eta
-\end{equation}
+\[
+  \Pi_{sal} = \beta \eta
+\]
 where $\beta$ (\np{rn\_scal\_load}, $\approx0.09$) is a spatially constant scalar,
 often chosen to minimize tidal prediction errors.
@@ -1207,6 +1219,5 @@
 \label{subsec:SBC_wave_cdgw}
 
-The neutral surface drag coefficient provided from an external data source ($i.e.$ a wave
-model), 
+The neutral surface drag coefficient provided from an external data source ($i.e.$ a wave model), 
 can be used by setting the logical variable \np{ln\_cdgw} \forcode{= .true.} in \ngn{namsbc} namelist. 
 Then using the routine \rou{turb\_ncar} and starting from the neutral drag coefficent provided, 
@@ -1231,9 +1242,10 @@
 The Stokes drift velocity $\mathbf{U}_{st}$ in deep water can be computed from the wave spectrum and may be written as: 
 
-\begin{equation} \label{eq:sbc_wave_sdw}
-\mathbf{U}_{st} = \frac{16{\pi^3}} {g} 
-                \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3}
-                \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw}
+  \mathbf{U}_{st} = \frac{16{\pi^3}} {g}
+  \int_0^\infty \int_{-\pi}^{\pi} (cos{\theta},sin{\theta}) {f^3}
+  \mathrm{S}(f,\theta) \mathrm{e}^{2kz}\,\mathrm{d}\theta {d}f
+\]
 
 where: ${\theta}$ is the wave direction, $f$ is the wave intrinsic frequency, 
@@ -1255,15 +1267,17 @@
 \citet{Breivik_al_JPO2014}:
 
-\begin{equation} \label{eq:sbc_wave_sdw_0a}
-\mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_0a}
+  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \frac{\mathrm{e}^{-2k_ez}} {1-8k_ez} 
+\]
 
 where $k_e$ is the effective wave number which depends on the Stokes transport $T_{st}$ defined as follows:
 
-\begin{equation} \label{eq:sbc_wave_sdw_0b}
-k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|} 
-\quad \text{and }\
-T_{st} = \frac{1}{16} \bar{\omega} H_s^2 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_0b}
+  k_e = \frac{|\mathbf{U}_{\left.st\right|_{z=0}}|} {|T_{st}|}
+  \quad \text{and }\
+  T_{st} = \frac{1}{16} \bar{\omega} H_s^2 
+\]
 
 where $H_s$ is the significant wave height and $\omega$ is the wave frequency.
@@ -1273,8 +1287,9 @@
 \citep{Breivik_al_OM2016}:
 
-\begin{equation} \label{eq:sbc_wave_sdw_1}
-\mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz} 
-\textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big]
-\end{equation}
+\[
+  % \label{eq:sbc_wave_sdw_1}
+  \mathbf{U}_{st} \cong \mathbf{U}_{st |_{z=0}} \Big[exp(2k_pz)-\beta \sqrt{-2 \pi k_pz}
+  \textit{ erf } \Big(\sqrt{-2 k_pz}\Big)\Big]
+\]
 
 where $erf$ is the complementary error function and $k_p$ is the peak wavenumber.
@@ -1288,8 +1303,9 @@
 and its effect on the evolution of the sea-surface height ${\eta}$ is considered as follows: 
 
-\begin{equation} \label{eq:sbc_wave_eta_sdw}
-\frac{\partial{\eta}}{\partial{t}} = 
--\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz 
-\end{equation}
+\[
+  % \label{eq:sbc_wave_eta_sdw}
+  \frac{\partial{\eta}}{\partial{t}} =
+  -\nabla_h \int_{-H}^{\eta} (\mathbf{U} + \mathbf{U}_{st}) dz
+\]
 
 The tracer advection equation is also modified in order for Eulerian ocean models to properly account 
@@ -1299,8 +1315,9 @@
 can be formulated as follows: 
 
-\begin{equation} \label{eq:sbc_wave_tra_sdw}
-\frac{\partial{c}}{\partial{t}} = 
-- (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c}
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tra_sdw}
+  \frac{\partial{c}}{\partial{t}} =
+  - (\mathbf{U} + \mathbf{U}_{st}) \cdot \nabla{c}
+\]
 
 
@@ -1333,14 +1350,16 @@
 So the atmospheric stress felt by the ocean circulation $\tau_{oc,a}$ can be expressed as: 
 
-\begin{equation} \label{eq:sbc_wave_tauoc}
-\tau_{oc,a} = \tau_a - \tau_w
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tauoc}
+  \tau_{oc,a} = \tau_a - \tau_w
+\]
 
 where $\tau_a$ is the atmospheric surface stress;
 $\tau_w$ is the atmospheric stress going into the waves defined as:
 
-\begin{equation} \label{eq:sbc_wave_tauw}
-\tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})}
-\end{equation}
+\[
+  % \label{eq:sbc_wave_tauw}
+  \tau_w = \rho g \int {\frac{dk}{c_p} (S_{in}+S_{nl}+S_{diss})}
+\]
 
 where: $c_p$ is the phase speed of the gravity waves,
@@ -1375,13 +1394,17 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]    \begin{center}
-\includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal}
-\caption{ \protect\label{fig:SBC_diurnal}
-  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}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.8\textwidth]{Fig_SBC_diurnal}
+    \caption{
+      \protect\label{fig:SBC_diurnal}
+      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}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1409,11 +1432,15 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]  \begin{center}
-\includegraphics[width=0.7\textwidth]{Fig_SBC_dcy}
-\caption{ \protect\label{fig:SBC_dcy}
-  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}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.7\textwidth]{Fig_SBC_dcy}
+    \caption{
+      \protect\label{fig:SBC_dcy}
+      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}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -1455,7 +1482,8 @@
 On forced mode using a flux formulation (\np{ln\_flx}\forcode{ = .true.}),
 a feedback term \emph{must} be added to the surface heat flux $Q_{ns}^o$:
-\begin{equation} \label{eq:sbc_dmp_q}
-Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
-\end{equation}
+\[
+  % \label{eq:sbc_dmp_q}
+  Q_{ns} = Q_{ns}^o + \frac{dQ}{dT} \left( \left. T \right|_{k=1} - SST_{Obs} \right)
+\]
 where SST is a sea surface temperature field (observed or climatological),
 $T$ is the model surface layer temperature and
@@ -1467,7 +1495,8 @@
 Converted into an equivalent freshwater flux, it takes the following expression :
 
-\begin{equation} \label{eq:sbc_dmp_emp}
-\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}}
+\begin{equation}
+  \label{eq:sbc_dmp_emp}
+  \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}
 
@@ -1599,4 +1628,5 @@
 % in ocean-ice models. 
 
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_STO.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_STO.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_STO.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -13,5 +14,4 @@
 \newpage
 
-
 The stochastic parametrization module aims to explicitly simulate uncertainties in the model.
 More particularly, \cite{Brankart_OM2013} has shown that,
@@ -23,5 +23,5 @@
 The stochastic formulation of the equation of state can be written as:
 \begin{equation}
- \label{eq:eos_sto}
+  \label{eq:eos_sto}
   \rho = \frac{1}{2} \sum_{i=1}^m\{ \rho[T+\Delta T_i,S+\Delta S_i,p_o(z)] + \rho[T-\Delta T_i,S-\Delta S_i,p_o(z)] \}
 \end{equation}
@@ -30,6 +30,6 @@
 the scalar product of the respective local T/S gradients with random walks $\mathbf{\xi}$:
 \begin{equation}
- \label{eq:sto_pert}
- \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S
+  \label{eq:sto_pert}
+  \Delta T_i = \mathbf{\xi}_i \cdot \nabla T \qquad \hbox{and} \qquad \Delta S_i = \mathbf{\xi}_i \cdot \nabla S
 \end{equation}
 $\mathbf{\xi}_i$ are produced by a first-order autoregressive processes (AR-1) with
@@ -53,6 +53,6 @@
 
 \begin{equation}
-\label{eq:autoreg}
-\xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)}
+  \label{eq:autoreg}
+  \xi^{(i)}_{k+1} = a^{(i)} \xi^{(i)}_k + b^{(i)} w^{(i)} + c^{(i)}
 \end{equation}
 
@@ -67,16 +67,14 @@
   and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
 
-\begin{equation}
-\label{eq:ord1}
-\left\{
-\begin{array}{l}
-a^{(i)} = \varphi \\
-b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 } 
- \qquad\qquad\mbox{with}\qquad\qquad
-\varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
-c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
-\end{array}
-\right.
-\end{equation}
+  \[
+    % \label{eq:ord1}
+    \left\{
+      \begin{array}{l}
+        a^{(i)} = \varphi \\
+        b^{(i)} = \sigma^{(i)} \sqrt{ 1 - \varphi^2 }        \qquad\qquad\mbox{with}\qquad\qquad \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
+        c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
+      \end{array}
+    \right.
+  \]
 
 \item
@@ -86,16 +84,16 @@
   and the parameters $a^{(i)}$, $b^{(i)}$, $c^{(i)}$ are given by:
 
-\begin{equation}
-\label{eq:ord2}
-\left\{
-\begin{array}{l}
-a^{(i)} = \varphi \\
-b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 } 
- \qquad\qquad\mbox{with}\qquad\qquad
-\varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
-c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
-\end{array}
-\right.
-\end{equation}
+  \begin{equation}
+    \label{eq:ord2}
+    \left\{
+      \begin{array}{l}
+        a^{(i)} = \varphi \\
+        b^{(i)} = \frac{n-1}{2(4n-3)} \sqrt{ 1 - \varphi^2 }
+        \qquad\qquad\mbox{with}\qquad\qquad
+        \varphi = \exp \left( - 1 / \tau^{(i)} \right) \\
+        c^{(i)} = \mu^{(i)} \left( 1 - \varphi \right) \\
+      \end{array}
+    \right.
+  \end{equation}
 
 \end{itemize}
@@ -173,4 +171,5 @@
 $i.e.$ when the state of the random number generator is read in the restart file.
 
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_TRA.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_TRA.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_TRA.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,4 +7,5 @@
 \chapter{Ocean Tracers (TRA)}
 \label{chap:TRA}
+
 \minitoc
 
@@ -14,6 +16,4 @@
 
 %\newpage
-\vspace{2.cm}
-%$\ $\newline    % force a new ligne
 
 Using the representation described in \autoref{chap:DOM},
@@ -27,8 +27,8 @@
 The two active tracers are potential temperature and salinity.
 Their prognostic equations can be summarized as follows:
-\begin{equation*}
-\text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
-                   \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})
-\end{equation*}
+\[
+  \text{NXT} = \text{ADV}+\text{LDF}+\text{ZDF}+\text{SBC}
+  \ (+\text{QSR})\ (+\text{BBC})\ (+\text{BBL})\ (+\text{DMP})
+\]
 
 NXT stands for next, referring to the time-stepping.
@@ -59,5 +59,4 @@
 (\np{ln\_tra\_trd} or \np{ln\_tra\_mxl}\forcode{ = .true.}), as described in \autoref{chap:DIA}.
 
-$\ $\newline    % force a new ligne
 % ================================================================
 % Tracer Advection
@@ -74,9 +73,10 @@
 $i.e.$ as the divergence of the advective fluxes.
 Its discrete expression is given by :
-\begin{equation} \label{eq:tra_adv}
-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]
+\begin{equation}
+  \label{eq:tra_adv}
+  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.
@@ -95,5 +95,6 @@
   \begin{center}
     \includegraphics[width=0.9\textwidth]{Fig_adv_scheme}
-    \caption{	\protect\label{fig:adv_scheme}
+    \caption{
+      \protect\label{fig:adv_scheme}
       Schematic representation of some ways used to evaluate the tracer value at $u$-point and
       the amount of tracer exchanged between two neighbouring grid points.
@@ -125,5 +126,5 @@
   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
-  $\left. {\tau _w } \right|_{k=1/2} =T_{k=1} $,
+  $\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:]
@@ -193,6 +194,7 @@
 the two neighbouring $T$-point values.
 For example, in the $i$-direction :
-\begin{equation} \label{eq:tra_adv_cen2}
-\tau _u^{cen2} =\overline T ^{i+1/2}
+\begin{equation}
+  \label{eq:tra_adv_cen2}
+  \tau_u^{cen2} =\overline T ^{i+1/2}
 \end{equation}
 
@@ -212,7 +214,7 @@
 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}
-\tau _u^{cen4} 
-=\overline{   T - \frac{1}{6}\,\delta _i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
+\begin{equation}
+  \label{eq:tra_adv_cen4}
+  \tau_u^{cen4} =\overline{   T - \frac{1}{6}\,\delta_i \left[ \delta_{i+1/2}[T] \,\right]   }^{\,i+1/2}
 \end{equation}
 In the vertical direction (\np{nn\_cen\_v}\forcode{ = 4}),
@@ -238,5 +240,5 @@
 
 At a $T$-grid cell adjacent to a boundary (coastline, bottom and surface),
-an additional hypothesis must be made to evaluate $\tau _u^{cen4}$.
+an additional hypothesis must be made to evaluate $\tau_u^{cen4}$.
 This hypothesis usually reduces the order of the scheme.
 Here we choose to set the gradient of $T$ across the boundary to zero.
@@ -258,13 +260,15 @@
 a centred scheme.
 For example, in the $i$-direction :
-\begin{equation} \label{eq:tra_adv_fct}
-\begin{split}
-\tau _u^{ups}&= \begin{cases}
- 					T_{i+1} 	& \text{if $\ u_{i+1/2} <     0$} \hfill \\
- 					T_i   		& \text{if $\ u_{i+1/2} \geq 0$} \hfill \\
-				  \end{cases}     \\
-\\
-\tau _u^{fct}&=\tau _u^{ups} +c_u \;\left( {\tau _u^{cen} -\tau _u^{ups} } \right)
-\end{split}
+\begin{equation}
+  \label{eq:tra_adv_fct}
+  \begin{split}
+    \tau_u^{ups}&=
+    \begin{cases}
+      T_{i+1} 	& \text{if $\ u_{i+1/2} <     0$} \hfill \\
+      T_i   		& \text{if $\ u_{i+1/2} \geq 0$} \hfill \\
+    \end{cases}
+    \\ \\
+    \tau_u^{fct}&=\tau_u^{ups} +c_u \;\left( {\tau_u^{cen} -\tau_u^{ups} } \right)
+  \end{split}
 \end{equation}
 where $c_u$ is a flux limiter function taking values between 0 and 1.
@@ -287,6 +291,6 @@
 
 For stability reasons (see \autoref{chap:STP}),
-$\tau _u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
-$\tau _u^{ups}$ is evaluated using the \textit{before} tracer.
+$\tau_u^{cen}$ is evaluated in (\autoref{eq:tra_adv_fct}) using the \textit{now} tracer while
+$\tau_u^{ups}$ is evaluated using the \textit{before} tracer.
 In other words, the advective part of the scheme is time stepped with a leap-frog scheme
 while a forward scheme is used for the diffusive part. 
@@ -305,17 +309,20 @@
 two $T$-points (\autoref{fig:adv_scheme}).
 For example, in the $i$-direction :
-\begin{equation} \label{eq:tra_adv_mus}
-   \tau _u^{mus} = \left\{      \begin{aligned}
-         &\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} \;\rdt}{e_{1u} } \right)
-         &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
-   \end{aligned}    \right.
-\end{equation}
+\[
+  % \label{eq:tra_adv_mus}
+  \tau_u^{mus} = \left\{
+    \begin{aligned}
+      &\tau_i  &+ \frac{1}{2} \;\left( 1-\frac{u_{i+1/2} \;\rdt}{e_{1u}} \right)
+      &\ \widetilde{\partial _i \tau}  & \quad \text{if }\;u_{i+1/2} \geqslant 0      \\
+      &\tau_{i+1/2} &+\frac{1}{2}\;\left( 1+\frac{u_{i+1/2} \;\rdt}{e_{1u} } \right)
+      &\ \widetilde{\partial_{i+1/2} \tau } & \text{if }\;u_{i+1/2} <0
+    \end{aligned}
+  \right.
+\]
 where $\widetilde{\partial _i \tau}$ is the slope of the tracer on which a limitation is imposed to
 ensure the \textit{positive} character of the scheme.
 
 The time stepping is performed using a forward scheme,
-that is the \textit{before} tracer field is used to evaluate $\tau _u^{mus}$.
+that is the \textit{before} tracer field is used to evaluate $\tau_u^{mus}$.
 
 For an ocean grid point adjacent to land and where the ocean velocity is directed toward land,
@@ -338,12 +345,14 @@
 It is an upstream-biased third order scheme based on an upstream-biased parabolic interpolation.
 For example, in the $i$-direction:
-\begin{equation} \label{eq:tra_adv_ubs}
-   \tau _u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{      
-   \begin{aligned}
-         &\tau"_i        	& \quad \text{if }\ u_{i+1/2} \geqslant 0      \\
-         &\tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
-   \end{aligned}    \right.
+\begin{equation}
+  \label{eq:tra_adv_ubs}
+  \tau_u^{ubs} =\overline T ^{i+1/2}-\;\frac{1}{6} \left\{
+    \begin{aligned}
+      &\tau"_i        	& \quad \text{if }\ u_{i+1/2} \geqslant 0      \\
+      &\tau"_{i+1}	& \quad \text{if }\ u_{i+1/2}       <       0
+    \end{aligned}
+  \right.
 \end{equation}
-where $\tau "_i =\delta _i \left[ {\delta _{i+1/2} \left[ \tau \right]} \right]$.
+where $\tau "_i =\delta_i \left[ {\delta_{i+1/2} \left[ \tau \right]} \right]$.
 
 This results in a dissipatively dominant (i.e. hyper-diffusive) truncation error
@@ -373,17 +382,20 @@
 
 Note that it is straightforward to rewrite \autoref{eq:tra_adv_ubs} as follows:
-\begin{equation} \label{eq:traadv_ubs2}
-\tau _u^{ubs} = \tau _u^{cen4} + \frac{1}{12} \left\{	 
-   \begin{aligned}
-	& + \tau"_i			& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
-	&  - \tau"_{i+1}		& \quad \text{if }\ u_{i+1/2}       <       0
-   \end{aligned}    \right.
-\end{equation}
+\[
+  \label{eq:traadv_ubs2}
+  \tau_u^{ubs} = \tau_u^{cen4} + \frac{1}{12} \left\{
+    \begin{aligned}
+      & + \tau"_i			& \quad \text{if }\ u_{i+1/2} \geqslant 0 \\
+      &  - \tau"_{i+1}		& \quad \text{if }\ u_{i+1/2}       <       0
+    \end{aligned}
+  \right.
+\]
 or equivalently 
-\begin{equation} \label{eq:traadv_ubs2b}
-u_{i+1/2} \ \tau _u^{ubs} 
-=u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta _i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}
-- \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
-\end{equation}
+\[
+  % \label{eq:traadv_ubs2b}
+  u_{i+1/2} \ \tau_u^{ubs}
+  =u_{i+1/2} \ \overline{ T - \frac{1}{6}\,\delta_i\left[ \delta_{i+1/2}[T] \,\right] }^{\,i+1/2}
+  - \frac{1}{2} |u|_{i+1/2} \;\frac{1}{6} \;\delta_{i+1/2}[\tau"_i]
+\]
 
 \autoref{eq:traadv_ubs2} has several advantages.
@@ -519,8 +531,9 @@
 
 The laplacian diffusion operator acting along the model (\textit{i,j})-surfaces is given by: 
-\begin{equation} \label{eq:tra_ldf_lap}
-D_t^{lT} =\frac{1}{b_t} \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)
+\begin{equation}
+  \label{eq:tra_ldf_lap}
+  D_t^{lT} =\frac{1}{b_t} \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 and
@@ -554,25 +567,27 @@
 \subsubsection{Standard rotated (bi-)laplacian operator (\protect\mdl{traldf\_iso})}
 \label{subsec:TRA_ldf_iso}
+
 The general form of the second order lateral tracer subgrid scale physics (\autoref{eq:PE_zdf})
 takes the following semi-discrete space form in $z$- and $s$-coordinates:
-\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]
-	- e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k}
-                                                     \right)   \right]   \right.    \\ 
-&             +\delta_j \left[ A_v^{lT} \left( 
-          \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T] 
-        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k} 
-                                                    \right)   \right]                 \\ 
-& +\delta_k \left[ A_w^{lT} \left( 
-       -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2}
-                                                    \right.   \right.                 \\ 
-& \qquad \qquad \quad 
-        - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\
-& \left. {\left. {   \qquad \qquad \ \ \ \left. {
-        +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right)
-        \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\}
-\end{split}
+\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]
+          - e_{2u}\;r_{1u} \,\overline{\overline{ \delta_{k+1/2}[T] }}^{\,i+1/2,k}
+        \right)   \right]   \right.    \\
+    &             +\delta_j \left[ A_v^{lT} \left(
+        \frac{e_{1v}\,e_{3v}}{e_{2v}}  \,\delta_{j+1/2} [T]
+        - e_{1v}\,r_{2v} \,\overline{\overline{ \delta_{k+1/2} [T] }}^{\,j+1/2,k}
+      \right)   \right]                 \\
+    & +\delta_k \left[ A_w^{lT} \left(
+        -\;e_{2w}\,r_{1w} \,\overline{\overline{ \delta_{i+1/2} [T] }}^{\,i,k+1/2}
+      \right.   \right.                 \\
+    & \qquad \qquad \quad
+    - e_{1w}\,r_{2w} \,\overline{\overline{ \delta_{j+1/2} [T] }}^{\,j,k+1/2}     \\
+    & \left. {\left. {   \qquad \qquad \ \ \ \left. {
+                +\;\frac{e_{1w}\,e_{2w}}{e_{3w}} \,\left( r_{1w}^2 + r_{2w}^2 \right)
+                \,\delta_{k+1/2} [T] } \right) } \right] \quad } \right\}
+  \end{split}
 \end{equation}
 where $b_t$=$e_{1t}\,e_{2t}\,e_{3t}$  is the volume of $T$-cells,
@@ -652,11 +667,11 @@
 and is based on a laplacian operator.
 The vertical diffusion operator given by (\autoref{eq:PE_zdf}) takes the following semi-discrete space form:
-\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^{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}
+\[
+  % \label{eq:tra_zdf}
+  \begin{split}
+    D^{vT}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vT}_w}{e_{3w}}  \delta_{k+1/2}[T] \;\right]    \\
+    D^{vS}_T &= \frac{1}{e_{3t}} \; \delta_k \left[ \;\frac{A^{vS}_w}{e_{3w}}  \delta_{k+1/2}[S] \;\right]
+  \end{split}
+\]
 where $A_w^{vT}$ and $A_w^{vS}$ are the vertical eddy diffusivity coefficients on temperature and salinity,
 respectively.
@@ -726,9 +741,10 @@
 
 The surface boundary condition on temperature and salinity is applied as follows:
-\begin{equation} \label{eq:tra_sbc}
-\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{ \textit{sfx} }^t   & \\   
- \end{aligned}
+\begin{equation}
+  \label{eq:tra_sbc}
+  \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{ \textit{sfx} }^t   & \\
+  \end{aligned}
 \end{equation} 
 where $\overline{x }^t$ means that $x$ is averaged over two consecutive time steps ($t-\rdt/2$ and $t+\rdt/2$).
@@ -741,12 +757,13 @@
 the volume of the first level.
 The resulting surface boundary condition is applied as follows:
-\begin{equation} \label{eq:tra_sbc_lin}
-\begin{aligned}
- &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{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\   
- \end{aligned}
+\begin{equation}
+  \label{eq:tra_sbc_lin}
+  \begin{aligned}
+    &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{sfx} - \textit{emp} \;\left. S \right|_{k=1}  \right) }^t   & \\
+  \end{aligned}
 \end{equation} 
 Note that an exact conservation of heat and salt content is only achieved with non-linear free surface.
@@ -772,15 +789,17 @@
 the surface boundary condition is modified to take into account only the non-penetrative part of the surface 
 heat flux:
-\begin{equation} \label{eq:PE_qsr}
-\begin{split}
-\frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}	\\
-Q_{ns} &= Q_\text{Total} - Q_{sr}
-\end{split}
+\begin{equation}
+  \label{eq:PE_qsr}
+  \begin{split}
+    \frac{\partial T}{\partial t} &= {\ldots} + \frac{1}{\rho_o\, C_p \,e_3} \; \frac{\partial I}{\partial k}	\\
+    Q_{ns} &= Q_\text{Total} - Q_{sr}
+  \end{split}
 \end{equation}
 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 \autoref{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]
+\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}
 
@@ -798,7 +817,8 @@
 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_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
-\end{equation}
+\[
+  % \label{eq:traqsr_iradiance}
+  I(z) = Q_{sr} \left[Re^{-z / \xi_0} + \left( 1-R\right) e^{-z / \xi_1} \right]
+\]
 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.
@@ -857,5 +877,6 @@
   \begin{center}
     \includegraphics[width=1.0\textwidth]{Fig_TRA_Irradiance}
-    \caption{ 	 \protect\label{fig:traqsr_irradiance}
+    \caption{
+      \protect\label{fig:traqsr_irradiance}
       Penetration profile of the downward solar irradiance calculated by four models.
       Two waveband chlorophyll-independent formulation (blue),
@@ -883,5 +904,6 @@
   \begin{center}
     \includegraphics[width=1.0\textwidth]{Fig_TRA_geoth}
-    \caption{ 	\protect\label{fig:geothermal}
+    \caption{
+      \protect\label{fig:geothermal}
       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}.
@@ -947,17 +969,20 @@
 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} 
+\[
+  % \label{eq:tra_bbl_diff}
+  {\rm {\bf F}}_\sigma=A_l^\sigma \; \nabla_\sigma T
+\]
 with $\nabla_\sigma$ the lateral gradient operator taken between bottom cells,
 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}
- A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ 
- \\
- 0\quad \quad \;\,\mbox{otherwise} \\ 
- \end{array}} \right.
+\begin{equation}
+  \label{eq:tra_bbl_coef}
+  A_l^\sigma (i,j,t)=\left\{ {
+      \begin{array}{l}
+        A_{bbl}  \quad \quad   \mbox{if}  \quad   \nabla_\sigma \rho  \cdot  \nabla H<0 \\ \\
+        0\quad \quad \;\,\mbox{otherwise} \\
+      \end{array}}
+  \right.
 \end{equation} 
 where $A_{bbl}$ is the BBL diffusivity coefficient, given by the namelist parameter \np{rn\_ahtbbl} and
@@ -968,7 +993,8 @@
 In practice, this constraint is applied separately in the two horizontal directions,
 and the density gradient in \autoref{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} 
+\[
+  % \label{eq:tra_bbl_Drho}
+  \nabla_\sigma \rho / \rho = \alpha \,\nabla_\sigma T + \beta   \,\nabla_\sigma S
+\]
 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.
@@ -987,5 +1013,6 @@
   \begin{center}
     \includegraphics[width=0.7\textwidth]{Fig_BBL_adv}
-    \caption{ 	\protect\label{fig:bbl}
+    \caption{
+      \protect\label{fig:bbl}
       Advective/diffusive Bottom Boundary Layer.
       The BBL parameterisation is activated when $\rho^i_{kup}$ is larger than $\rho^{i+1}_{kdnw}$.
@@ -1024,7 +1051,8 @@
 For example, the resulting transport of the downslope flow, here in the $i$-direction (\autoref{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}
+\[
+  % \label{eq:bbl_Utr}
+  u^{tr}_{bbl} = \gamma \, g \frac{\Delta \rho}{\rho_o}  e_{1u} \; min \left( {e_{3u}}_{kup},{e_{3u}}_{kdwn} \right)
+\]
 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,
@@ -1043,15 +1071,15 @@
 the downslope flow \autoref{eq:bbl_dw}, the horizontal \autoref{eq:bbl_hor} and
 the upward \autoref{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}
+\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. 
@@ -1071,9 +1099,10 @@
 
 In some applications it can be useful to add a Newtonian damping term into the temperature and salinity equations:
-\begin{equation} \label{eq:tra_dmp}
-\begin{split}
- \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}
+\begin{equation}
+  \label{eq:tra_dmp}
+  \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}
 \end{equation} 
 where $\gamma$ is the inverse of a time scale, and $T_o$ and $S_o$ are given temperature and salinity fields
@@ -1173,12 +1202,12 @@
 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. \autoref{sec:STP_mLF}):
-\begin{equation} \label{eq:tra_nxt}
-\begin{aligned}
-(e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t &	\\
-\\
-(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}
+\begin{equation}
+  \label{eq:tra_nxt}
+  \begin{aligned}
+    (e_{3t}T)^{t+\rdt} &= (e_{3t}T)_f^{t-\rdt} &+ 2 \, \rdt  \,e_{3t}^t\ \text{RHS}^t &	\\ \\
+    (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 RHS is the right hand side of the temperature equation, the subscript $f$ denotes filtered values,
@@ -1288,5 +1317,6 @@
   as well as between \textit{absolute} and \textit{practical} salinity.
   S-EOS takes the following expression:
-  \begin{equation} \label{eq:tra_S-EOS}
+  \[
+    % \label{eq:tra_S-EOS}
     \begin{split}
       d_a(T,S,z)  =  ( & - a_0 \; ( 1 + 0.5 \; \lambda_1 \; T_a + \mu_1 \; z ) * T_a  \\
@@ -1295,5 +1325,5 @@
       with \ \  T_a = T-10  \; ;  & \;  S_a = S-35  \; ;\;  \rho_o = 1026~Kg/m^3
     \end{split}
-  \end{equation}
+  \]
   where the computer name of the coefficients as well as their standard value are given in \autoref{tab:SEOS}.
   In fact, when choosing S-EOS, various approximation of EOS can be specified simply by changing the associated coefficients.
@@ -1306,19 +1336,21 @@
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{table}[!tb]
-\begin{center} \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|}
-\hline
-coeff.	& computer name   & S-EOS		&  description		                  \\ \hline
-$a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff. 	\\ \hline
-$b_0$	      & \np{rn\_b0} 	   & 7.6554 $10^{-1}$ &  linear haline  expansion coeff. 	\\ \hline
-$\lambda_1$	& \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$ 	      \\ \hline
-$\lambda_2$	& \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$	 	   \\ \hline
-$\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$ 	   \\ \hline
-$\mu_1$     & \np{rn\_mu1} 	& 1.4970 $10^{-4}$ &  thermobaric coeff. in T    	   \\ \hline
-$\mu_2$     & \np{rn\_mu2} 	& 1.1090 $10^{-5}$ &  thermobaric coeff. in S   	      \\ \hline
-\end{tabular}
-\caption{ \protect\label{tab:SEOS}
-  Standard value of S-EOS coefficients.
-}
-\end{center}
+  \begin{center}
+    \begin{tabular}{|p{26pt}|p{72pt}|p{56pt}|p{136pt}|}
+      \hline
+      coeff.	& computer name   & S-EOS		&  description		                  \\ \hline
+      $a_0$       & \np{rn\_a0}     & 1.6550 $10^{-1}$ &  linear thermal expansion coeff. 	\\ \hline
+      $b_0$	      & \np{rn\_b0} 	   & 7.6554 $10^{-1}$ &  linear haline  expansion coeff. 	\\ \hline
+      $\lambda_1$	& \np{rn\_lambda1}& 5.9520 $10^{-2}$ &  cabbeling coeff. in $T^2$ 	      \\ \hline
+      $\lambda_2$	& \np{rn\_lambda2}& 5.4914 $10^{-4}$ &  cabbeling coeff. in $S^2$	 	   \\ \hline
+      $\nu$       & \np{rn\_nu}     & 2.4341 $10^{-3}$ &  cabbeling coeff. in $T \, S$ 	   \\ \hline
+      $\mu_1$     & \np{rn\_mu1} 	& 1.4970 $10^{-4}$ &  thermobaric coeff. in T    	   \\ \hline
+      $\mu_2$     & \np{rn\_mu2} 	& 1.1090 $10^{-5}$ &  thermobaric coeff. in S   	      \\ \hline
+    \end{tabular}
+    \caption{
+      \protect\label{tab:SEOS}
+      Standard value of S-EOS coefficients.
+    }
+  \end{center}
 \end{table}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -1338,7 +1370,8 @@
 (pressure in decibar being approximated by the depth in meters).
 The expression for $N^2$  is given by: 
-\begin{equation} \label{eq:tra_bn2}
-N^2 =	\frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
-\end{equation} 
+\[
+  % \label{eq:tra_bn2}
+  N^2 =	\frac{g}{e_{3w}} \left(   \beta \;\delta_{k+1/2}[S] - \alpha \;\delta_{k+1/2}[T]   \right)
+\]
 where $(T,S) = (\Theta, S_A)$ for TEOS10, $= (\theta, S_p)$ for TEOS-80, or $=(T,S)$ for S-EOS,
 and, $\alpha$ and $\beta$ are the thermal and haline expansion coefficients.
@@ -1354,10 +1387,10 @@
 
 The freezing point of seawater is a function of salinity and pressure \citep{UNESCO1983}:
-\begin{equation} \label{eq:tra_eos_fzp}
-   \begin{split}
-T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} 
-						     -  2.154996 \;10^{-4} \,S  \right) \ S    \\
-               - 7.53\,10^{-3} \ \ p 
-   \end{split}
+\begin{equation}
+  \label{eq:tra_eos_fzp}
+  \begin{split}
+    T_f (S,p) = \left( -0.0575 + 1.710523 \;10^{-3} \, \sqrt{S} -  2.154996 \;10^{-4} \,S  \right) \ S    \\
+    - 7.53\,10^{-3} \ \ p
+  \end{split}
 \end{equation}
 
@@ -1405,10 +1438,11 @@
   \begin{center}
     \includegraphics[width=0.9\textwidth]{Fig_partial_step_scheme}
-    \caption{ 	\protect\label{fig:Partial_step_scheme}
+    \caption{
+      \protect\label{fig:Partial_step_scheme}
       Discretisation of the horizontal difference and average of tracers in the $z$-partial step coordinate
       (\protect\np{ln\_zps}\forcode{ = .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 horizontal difference is then given by: $\delta_{i+1/2} T_k=  \widetilde{T}_k^{\,i+1} -T_k^{\,i}$ and
       the average by: $\overline{T}_k^{\,i+1/2}= ( \widetilde{T}_k^{\,i+1/2} - T_k^{\,i} ) / 2$.
     }
@@ -1416,28 +1450,30 @@
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{equation*}
-\widetilde{T}= \left\{  \begin{aligned}  
-&T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta _k T^{i+1}	
- 								&& \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   } 	\\
-										\\
-&T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta _k T^{i+1}
-			 					&& \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
-            \end{aligned}   \right.
-\end{equation*}
+\[
+  \widetilde{T}= \left\{
+    \begin{aligned}
+      &T^{\,i+1}      -\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right)}{ e_{3w}^{i+1} }\;\delta_k T^{i+1}
+      && \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$   } 	\\ \\
+      &T^{\,i} \ \ \ \,+\frac{ \left( e_{3w}^{i+1} -e_{3w}^i \right) }{e_{3w}^i       }\;\delta_k T^{i+1}
+      && \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }
+    \end{aligned}
+  \right.
+\]
 and the resulting forms for the horizontal difference and the horizontal average value of $T$ at a $U$-point are: 
-\begin{equation} \label{eq:zps_hde}
-\begin{aligned}
- \delta _{i+1/2} T= 	\begin{cases}
-\ \ \ \widetilde {T}\quad\ -T^i	 	& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
-										\\
-\ \ \ T^{\,i+1}-\widetilde{T}		& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
-            		\end{cases}     \\
-\\
-\overline {T}^{\,i+1/2} \ = 	\begin{cases}
-( \widetilde {T}\ \ \;\,-T^{\,i})	 / 2	& \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\
-										\\
-( T^{\,i+1}-\widetilde{T} ) / 2		& \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   } 
-            \end{cases}
-\end{aligned}
+\begin{equation}
+  \label{eq:zps_hde}
+  \begin{aligned}
+    \delta_{i+1/2} T=
+    \begin{cases}
+      \ \ \ \widetilde {T}\quad\ -T^i	 	& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\
+      \ \ \ T^{\,i+1}-\widetilde{T}		& \ \ \quad\quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }
+    \end{cases}
+    \\ \\
+    \overline {T}^{\,i+1/2} \ =
+    \begin{cases}
+      ( \widetilde {T}\ \ \;\,-T^{\,i})	 / 2	& \;\ \ \quad\text{if  $\ e_{3w}^{i+1} \geq e_{3w}^i$ } \\ \\
+      ( T^{\,i+1}-\widetilde{T} ) / 2		& \;\ \ \quad\text{if  $\ e_{3w}^{i+1}    <   e_{3w}^i$   }
+    \end{cases}
+  \end{aligned}
 \end{equation}
 
@@ -1449,8 +1485,9 @@
 $T$ and $S$, and the pressure at a $u$-point
 (in the equation of state pressure is approximated by depth, see \autoref{subsec:TRA_eos} ): 
-\begin{equation} \label{eq:zps_hde_rho}
-\widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u }) 
-\quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right)
-\end{equation} 
+\[
+  % \label{eq:zps_hde_rho}
+  \widetilde{\rho } = \rho ( {\widetilde{T},\widetilde {S},z_u })
+  \quad \text{where }\  z_u = \min \left( {z_T^{i+1} ,z_T^i } \right)
+\]
 
 This is a much better approximation as the variation of $\rho$ with depth (and thus pressure)
@@ -1471,3 +1508,6 @@
 \gmcomment{gm :   this last remark has to be done}
 %%%
+
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ZDF.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ZDF.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_ZDF.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
@@ -6,12 +7,10 @@
 \chapter{Vertical Ocean Physics (ZDF)}
 \label{chap:ZDF}
+
 \minitoc
 
 %gm% Add here a small introduction to ZDF and naming of the different physics (similar to what have been written for TRA and DYN.
 
-
 \newpage
-$\ $\newline    % force a new ligne
-
 
 % ================================================================
@@ -60,7 +59,7 @@
 It is recommended that this option is only used in process studies, not in basin scale simulations.
 Typical values used in this case are:
-\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}
+\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*}
 
@@ -69,5 +68,4 @@
 that is $\sim10^{-6}~m^2.s^{-1}$ for momentum, $\sim10^{-7}~m^2.s^{-1}$ for temperature and
 $\sim10^{-9}~m^2.s^{-1}$ for salinity.
-
 
 % -------------------------------------------------------------------------------------------------------------
@@ -90,10 +88,13 @@
 ($i.e.$ the 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.
-\end{equation}
+\[
+  % \label{eq:zdfric}
+  \left\{
+    \begin{aligned}
+      A^{vT} &= \frac {A_{ric}^{vT}}{\left( 1+a \; Ri \right)^n} + A_b^{vT}       \\
+      A^{vm} &= \frac{A^{vT}        }{\left( 1+ a \;Ri  \right)   } + A_b^{vm}
+    \end{aligned}
+  \right.
+\]
 where $Ri = N^2 / \left(\partial_z \textbf{U}_h \right)^2$ is the local Richardson number,
 $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 
@@ -111,13 +112,13 @@
 
 This depth is assumed proportional to the "depth of frictional influence" that is limited by rotation:
-\begin{equation}
-h_{e} = Ek \frac {u^{*}} {f_{0}}
-\end{equation}
+\[
+  h_{e} = Ek \frac {u^{*}} {f_{0}}
+\]
 where, $Ek$ is an empirical parameter, $u^{*}$ is the friction velocity and $f_{0}$ is the Coriolis parameter.
 
 In this similarity height relationship, the turbulent friction velocity:
-\begin{equation}
-u^{*} = \sqrt \frac {|\tau|} {\rho_o}
-\end{equation}
+\[
+  u^{*} = \sqrt \frac {|\tau|} {\rho_o}
+\]
 is computed from the wind stress vector $|\tau|$ and the reference density $ \rho_o$.
 The final $h_{e}$ is further constrained by the adjustable bounds \np{rn\_mldmin} and \np{rn\_mldmax}.
@@ -146,19 +147,21 @@
 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{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]
-- c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
+\begin{equation}
+  \label{eq:zdftke_e}
+  \frac{\partial \bar{e}}{\partial t} =
+  \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]
+  - c_\epsilon \;\frac{\bar {e}^{3/2}}{l_\epsilon }
 \end{equation}
-\begin{equation} \label{eq:zdftke_kz}
-   \begin{split}
-         K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }  	\\
-         K_\rho &= A^{vm} / P_{rt}
-   \end{split}
-\end{equation}
+\[
+  % \label{eq:zdftke_kz}
+  \begin{split}
+    K_m &= C_k\  l_k\  \sqrt {\bar{e}\; }  	\\
+    K_\rho &= A^{vm} / P_{rt}
+  \end{split}
+\]
 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}), 
 $l_{\epsilon }$ and $l_{\kappa }$ are the dissipation and mixing length scales, 
@@ -168,10 +171,12 @@
 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}
-                    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$} 	\\
-                    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$} 	\\
-                    \ \ 10 &      \text{if $\ 2 \leq R_i$} 
-            \end{cases}
+\begin{align*}
+  % \label{eq:prt}
+  P_{rt} =
+  \begin{cases}
+    \ \ \ 1 &      \text{if $\ R_i \leq 0.2$} 	\\
+    5\,R_i &      \text{if $\ 0.2 \leq R_i \leq 2$} 	\\
+    \ \ 10 &      \text{if $\ 2 \leq R_i$}
+  \end{cases}
 \end{align*}
 Options are defined through the  \ngn{namzdfy\_tke} namelist variables.
@@ -195,10 +200,12 @@
 
 \subsubsection{Turbulent length scale}
+
 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{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
+\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-Vais\"{a}l\"{a} frequency.
@@ -211,7 +218,8 @@
 which add an extra assumption concerning the vertical gradient of the computed length scale.
 So, the length scales are first evaluated as in \autoref{eq:tke_mxl0_1} and then bounded such that:
-\begin{equation} \label{eq:tke_mxl_constraint}
-\frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
-\qquad \text{with }\  l =  l_k = l_\epsilon
+\begin{equation}
+  \label{eq:tke_mxl_constraint}
+  \frac{1}{e_3 }\left| {\frac{\partial l}{\partial k}} \right| \leq 1
+  \qquad \text{with }\  l =  l_k = l_\epsilon
 \end{equation}
 \autoref{eq:tke_mxl_constraint} means that the vertical variations of the length scale cannot be larger than
@@ -227,19 +235,23 @@
 (and note that here we use numerical indexing):
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t] \begin{center}
-\includegraphics[width=1.00\textwidth]{Fig_mixing_length}
-\caption{ \protect\label{fig:mixing_length} 
-Illustration of the mixing length computation. }
-\end{center}  
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.00\textwidth]{Fig_mixing_length}
+    \caption{
+      \protect\label{fig:mixing_length}
+      Illustration of the mixing length computation.
+    }
+  \end{center}
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{equation} \label{eq:tke_mxl2}
-\begin{aligned}
- l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
+\[
+  % \label{eq:tke_mxl2}
+  \begin{aligned}
+    l_{up\ \ }^{(k)} &= \min \left(  l^{(k)} \ , \ l_{up}^{(k+1)} + e_{3t}^{(k)}\ \ \ \;  \right)
     \quad &\text{ from $k=1$ to $jpk$ }\ \\
- l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)   
+    l_{dwn}^{(k)} &= \min \left(  l^{(k)} \ , \ l_{dwn}^{(k-1)} + e_{3t}^{(k-1)}  \right)
     \quad &\text{ from $k=jpk$ to $1$ }\ \\
-\end{aligned}
-\end{equation}
+  \end{aligned}
+\]
 where $l^{(k)}$ is computed using \autoref{eq:tke_mxl0_1}, $i.e.$ $l^{(k)} = \sqrt {2 {\bar e}^{(k)} / {N^2}^{(k)} }$.
 
@@ -247,10 +259,11 @@
 $ l_k=  l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)$, while in the \np{nn\_mxl}\forcode{ = 3} case,
 the dissipation and mixing turbulent length scales are give as in \citet{Gaspar1990}:
-\begin{equation} \label{eq:tke_mxl_gaspar}
-\begin{aligned}
-& l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }  	\\
-& l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right) 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:tke_mxl_gaspar}
+  \begin{aligned}
+    & l_k          = \sqrt{\  l_{up} \ \ l_{dwn}\ }  	\\
+    & l_\epsilon = \min \left(\ l_{up} \;,\;  l_{dwn}\ \right)
+  \end{aligned}
+\]
 
 At the ocean surface, a non zero length scale is set through the  \np{rn\_mxl0} namelist parameter.
@@ -261,7 +274,7 @@
 $\bar{e}$ reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ).
 
-
 \subsubsection{Surface wave breaking parameterization}
 %-----------------------------------------------------------------------%
+
 Following \citet{Mellor_Blumberg_JPO04}, the TKE turbulence closure model has been modified to
 include the effect of surface wave breaking energetics.
@@ -272,12 +285,14 @@
 
 Following \citet{Craig_Banner_JPO94}, the boundary condition on surface TKE value is :
-\begin{equation}  \label{eq:ZDF_Esbc}
-\bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
+\begin{equation}
+  \label{eq:ZDF_Esbc}
+  \bar{e}_o = \frac{1}{2}\,\left(  15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o}
 \end{equation}
 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality which depends on the ''wave age'',
 ranging from 57 for mature waves to 146 for younger waves \citep{Mellor_Blumberg_JPO04}. 
 The boundary condition on the turbulent length scale follows the Charnock's relation:
-\begin{equation} \label{eq:ZDF_Lsbc}
-l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
+\begin{equation}
+  \label{eq:ZDF_Lsbc}
+  l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o}
 \end{equation}
 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant.
@@ -293,7 +308,7 @@
 surface $\bar{e}$ value.
 
-
 \subsubsection{Langmuir cells}
 %--------------------------------------%
+
 Langmuir circulations (LC) can be described as ordered large-scale vertical motions in
 the surface layer of the oceans.
@@ -313,7 +328,7 @@
 By making an analogy with the characteristic convective velocity scale ($e.g.$, \citet{D'Alessio_al_JPO98}),
 $P_{LC}$ is assumed to be : 
-\begin{equation}
+\[
 P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
-\end{equation}
+\]
 where $w_{LC}(z)$ is the vertical velocity profile of LC, and $H_{LC}$ is the LC depth.
 With no information about the wave field, $w_{LC}$ is assumed to be proportional to 
@@ -322,15 +337,17 @@
   $u_s =  0.016 \,|U_{10m}|$.
   Assuming an air density of $\rho_a=1.22 \,Kg/m^3$ and a drag coefficient of
-  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress}.
+  $1.5~10^{-3}$ give the expression used of $u_s$ as a function of the module of surface stress
+}.
 For the vertical variation, $w_{LC}$ is assumed to be zero at the surface as well as at
 a finite depth $H_{LC}$ (which is often close to the mixed layer depth),
 and simply varies as a sine function in between (a first-order profile for the Langmuir cell structures). 
 The resulting expression for $w_{LC}$ is :
-\begin{equation}
-w_{LC}  = \begin{cases}
-                   c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$} 	\\
-                   0                 				 &      \text{otherwise} 
-                 \end{cases}
-\end{equation}
+\[
+  w_{LC}  =
+  \begin{cases}
+    c_{LC} \,u_s \,\sin(- \pi\,z / H_{LC} )    &      \text{if $-z \leq H_{LC}$} 	\\
+    0                 				 &      \text{otherwise}
+  \end{cases}
+\]
 where $c_{LC} = 0.15$ has been chosen by \citep{Axell_JGR02} as a good compromise to fit LES data.
 The chosen value yields maximum vertical velocities $w_{LC}$ of the order of a few centimeters per second.
@@ -341,8 +358,7 @@
 $H_{LC}$ is depth to which a water parcel with kinetic energy due to Stoke drift can reach on its own by
 converting its kinetic energy to potential energy, according to 
-\begin{equation}
+\[
 - \int_{-H_{LC}}^0 { N^2\;z  \;dz} = \frac{1}{2} u_s^2
-\end{equation}
-
+\]
 
 \subsubsection{Mixing just below the mixed layer}
@@ -362,6 +378,7 @@
 swell and waves is parameterized by \autoref{eq:ZDF_Esbc} the standard TKE surface boundary condition,
 plus a depth depend one given by:
-\begin{equation}  \label{eq:ZDF_Ehtau}
-S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau} 
+\begin{equation}
+  \label{eq:ZDF_Ehtau}
+  S = (1-f_i) \; f_r \; e_s \; e^{-z / h_\tau}
 \end{equation}
 where $z$ is the depth, $e_s$ is TKE surface boundary condition, $f_r$ is the fraction of the surface TKE that
@@ -380,6 +397,4 @@
 They will be removed in the next release. 
 
-
-
 % from Burchard et al OM 2008 : 
 % the most critical process not reproduced by statistical turbulence models is the activity of 
@@ -390,6 +405,4 @@
 % (e.g. Mellor, 1989; Large et al., 1994; Meier, 2001; Axell, 2002; St. Laurent and Garrett, 2002).
 
-
-
 % -------------------------------------------------------------------------------------------------------------
 %        TKE discretization considerations
@@ -399,9 +412,12 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]   \begin{center}
-\includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme}
-\caption{ \protect\label{fig:TKE_time_scheme} 
-Illustration of the TKE time integration and its links to the momentum and tracer time integration. }
-\end{center}  
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=1.00\textwidth]{Fig_ZDF_TKE_time_scheme}
+    \caption{
+      \protect\label{fig:TKE_time_scheme}
+      Illustration of the TKE time integration and its links to the momentum and tracer time integration.
+    }
+  \end{center}  
 \end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
@@ -419,10 +435,11 @@
 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}
+\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, $K_m$,
@@ -443,12 +460,13 @@
 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}
+\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 $N^2 = -g \,\partial_k \rho / (e_3 \rho)$. 
@@ -468,20 +486,21 @@
 
 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}
+\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 \autoref{eq:zdftke_ene} (vertical diffusion and Kolmogorov dissipation)
@@ -511,34 +530,38 @@
 $k$-$\omega$ \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
+\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}
+\[
+  % \label{eq:zdfgls_psi}
+  \begin{split}
+    \frac{\partial \psi}{\partial t} =& \frac{\psi}{\bar{e}} \left\{
+      \frac{C_1\,K_m}{\sigma_{\psi} {e_3}}\;\left[ {\left( \frac{\partial u}{\partial k} \right)^2
+          +\left( \frac{\partial v}{\partial k} \right)^2} \right]
+      - C_3 \,K_\rho\,N^2   - C_2 \,\epsilon \,Fw   \right\}             \\
+    &+\frac{1}{e_3}  \;\frac{\partial }{\partial k}\left[ {\frac{K_m}{e_3 }
+        \;\frac{\partial \psi}{\partial k}} \right]\;
+  \end{split}
+\]
+
+\[
+  % \label{eq:zdfgls_kz}
+  \begin{split}
+    K_m    &= C_{\mu} \ \sqrt {\bar{e}} \ l         \\
+    K_\rho &= C_{\mu'}\ \sqrt {\bar{e}} \ l
+  \end{split}
+\]
+
+\[
+  % \label{eq:zdfgls_eps}
+  {\epsilon} = C_{0\mu} \,\frac{\bar {e}^{3/2}}{l} \;
+\]
 where $N$ is the local Brunt-Vais\"{a}l\"{a} frequency (see \autoref{subsec:TRA_bn2}) and
 $\epsilon$ the dissipation rate. 
@@ -549,26 +572,31 @@
 
 %--------------------------------------------------TABLE--------------------------------------------------
-\begin{table}[htbp]	\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{   \protect\label{tab:GLS} 
-  Set of predefined GLS parameters, or equivalently predefined turbulence models available with
-  \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.}
-\end{center}	\end{table}
+\begin{table}[htbp]
+  \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{
+      \protect\label{tab:GLS}
+      Set of predefined GLS parameters, or equivalently predefined turbulence models available with
+      \protect\key{zdfgls} and controlled by the \protect\np{nn\_clos} namelist variable in \protect\ngn{namzdf\_gls}.
+    }
+  \end{center}
+\end{table}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -646,19 +674,23 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!htb]  	\begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_npc}
-\caption{  \protect\label{fig:npc} 
-  Example of an unstable density profile treated by the non penetrative convective adjustment algorithm.
-  $1^{st}$ step: the initial profile is checked from the surface to the bottom.
-  It is found to be unstable between levels 3 and 4.
-  They are mixed.
-  The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
-  The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
-  The $1^{st}$ step ends since the density profile is then stable below the level 3.
-  $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
-  levels 2 to 5 are mixed.
-  The new density profile is checked.
-  It is found stable: end of algorithm.}
-\end{center}  	\end{figure}
+\begin{figure}[!htb]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_npc}
+    \caption{
+      \protect\label{fig:npc}
+      Example of an unstable density profile treated by the non penetrative convective adjustment algorithm.
+      $1^{st}$ step: the initial profile is checked from the surface to the bottom.
+      It is found to be unstable between levels 3 and 4.
+      They are mixed.
+      The resulting $\rho$ is still larger than $\rho$(5): levels 3 to 5 are mixed.
+      The resulting $\rho$ is still larger than $\rho$(6): levels 3 to 6 are mixed.
+      The $1^{st}$ step ends since the density profile is then stable below the level 3.
+      $2^{nd}$ step: the new $\rho$ profile is checked following the same procedure as in $1^{st}$ step:
+      levels 2 to 5 are mixed.
+      The new density profile is checked.
+      It is found stable: end of algorithm.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -781,7 +813,8 @@
 
 Diapycnal mixing of S and T are described by diapycnal diffusion coefficients 
-\begin{align*} % \label{eq:zdfddm_Kz}
-    &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}	\\
-    &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
+\begin{align*}
+  % \label{eq:zdfddm_Kz}
+  &A^{vT} = A_o^{vT}+A_f^{vT}+A_d^{vT}	\\
+  &A^{vS} = A_o^{vS}+A_f^{vS}+A_d^{vS}
 \end{align*}
 where subscript $f$ represents mixing by salt fingering, $d$ by diffusive convection,
@@ -792,24 +825,31 @@
 To represent mixing of $S$ and $T$ by salt fingering, we adopt the diapycnal diffusivities suggested by Schmitt
 (1981):
-\begin{align} \label{eq:zdfddm_f}
-A_f^{vS} &= 	\begin{cases}
-	\frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
-	0 				  					    &\text{otherwise} 
-				\end{cases}   
-\\ 		    \label{eq:zdfddm_f_T}
-A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
+\begin{align}
+  \label{eq:zdfddm_f}
+  A_f^{vS} &=
+             \begin{cases}
+               \frac{A^{\ast v}}{1+(R_\rho / R_c)^n   } &\text{if  $R_\rho > 1$ and $N^2>0$ } \\
+               0 				  					    &\text{otherwise}
+             \end{cases}
+  \\ 		    \label{eq:zdfddm_f_T}
+  A_f^{vT} &= 0.7 \ A_f^{vS} / R_\rho 
 \end{align}
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t]   \begin{center}
-\includegraphics[width=0.99\textwidth]{Fig_zdfddm}
-\caption{  \protect\label{fig:zdfddm}
-  From \citet{Merryfield1999} :
-  (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering.
-  Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
-  (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of diffusive convection.
-  Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
-  The latter is not implemented in \NEMO. }
-\end{center}    \end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.99\textwidth]{Fig_zdfddm}
+    \caption{
+      \protect\label{fig:zdfddm}
+      From \citet{Merryfield1999} :
+      (a) Diapycnal diffusivities $A_f^{vT}$ and $A_f^{vS}$ for temperature and salt in regions of salt fingering.
+      Heavy curves denote $A^{\ast v} = 10^{-3}~m^2.s^{-1}$ and thin curves $A^{\ast v} = 10^{-4}~m^2.s^{-1}$;
+      (b) diapycnal diffusivities $A_d^{vT}$ and $A_d^{vS}$ for temperature and salt in regions of
+      diffusive convection.
+      Heavy curves denote the Federov parameterisation and thin curves the Kelley parameterisation.
+      The latter is not implemented in \NEMO.
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 
@@ -820,18 +860,20 @@
 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}
-	1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
-									&\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
-	0 								&\text{otherwise} 
-				\end{cases}   
-\\   			\label{eq:zdfddm_d_S}
-A_d^{vS} &= 	\begin{cases}
-	A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right)
-									&\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
-	A_d^{vT} \ 0.15 \ R_\rho
-									&\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
-	0 								&\text{otherwise} 
-				\end{cases}   
+\begin{align}
+  % \label{eq:zdfddm_d}
+  A_d^{vT} &=
+             \begin{cases}
+               1.3635 \, \exp{\left( 4.6\, \exp{ \left[  -0.54\,( R_{\rho}^{-1} - 1 )  \right] }    \right)}
+               &\text{if  $0<R_\rho < 1$ and $N^2>0$ } \\
+               0 								&\text{otherwise}
+             \end{cases}
+                                       \nonumber \\
+  \label{eq:zdfddm_d_S}
+  A_d^{vS} &=
+             \begin{cases}
+               A_d^{vT}\ \left( 1.85\,R_{\rho} - 0.85 \right) &\text{if  $0.5 \leq R_\rho<1$ and $N^2>0$ } \\
+               A_d^{vT} \ 0.15 \ R_\rho               &\text{if  $\ \ 0 < R_\rho<0.5$ and $N^2>0$ } \\
+               0 								&\text{otherwise}
+             \end{cases}
 \end{align}
 
@@ -863,7 +905,8 @@
 a condition on the vertical 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) = {{\cal F}}_h^{\textbf U}
-\end{equation}
+\[
+  % \label{eq:zdfbfr_flux}
+  A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U}
+\]
 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).
@@ -878,6 +921,7 @@
 bottom model layer.
 To illustrate 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[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}}
+\begin{equation}
+  \label{eq:zdfbfr_flux2}
+  \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.
@@ -897,6 +941,7 @@
 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} = 
+\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}
@@ -911,7 +956,8 @@
 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}
-{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
-\end{equation}
+\[
+  % \label{eq:zdfbfr_linear}
+  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
+\]
 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, 
@@ -927,10 +973,11 @@
 
 For the linear friction case the coefficients defined in the general expression \autoref{eq:zdfbfr_bdef} are: 
-\begin{equation} \label{eq:zdfbfr_linbfr_b}
-\begin{split}
- c_b^u &= - r\\
- c_b^v &= - r\\
-\end{split}
-\end{equation}
+\[
+  % \label{eq:zdfbfr_linbfr_b}
+  \begin{split}
+    c_b^u &= - r\\
+    c_b^v &= - r\\
+  \end{split}
+\]
 When \np{nn\_botfr}\forcode{ = 1}, the value of $r$ used is \np{rn\_bfri1}.
 Setting \np{nn\_botfr}\forcode{ = 0} is equivalent to setting $r=0$ and
@@ -950,8 +997,9 @@
 
 The non-linear bottom friction parameterisation assumes that the bottom friction is quadratic: 
-\begin{equation} \label{eq:zdfbfr_nonlinear}
-{\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}
+\[
+  % \label{eq:zdfbfr_nonlinear}
+  {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h
+  }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b
+\]
 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.
@@ -965,10 +1013,11 @@
 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}
- 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}
+\[
+  % \label{eq:zdfbfr_nonlinbfr}
+  \begin{split}
+    c_b^u &= - \; C_D\;\left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\
+    c_b^v &= - \; C_D\;\left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\
+  \end{split}
+\]
 
 The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters:
@@ -991,7 +1040,7 @@
 If  \np{ln\_loglayer} = .true., $C_D$ is no longer constant but is related to the thickness of
 the last wet layer in each column by:
-\begin{equation}
-C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2
-\end{equation}
+\[
+  C_D = \left ( {\kappa \over {\rm log}\left ( 0.5e_{3t}/rn\_bfrz0 \right ) } \right )^2
+\]
 
 \noindent where $\kappa$ is the von-Karman constant and \np{rn\_bfrz0} is a roughness length provided via
@@ -1001,7 +1050,7 @@
 the base \np{rn\_bfri2} value and it is not allowed to exceed the value of an additional namelist parameter:
 \np{rn\_bfri2\_max}, i.e.:
-\begin{equation}
-rn\_bfri2 \leq C_D \leq rn\_bfri2\_max
-\end{equation}
+\[
+  rn\_bfri2 \leq C_D \leq rn\_bfri2\_max
+\]
 
 \noindent Note also that a log-layer enhancement can also be applied to the top boundary friction if
@@ -1018,25 +1067,26 @@
 bottom friction does not induce numerical instability.
 For the purposes of stability analysis, an approximation to \autoref{eq:zdfbfr_flux2} is:
-\begin{equation} \label{eq:Eqn_bfrstab}
-\begin{split}
- \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2 \rdt    \\
-               &= -\frac{ru}{e_{3u}}\;2\rdt\\
-\end{split}
+\begin{equation}
+  \label{eq: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}
+\[
+  |\Delta u| < \;|u| 
+\]
 \noindent which, using \autoref{eq:Eqn_bfrstab}, gives:
-\begin{equation}
-r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
-\end{equation}
+\[
+  r\frac{2\rdt}{e_{3u}} < 1 \qquad  \Rightarrow \qquad r < \frac{e_{3u}}{2\rdt}\\
+\]
 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}
+\[
+  e_{3u} > 2\;r\;\rdt
+\]
 \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.
@@ -1070,8 +1120,9 @@
 the bottom boundary condition is implemented implicitly.
 
-\begin{equation} \label{eq:dynzdf_bfr}
-\left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk}
-	 = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}}
-\end{equation}
+\[
+  % \label{eq:dynzdf_bfr}
+  \left.{\left( {\frac{A^{vm} }{e_3 }\ \frac{\partial \textbf{U}_h}{\partial k}} \right)} \right|_{mbk}
+  = \binom{c_{b}^{u}u^{n+1}_{mbk}}{c_{b}^{v}v^{n+1}_{mbk}}
+\]
 
 where $mbk$ is the layer number of the bottom wet layer.
@@ -1089,13 +1140,14 @@
 The implementation of the implicit bottom friction in \mdl{dynspg\_ts} is done in two steps as the following:
 
-\begin{equation} \label{eq:dynspg_ts_bfr1}
-\frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b}
-\left(\textbf{U}_{med}-\textbf{U}^{m-1}\right)
-\end{equation}
-\begin{equation} \label{eq:dynspg_ts_bfr2}
-\frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+
-\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)-
-2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right)
-\end{equation}
+\[
+  % \label{eq:dynspg_ts_bfr1}
+  \frac{\textbf{U}_{med}-\textbf{U}^{m-1}}{2\Delta t}=-g\nabla\eta-f\textbf{k}\times\textbf{U}^{m}+c_{b}
+  \left(\textbf{U}_{med}-\textbf{U}^{m-1}\right)
+\]
+\[
+  \frac{\textbf{U}^{m+1}-\textbf{U}_{med}}{2\Delta t}=\textbf{T}+
+  \left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{U}^{'}\right)-
+  2\Delta t_{bc}c_{b}\left(g\nabla\eta^{'}+f\textbf{k}\times\textbf{u}_{b}\right)
+\]
 
 where $\textbf{T}$ is the vertical integrated 3-D momentum trend.
@@ -1106,7 +1158,4 @@
 the 3-D baroclinic mode.
 $\textbf{u}_{b}$ is the bottom layer horizontal velocity.
-
-
-
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1157,14 +1206,12 @@
 
 Otherwise, 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}
+\[
+  % \label{eq:zdfbfr_implicitts}
+  \bar{U}^{t+ \rdt} = \; \left [ \bar{U}^{t-\rdt}\; + 2 \rdt\;RHS \right ] / \left [ 1 - 2 \rdt \;c_b^{u} / H_e \right ]
+\]
 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.
-
-
-
 
 % ================================================================
@@ -1192,6 +1239,7 @@
 $A^{vT}_{tides}$ 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 }
+\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 \autoref{subsec:TRA_bn2}),
@@ -1209,7 +1257,8 @@
 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}
+\[
+  % \label{eq:Fz}
+  F(i,j,k) = \frac{ e^{ -\frac{H+z}{h_o} } }{ h_o \left( 1- e^{ -\frac{H}{h_o} } \right) }
+\]
 and is normalized so that vertical integral over the water column is unity. 
 
@@ -1234,9 +1283,13 @@
 
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
-\begin{figure}[!t] 	\begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx}
-\caption{  \protect\label{fig:ZDF_M2_K1_tmx} 
-(a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$). }
-\end{center}  	\end{figure}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_ZDF_M2_K1_tmx}
+    \caption{
+      \protect\label{fig:ZDF_M2_K1_tmx}
+      (a) M2 and (b) K1 internal wave drag energy from \citet{Carrere_Lyard_GRL03} ($W/m^2$).
+    }
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
  
@@ -1268,10 +1321,13 @@
 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}
+\[
+  % \label{eq:Fz_itf}
+  F(i,j,k) \sim     \left\{
+    \begin{aligned}
+      \frac{q\,\Gamma E(i,j) } {\rho N \, \int N     dz}    \qquad \text{when $\partial_z N < 0$} \\
+      \frac{q\,\Gamma E(i,j) } {\rho     \, \int N^2 dz}    \qquad \text{when $\partial_z N > 0$}
+    \end{aligned}
+  \right.
+\]
 
 Averaged over the ITF area, the resulting tidal mixing coefficient is $1.5\,cm^2/s$, 
@@ -1283,5 +1339,4 @@
 global coupled GCMs \citep{Koch-Larrouy_al_CD10}.
 
-
 % ================================================================
 % Internal wave-driven mixing
@@ -1299,7 +1354,8 @@
 A three-dimensional field of internal wave energy dissipation $\epsilon(x,y,z)$ is first constructed,
 and the resulting diffusivity is obtained as 
-\begin{equation} \label{eq:Kwave}
-A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
-\end{equation}
+\[
+  % \label{eq:Kwave}
+  A^{vT}_{wave} =  R_f \,\frac{ \epsilon }{ \rho \, N^2 }
+\]
 where $R_f$ is the mixing efficiency and $\epsilon$ is a specified three dimensional distribution of
 the energy available for mixing.
@@ -1323,13 +1379,13 @@
 (de Lavergne et al., in prep):
 \begin{align*}
-F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
-F_{pyc}(i,j,k) &\propto N^{n\_p}\\
-F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
+  F_{cri}(i,j,k) &\propto e^{-h_{ab} / h_{cri} }\\
+  F_{pyc}(i,j,k) &\propto N^{n\_p}\\
+  F_{bot}(i,j,k) &\propto N^2 \, e^{- h_{wkb} / h_{bot} }
 \end{align*} 
 In the above formula, $h_{ab}$ denotes the height above bottom,
 $h_{wkb}$ denotes the WKB-stretched height above bottom, defined by
-\begin{equation*}
-h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
-\end{equation*}
+\[
+  h_{wkb} = H \, \frac{ \int_{-H}^{z} N \, dz' } { \int_{-H}^{\eta} N \, dz'  } \; ,
+\]
 The $n_p$ parameter (given by \np{nn\_zpyc} in \ngn{namzdf\_tmx\_new} namelist)
 controls the stratification-dependence of the pycnocline-intensified dissipation.
@@ -1343,5 +1399,5 @@
 % ================================================================
 
-
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_conservation.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
@@ -61,18 +62,21 @@
 Let us define as either the relative, planetary and total potential vorticity, i.e. ?, ?, and ?, respectively.
 The continuous formulation of the vorticity term satisfies following integral constraints:
-\begin{equation} \label{eq:vor_vorticity}
-\int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma 
-\;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:vor_enstrophy}
-if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot 
-\frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv} 
-=0
-\end{equation}
-
-\begin{equation} \label{eq:vor_energy}
-\int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
-\end{equation}
+\[
+  % \label{eq:vor_vorticity}
+  \int_D {{\textbf {k}}\cdot \frac{1}{e_3 }\nabla \times \left( {\varsigma
+        \;{\rm {\bf k}}\times {\textbf {U}}_h } \right)\;dv} =0
+\]
+
+\[
+  % \label{eq:vor_enstrophy}
+  if\quad \chi =0\quad \quad \int\limits_D {\varsigma \;{\textbf{k}}\cdot
+    \frac{1}{e_3 }\nabla \times \left( {\varsigma {\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =-\int\limits_D {\frac{1}{2}\varsigma ^2\,\chi \;dv}
+  =0
+\]
+
+\[
+  % \label{eq:vor_energy}
+  \int_D {{\textbf{U}}_h \times \left( {\varsigma \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} =0
+\]
 where $dv = e_1\, e_2\, e_3\, di\, dj\, dk$ is the volume element.
 (II.4.1a) means that $\varsigma $ is conserved. (II.4.1b) is obtained by an integration by part.
@@ -122,7 +126,8 @@
 potential energy due to buoyancy forces:
 
-\begin{equation} \label{eq:hpg_pe}
-\int_D {-\frac{1}{\rho _o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
-\end{equation}
+\[
+  % \label{eq:hpg_pe}
+  \int_D {-\frac{1}{\rho_o }\left. {\nabla p^h} \right|_z \cdot {\textbf {U}}_h \;dv} \;=\;\int_D {\nabla .\left( {\rho \,{\textbf{U}}} \right)\;g\;z\;\;dv}
+\]
 
 Using the discrete form given in {\S}~II.2-a and the symmetry or anti-symmetry properties of
@@ -142,7 +147,8 @@
 In addition, with the rigid-lid approximation, the change of horizontal kinetic energy due to the work of 
 surface pressure forces is exactly zero:
-\begin{equation} \label{eq:spg}
-\int_D {-\frac{1}{\rho _o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
-\end{equation}
+\[
+  % \label{eq:spg}
+  \int_D {-\frac{1}{\rho_o }\nabla _h } \left( {p_s } \right)\cdot {\textbf{U}}_h \;dv=0
+\]
 
 (II.4.4) is satisfied in discrete form only if
@@ -159,9 +165,10 @@
 In continuous formulation, the advective terms of the tracer equations conserve the tracer content and
 the quadratic form of the tracer, $i.e.$
-\begin{equation} \label{eq:tra_tra2}
-\int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
-\;\text{and}
-\int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
-\end{equation}
+\[
+  % \label{eq:tra_tra2}
+  \int_D {\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+  \;\text{and}
+  \int_D {T\;\nabla .\left( {T\;{\textbf{U}}} \right)\;dv} =0
+\]
 
 The numerical scheme used ({\S}II.2-b) (equations in flux form, second order centred finite differences) satisfies
@@ -180,34 +187,39 @@
 
 The continuous formulation of the horizontal diffusion of momentum satisfies the following integral constraints~:
-\begin{equation} \label{eq:dynldf_dyn}
-\int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla 
-_h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta 
-\;{\rm {\bf k}}} \right)} \right]\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_div}
-\int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
-\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
-\right]\;dv} =0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_curl}
-\int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi } 
-\right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} 
-\right]\;dv} \leqslant 0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_curl2}
-\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot 
-\nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h 
-\times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} 
-\leqslant 0
-\end{equation}
-
-\begin{equation} \label{eq:dynldf_div2}
-\mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[ 
-{\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( 
-{A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
-\end{equation}
+\[
+  % \label{eq:dynldf_dyn}
+  \int\limits_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left[ {\nabla
+        _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left( {A^{lm}\;\zeta
+            \;{\rm {\bf k}}} \right)} \right]\;dv} =0
+\]
+
+\[
+  % \label{eq:dynldf_div}
+  \int\limits_D {\nabla _h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+    \right]\;dv} =0
+\]
+
+\[
+  % \label{eq:dynldf_curl}
+  \int_D {{\rm {\bf U}}_h \cdot \left[ {\nabla _h \left( {A^{lm}\;\chi }
+        \right)-\nabla _h \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)}
+    \right]\;dv} \leqslant 0
+\]
+
+\[
+  % \label{eq:dynldf_curl2}
+  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\zeta \;{\rm {\bf k}}\cdot
+    \nabla \times \left[ {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h
+        \times \left( {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv}
+  \leqslant 0
+\]
+
+\[
+  % \label{eq:dynldf_div2}
+  \mbox{if}\quad A^{lm}=cste\quad \quad \int_D {\chi \;\nabla _h \cdot \left[
+      {\nabla _h \left( {A^{lm}\;\chi } \right)-\nabla _h \times \left(
+          {A^{lm}\;\zeta \;{\rm {\bf k}}} \right)} \right]\;dv} \leqslant 0
+\]
 
 
@@ -237,32 +249,35 @@
 conservation of momentum, dissipation of horizontal kinetic energy
 
-\begin{equation} \label{eq:dynzdf_dyn}
-\begin{aligned}
-& \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\ 
-& \int_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 \leq 0 \\ 
- \end{aligned} 
- \end{equation}
+\[
+  % \label{eq:dynzdf_dyn}
+  \begin{aligned}
+    & \int_D {\frac{1}{e_3 }}  \frac{\partial }{\partial k}\left( \frac{A^{vm}}{e_3 }\frac{\partial {\textbf{U}}_h }{\partial k} \right) \;dv = \overrightarrow{\textbf{0}} \\
+    & \int_D \textbf{U}_h \cdot \frac{1}{e_3} \frac{\partial}{\partial k} \left( {\frac{A^{vm}}{e_3 }}{\frac{\partial \textbf{U}_h }{\partial k}} \right) \;dv \leq 0 \\
+  \end{aligned}
+\]
 conservation of vorticity, dissipation of enstrophy
-\begin{equation} \label{eq:dynzdf_vor}
-\begin{aligned}
-& \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
-}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
-{\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\ 
-& \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3 
-}\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm 
-{\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:dynzdf_vor}
+  \begin{aligned}
+    & \int_D {\frac{1}{e_3 }{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} =0 \\
+    & \int_D {\zeta \,{\rm {\bf k}}\cdot \nabla \times \left( {\frac{1}{e_3
+          }\frac{\partial }{\partial k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm
+                  {\bf U}}_h }{\partial k}} \right)} \right)\;dv} \leq 0 \\
+  \end{aligned}
+\]
 conservation of horizontal divergence, dissipation of square of the horizontal divergence
-\begin{equation} \label{eq:dynzdf_div}
-\begin{aligned}
- &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
-k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
-\right)} \right)\;dv} =0 \\ 
-& \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial 
-k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}} 
-\right)} \right)\;dv} \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:dynzdf_div}
+  \begin{aligned}
+    &\int_D {\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+          \right)} \right)\;dv} =0 \\
+    & \int_D {\chi \;\nabla \cdot \left( {\frac{1}{e_3 }\frac{\partial }{\partial
+            k}\left( {\frac{A^{vm}}{e_3 }\frac{\partial {\rm {\bf U}}_h }{\partial k}}
+          \right)} \right)\;dv} \leq 0 \\
+  \end{aligned}
+\]
 
 In discrete form, all these properties are satisfied in $z$-coordinate (see Appendix C).
@@ -286,19 +301,21 @@
 variance, i.e.
 
-\begin{equation} \label{eq:traldf_t_t2}
-\begin{aligned}
-&\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\ 
-&\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:traldf_t_t2}
+  \begin{aligned}
+    &\int_D \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv = 0 \\
+    &\int_D \,T\, \nabla\, \cdot\, \left( A^{lT} \,\Re \,\nabla \,T \right)\;dv \leq 0 \\
+  \end{aligned}
+\]
 
 \textbf{* vertical physics: }conservation of tracer, dissipation of tracer variance, $i.e.$
 
-\begin{equation} \label{eq:trazdf_t_t2}
-\begin{aligned}
-& \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\ 
-& \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\ 
-\end{aligned}
-\end{equation}
+\[
+  % \label{eq:trazdf_t_t2}
+  \begin{aligned}
+    & \int_D \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv = 0 \\
+    & \int_D \,T \frac{1}{e_3 } \frac{\partial }{\partial k}\left( \frac{A^{vT}}{e_3 }  \frac{\partial T}{\partial k}  \right)\;dv \leq 0 \\
+  \end{aligned}
+\]
 
 Using the symmetry or anti-symmetry properties of the mean and difference operators,
@@ -311,3 +328,5 @@
 It has not been implemented.
 
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_misc.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_misc.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_misc.tex	(revision 10419)
@@ -1,13 +1,14 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter --- Miscellaneous Topics
+% Chapter --- Miscellaneous Topics
 % ================================================================
 \chapter{Miscellaneous Topics}
 \label{chap:MISC}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -64,5 +65,6 @@
     \includegraphics[width=0.80\textwidth]{Fig_Gibraltar}
     \includegraphics[width=0.80\textwidth]{Fig_Gibraltar2}
-    \caption{	\protect\label{fig:MISC_strait_hand}
+    \caption{
+      \protect\label{fig:MISC_strait_hand}
       Example of the Gibraltar strait defined in a $1^{\circ} \times 1^{\circ}$ mesh.
       \textit{Top}: using partially open cells.
@@ -73,5 +75,6 @@
       The four fmask parameters along the strait coastlines are set to a value larger than 4,
       $i.e.$ "strong" no-slip case (see \autoref{fig:LBC_shlat}) creating a large viscous boundary layer that
-      allows a reduced transport through the strait.}
+      allows a reduced transport through the strait.
+    }
   \end{center}
 \end{figure}
@@ -146,5 +149,6 @@
   \begin{center}
     \includegraphics[width=0.90\textwidth]{Fig_LBC_zoom}
-    \caption{	\protect\label{fig:LBC_zoom}
+    \caption{
+      \protect\label{fig:LBC_zoom}
       Position of a model domain compared to the data input domain when the zoom functionality is used.
     }
@@ -290,7 +294,5 @@
 
 % ================================================================
+\biblio
+
 \end{document}
-
-
-
-
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics.tex	(revision 10419)
@@ -1,14 +1,15 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter 1 Ñ Model Basics
+% Chapter 1 Model Basics
 % ================================================================
 
 \chapter{Model Basics}
 \label{chap:PE}
+
 \minitoc
 
 \newpage
-$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -62,26 +63,33 @@
 (namely the momentum balance, the hydrostatic equilibrium, the incompressibility equation,
 the heat and salt conservation equations and an equation of state):
-\begin{subequations} \label{eq:PE}
-  \begin{equation}     \label{eq:PE_dyn}
-\frac{\partial {\rm {\bf U}}_h }{\partial t}=
--\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}}
-            +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h
- -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h 
--\frac{1}{\rho _o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
+\begin{subequations}
+  \label{eq:PE}
+  \begin{equation}
+    \label{eq:PE_dyn}
+    \frac{\partial {\rm {\bf U}}_h }{\partial t}=
+    -\left[    {\left( {\nabla \times {\rm {\bf U}}} \right)\times {\rm {\bf U}}
+        +\frac{1}{2}\nabla \left( {{\rm {\bf U}}^2} \right)}    \right]_h
+    -f\;{\rm {\bf k}}\times {\rm {\bf U}}_h
+    -\frac{1}{\rho_o }\nabla _h p + {\rm {\bf D}}^{\rm {\bf U}} + {\rm {\bf F}}^{\rm {\bf U}}
   \end{equation}
-  \begin{equation}     \label{eq:PE_hydrostatic}
-\frac{\partial p }{\partial z} = - \rho \ g
+  \begin{equation}
+    \label{eq:PE_hydrostatic}
+    \frac{\partial p }{\partial z} = - \rho \ g
   \end{equation}
-  \begin{equation}     \label{eq:PE_continuity}
-\nabla \cdot {\bf U}=  0
+  \begin{equation}
+    \label{eq:PE_continuity}
+    \nabla \cdot {\bf U}=  0
   \end{equation}
-\begin{equation} \label{eq:PE_tra_T}
-\frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T
+  \begin{equation}
+    \label{eq:PE_tra_T}
+    \frac{\partial T}{\partial t} = - \nabla \cdot  \left( T \ \rm{\bf U} \right) + D^T + F^T
   \end{equation}
-  \begin{equation}     \label{eq:PE_tra_S}
-\frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S
+  \begin{equation}
+    \label{eq:PE_tra_S}
+    \frac{\partial S}{\partial t} = - \nabla \cdot  \left( S \ \rm{\bf U} \right) + D^S + F^S
   \end{equation}
-  \begin{equation}     \label{eq:PE_eos}
-\rho = \rho \left( T,S,p \right)
+  \begin{equation}
+    \label{eq:PE_eos}
+    \rho = \rho \left( T,S,p \right)
   \end{equation}
 \end{subequations}
@@ -140,5 +148,6 @@
 \item[Solid earth - ocean interface:]
   heat and salt fluxes through the sea floor are small, except in special areas of little extent.
-  They are usually neglected in the model \footnote{
+  They are usually neglected in the model
+  \footnote{
     In fact, it has been shown that the heat flux associated with the solid Earth cooling
     ($i.e.$the geothermal heating) is not negligible for the thermohaline circulation of the world ocean
@@ -150,5 +159,6 @@
   the bottom velocity is parallel to solid boundaries). This kinematic boundary condition
   can be expressed as:
-  \begin{equation} \label{eq:PE_w_bbc}
+  \begin{equation}
+    \label{eq:PE_w_bbc}
     w = -{\rm {\bf U}}_h \cdot  \nabla _h \left( H \right)
   \end{equation}
@@ -162,9 +172,10 @@
   the kinematic surface condition plus the mass flux of fresh water PE (the precipitation minus evaporation budget)
   leads to:
-  \begin{equation} \label{eq:PE_w_sbc}
+  \[
+    % \label{eq:PE_w_sbc}
     w = \frac{\partial \eta }{\partial t}
     + \left. {{\rm {\bf U}}_h } \right|_{z=\eta } \cdot  \nabla _h \left( \eta \right)
     + P-E
-  \end{equation}
+  \]
   The dynamic boundary condition, neglecting the surface tension (which removes capillary waves from the system)
   leads to the continuity of pressure across the interface $z=\eta$.
@@ -179,5 +190,4 @@
 
 %\newpage
-%$\ $\newline    % force a new ligne
 
 % ================================================================
@@ -199,8 +209,9 @@
 assuming that pressure in decibars can be approximated by depth in meters in (\autoref{eq:PE_eos}).
 The hydrostatic pressure is then given by:
-\begin{equation} \label{eq:PE_pressure}
-p_h \left( {i,j,z,t} \right)
- = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma } 
-\end{equation}
+\[
+  % \label{eq:PE_pressure}
+  p_h \left( {i,j,z,t} \right)
+  = \int_{\varsigma =z}^{\varsigma =0} {g\;\rho \left( {T,S,\varsigma} \right)\;d\varsigma }
+\]
 Two strategies can be considered for the surface pressure term:
 $(a)$ introduce of a  new variable $\eta$, the free-surface elevation,
@@ -231,8 +242,9 @@
 This variable is solution of a prognostic equation which is established by forming the vertical average of
 the kinematic surface condition (\autoref{eq:PE_w_bbc}):
-\begin{equation} \label{eq:PE_ssh}
-\frac{\partial \eta }{\partial t}=-D+P-E
- 	\quad \text{where} \ 
-D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]
+\begin{equation}
+  \label{eq:PE_ssh}
+  \frac{\partial \eta }{\partial t}=-D+P-E
+  \quad \text{where} \
+  D=\nabla \cdot \left[ {\left( {H+\eta } \right) \; {\rm{\bf \overline{U}}}_h \,} \right]
 \end{equation}
 and using (\autoref{eq:PE_hydrostatic}) the surface pressure is given by: $p_s = \rho \, g \, \eta$.
@@ -275,5 +287,4 @@
 
 %\newpage
-%$\ $\newline    % force a new line
 
 % ================================================================
@@ -318,5 +329,6 @@
 The local deformation of the curvilinear coordinate system is given by $e_1$, $e_2$ and $e_3$,
 the three scale factors:
-\begin{equation} \label{eq:scale_factors}
+\begin{equation}
+  \label{eq:scale_factors}
   \begin{aligned}
     e_1 &=\left( {a+z} \right)\;\left[ {\left( {\frac{\partial \lambda}{\partial i}\cos \varphi } \right)^2
@@ -346,36 +358,42 @@
 (\autoref{eq:PE_dyn} to \autoref{eq:PE_eos}) can be written in the tensorial form,
 invariant in any orthogonal horizontal curvilinear coordinate system transformation:
-\begin{subequations} \label{eq:PE_discrete_operators}
-\begin{equation} \label{eq:PE_grad}
-\nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf 
-i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3 
-}\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\
-\end{equation}
-\begin{equation} \label{eq:PE_div}
-\nabla \cdot {\rm {\bf A}} 
-= \frac{1}{e_1 \; e_2} \left[ 
-  \frac{\partial \left(e_2 \; a_1\right)}{\partial i }
-+\frac{\partial \left(e_1 \; a_2\right)}{\partial j }    	\right]
-+ \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right]
-\end{equation}
-\begin{equation} \label{eq:PE_curl}
-   \begin{split}
-\nabla \times \vect{A} = 
-    \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j}
-            -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i}
-&+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k}
-           -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} 		\\
-&+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i}
-                                       -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k} 
-   \end{split}
-\end{equation}
-\begin{equation} \label{eq:PE_lap}
-\Delta q = \nabla \cdot \left(  \nabla q \right)
-\end{equation}
-\begin{equation} \label{eq:PE_lap_vector}
-\Delta {\rm {\bf A}} =
-  \nabla \left( \nabla \cdot {\rm {\bf A}} \right)
-- \nabla \times \left(  \nabla \times {\rm {\bf A}} \right)
-\end{equation}
+\begin{subequations}
+  % \label{eq:PE_discrete_operators}
+  \begin{equation}
+    \label{eq:PE_grad}
+    \nabla q=\frac{1}{e_1 }\frac{\partial q}{\partial i}\;{\rm {\bf
+        i}}+\frac{1}{e_2 }\frac{\partial q}{\partial j}\;{\rm {\bf j}}+\frac{1}{e_3
+    }\frac{\partial q}{\partial k}\;{\rm {\bf k}}    \\
+  \end{equation}
+  \begin{equation}
+    \label{eq:PE_div}
+    \nabla \cdot {\rm {\bf A}}
+    = \frac{1}{e_1 \; e_2} \left[
+      \frac{\partial \left(e_2 \; a_1\right)}{\partial i }
+      +\frac{\partial \left(e_1 \; a_2\right)}{\partial j }    	\right]
+    + \frac{1}{e_3} \left[ \frac{\partial a_3}{\partial k }   \right]
+  \end{equation}
+  \begin{equation}
+    \label{eq:PE_curl}
+    \begin{split}
+      \nabla \times \vect{A} =
+      \left[ {\frac{1}{e_2 }\frac{\partial a_3}{\partial j}
+          -\frac{1}{e_3 }\frac{\partial a_2 }{\partial k}} \right] \; \vect{i}
+      &+\left[ {\frac{1}{e_3 }\frac{\partial a_1 }{\partial k}
+          -\frac{1}{e_1 }\frac{\partial a_3 }{\partial i}} \right] \; \vect{j} 		\\
+      &+\frac{1}{e_1 e_2 } \left[ {\frac{\partial \left( {e_2 a_2 } \right)}{\partial i}
+          -\frac{\partial \left( {e_1 a_1 } \right)}{\partial j}} \right] \; \vect{k}
+    \end{split}
+  \end{equation}
+  \begin{equation}
+    \label{eq:PE_lap}
+    \Delta q = \nabla \cdot \left(  \nabla q \right)
+  \end{equation}
+  \begin{equation}
+    \label{eq:PE_lap_vector}
+    \Delta {\rm {\bf A}} =
+    \nabla \left( \nabla \cdot {\rm {\bf A}} \right)
+    - \nabla \times \left(  \nabla \times {\rm {\bf A}} \right)
+  \end{equation}
 \end{subequations}
 where $q$ is a scalar quantity and ${\rm {\bf A}}=(a_1,a_2,a_3)$ a vector in the $(i,j,k)$ coordinate system.
@@ -392,13 +410,15 @@
 Let us set $\vect U=(u,v,w)={\vect{U}}_h +w\;\vect{k}$, the velocity in the $(i,j,k)$ coordinate system and
 define the relative vorticity $\zeta$ and the divergence of the horizontal velocity field $\chi$, by:
-\begin{equation} \label{eq:PE_curl_Uh}
-\zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v} 
-\right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}} 
-\right]
+\begin{equation}
+  \label{eq:PE_curl_Uh}
+  \zeta =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,v}
+        \right)}{\partial i}-\frac{\partial \left( {e_1 \,u} \right)}{\partial j}}
+  \right]
 \end{equation}
-\begin{equation} \label{eq:PE_div_Uh}
-\chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u} 
-\right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}} 
-\right]
+\begin{equation}
+  \label{eq:PE_div_Uh}
+  \chi =\frac{1}{e_1 e_2 }\left[ {\frac{\partial \left( {e_2 \,u}
+        \right)}{\partial i}+\frac{\partial \left( {e_1 \,v} \right)}{\partial j}}
+  \right]
 \end{equation}
 
@@ -407,49 +427,60 @@
 the nonlinear term of \autoref{eq:PE_dyn} can be transformed as follows:
 \begin{flalign*}
-&\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
-+\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h 			&
+  &\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
+      +\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h 			&
 \end{flalign*}
 \begin{flalign*}
-&\qquad=\left( {{\begin{array}{*{20}c}
- {\left[ 	{	 \frac{1}{e_3}	\frac{\partial u	}{\partial k}
-			-\frac{1}{e_1}	\frac{\partial w	}{\partial i} } \right] w - \zeta \; v } 	 	\\
- 		{\zeta \; u - \left[ {	 \frac{1}{e_2} \frac{\partial w}{\partial j}
-							-\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}	 \\
-		 \end{array} }} \right)			
-+\frac{1}{2}	\left( {{\begin{array}{*{20}c}
-		 {	\frac{1}{e_1}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} 	\hfill	 \\
-		 {	\frac{1}{e_2}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} 	\hfill	 \\
-		 \end{array} }} \right)			&
+  &\qquad=\left( {{
+        \begin{array}{*{20}c}
+          {\left[ 	{	 \frac{1}{e_3}	\frac{\partial u	}{\partial k}
+          -\frac{1}{e_1}	\frac{\partial w	}{\partial i} } \right] w - \zeta \; v } 	 	\\
+          {\zeta \; u - \left[ {	 \frac{1}{e_2} \frac{\partial w}{\partial j}
+          -\frac{1}{e_3} \frac{\partial v}{\partial k} } \right] \ w}	 \\
+        \end{array}
+      }} \right)
+  +\frac{1}{2}	\left( {{
+        \begin{array}{*{20}c}
+          {	\frac{1}{e_1}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial i}} 	\hfill	 \\
+          {	\frac{1}{e_2}	\frac{\partial \left( u^2+v^2+w^2 \right)}{\partial j}} 	\hfill	 \\
+        \end{array}
+      }} \right)			&
 \end{flalign*}
 \begin{flalign*}
-& \qquad =\left( {{	\begin{array}{*{20}c}
- {-\zeta \; v} \hfill \\
- { \zeta \; u} \hfill \\
-			\end{array} }} \right)
-+\frac{1}{2}\left( {{	\begin{array}{*{20}c}
- {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill 	\\
- {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill 	\\
-						\end{array} }} \right)			
-+\frac{1}{e_3 }\left( {{		\begin{array}{*{20}c}
- { w \; \frac{\partial u}{\partial k}} 	\\
- { w \; \frac{\partial v}{\partial k}} 	\\
-							\end{array} }} \right)	
--\left( {{	\begin{array}{*{20}c}
- {\frac{w}{e_1}\frac{\partial w}{\partial i}
- -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\
- {\frac{w}{e_2}\frac{\partial w}{\partial j}
-  -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\
-			\end{array} }} \right) 			&
+  & \qquad =\left( {{
+        \begin{array}{*{20}c}
+          {-\zeta \; v} \hfill \\
+          { \zeta \; u} \hfill \\
+        \end{array}
+      }} \right)
+  +\frac{1}{2}\left( {{
+        \begin{array}{*{20}c}
+          {\frac{1}{e_1 }\frac{\partial \left( {u^2+v^2} \right)}{\partial i}} \hfill 	\\
+          {\frac{1}{e_2 }\frac{\partial \left( {u^2+v^2} \right)}{\partial j}} \hfill 	\\
+        \end{array}
+      }} \right)
+  +\frac{1}{e_3 }\left( {{
+        \begin{array}{*{20}c}
+          { w \; \frac{\partial u}{\partial k}} 	\\
+          { w \; \frac{\partial v}{\partial k}} 	\\
+        \end{array}
+      }} \right)
+  -\left( {{
+        \begin{array}{*{20}c}
+          {\frac{w}{e_1}\frac{\partial w}{\partial i} -\frac{1}{2e_1}\frac{\partial w^2}{\partial i}} \hfill \\
+          {\frac{w}{e_2}\frac{\partial w}{\partial j} -\frac{1}{2e_2}\frac{\partial w^2}{\partial j}} \hfill \\
+        \end{array}
+      }} \right) 			&
 \end{flalign*}
 
 The last term of the right hand side is obviously zero, and thus the nonlinear term of
 \autoref{eq:PE_dyn} is written in the $(i,j,k)$ coordinate system:
-\begin{equation} \label{eq:PE_vector_form}
-\left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
-+\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h 
-=\zeta 
-\;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm 
-{\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h 
-}{\partial k}   	
+\begin{equation}
+  \label{eq:PE_vector_form}
+  \left[ {\left( {	\nabla \times {\rm {\bf U}}    } \right) \times {\rm {\bf U}}
+      +\frac{1}{2}	\nabla \left( {{\rm {\bf U}}^2} \right)}	 \right]_h
+  =\zeta
+  \;{\rm {\bf k}}\times {\rm {\bf U}}_h +\frac{1}{2}\nabla _h \left( {{\rm
+        {\bf U}}_h^2 } \right)+\frac{1}{e_3 }w\frac{\partial {\rm {\bf U}}_h
+  }{\partial k}
 \end{equation}
 
@@ -459,85 +490,98 @@
 For example, the first component of \autoref{eq:PE_vector_form} (the $i$-component) is transformed as follows:
 \begin{flalign*}
-&{	\begin{array}{*{20}l}
-\left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
-          +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\
-%\\
-     = - \zeta \;v 
-     + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
-     + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} 			\\
-\\
-\qquad =\frac{1}{e_1 \; e_2} \left( 	-v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
-							+v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}	 \right)
-+\frac{1}{e_1 e_2 }\left( 	+e_2 \; u\frac{\partial u}{\partial i}
- 							+e_2 \; v\frac{\partial v}{\partial i}	  				 \right) 
-+\frac{1}{e_3}       \left( 	w\;\frac{\partial u}{\partial k} 		\right)   \\
-\end{array} } 			&
+  &{
+    \begin{array}{*{20}l}
+      \left[ {\left( {\nabla \times \vect{U}} \right)\times \vect{U}
+      +\frac{1}{2}\nabla \left( {\vect{U}}^2 \right)} \right]_i   % \\
+  % \\
+      = - \zeta \;v
+      + \frac{1}{2\;e_1 } \frac{\partial \left( {u^2+v^2} \right)}{\partial i}
+      + \frac{1}{e_3}w \ \frac{\partial u}{\partial k} 			\\ \\
+      \qquad =\frac{1}{e_1 \; e_2} \left( 	-v\frac{\partial \left( {e_2 \,v} \right)}{\partial i}
+      +v\frac{\partial \left( {e_1 \,u} \right)}{\partial j}	 \right)
+      +\frac{1}{e_1 e_2 }\left( 	+e_2 \; u\frac{\partial u}{\partial i}
+      +e_2 \; v\frac{\partial v}{\partial i}	  				 \right)
+      +\frac{1}{e_3}       \left( 	w\;\frac{\partial u}{\partial k} 		\right)   \\
+    \end{array}
+  } 			&
 \end{flalign*}
 \begin{flalign*}
-&{	\begin{array}{*{20}l}
-\qquad =\frac{1}{e_1 \; e_2} 	\left\{ 
- -\left(   	    v^2 	\frac{\partial e_2                                }{\partial i} 
+  &{
+    \begin{array}{*{20}l}
+      \qquad =\frac{1}{e_1 \; e_2} 	\left\{
+      -\left(   	    v^2 	\frac{\partial e_2                                }{\partial i}
  		+e_2 \,v		\frac{\partial v                                   }{\partial i}  	\right)
-+\left( 				\frac{\partial \left( {e_1 \,u\,v} 	\right)}{\partial j}
-		-e_1 \,u		\frac{\partial v                                   }{\partial j} 	\right) 	\right. 
-\\  \left.	\qquad \qquad \quad
-+\left( 				\frac{\partial \left( {e_2 u\,u}     \right)}{\partial i}
+      +\left( 				\frac{\partial \left( {e_1 \,u\,v} 	\right)}{\partial j}
+		-e_1 \,u		\frac{\partial v                                   }{\partial j} 	\right) 	\right. \\
+      \left.	\qquad \qquad \quad
+      +\left( 				\frac{\partial \left( {e_2 u\,u}     \right)}{\partial i}
 		-u			\frac{\partial \left( {e_2 u}         \right)}{\partial i} 	\right)
-+e_2 v		 		\frac{\partial v                                    }{\partial i}
- 						\right\} 
-+\frac{1}{e_3} \left(
- 					\frac{\partial \left( {w\,u} \right)         }{\partial k}
-		 -u			\frac{\partial w 						   }{\partial k} 	\right) \\
-\end{array} } 		&
+      +e_2 v		 		\frac{\partial v                                    }{\partial i}
+      \right\}
+      +\frac{1}{e_3} \left(
+      \frac{\partial \left( {w\,u} \right)         }{\partial k}
+      -u			\frac{\partial w 						   }{\partial k} 	\right) \\
+    \end{array}
+  } 		&
 \end{flalign*}
 \begin{flalign*}
-&{	\begin{array}{*{20}l}
-\qquad =\frac{1}{e_1 \; e_2} 	\left( 
-					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+  &
+  {
+    \begin{array}{*{20}l}
+      \qquad =\frac{1}{e_1 \; e_2} 	\left(
+      \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
 		+			\frac{\partial \left( {e_1 \,u\,v} \right)}{\partial j}	\right)
-+\frac{1}{e_3 }		\frac{\partial \left( {w\,u		 } \right)}{\partial k}
-\\  \qquad \qquad \quad
-+\frac{1}{e_1 e_2 }		\left( 
+      +\frac{1}{e_3 }		\frac{\partial \left( {w\,u		 } \right)}{\partial k} \\
+      \qquad \qquad \quad
+      +\frac{1}{e_1 e_2 }		\left(
  		-u	\left( 	\frac{\partial \left( {e_1 v	 } \right)}{\partial j}
-	 				-v\,\frac{\partial e_1 }{\partial j}				 \right)
+      -v\,\frac{\partial e_1 }{\partial j}				 \right)
 		-u			\frac{\partial \left( {e_2 u	 } \right)}{\partial i}
-						\right)
- -\frac{1}{e_3 }	 	\frac{\partial w}{\partial k}	u
- +\frac{1}{e_1 e_2 }\left( 	-v^2\frac{\partial e_2	 }{\partial i}		 \right)	
-\end{array} } 		&
+      \right)
+      -\frac{1}{e_3 }	 	\frac{\partial w}{\partial k}	u
+      +\frac{1}{e_1 e_2 }\left( 	-v^2\frac{\partial e_2	 }{\partial i}		 \right)
+    \end{array}
+  } 		&
 \end{flalign*}
 \begin{flalign*}
-&{	\begin{array}{*{20}l}
-\qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u}	\right)
--   \left( \nabla \cdot {\rm {\bf U}} \right) \ u
-+\frac{1}{e_1 e_2 }\left( 
+  &{
+    \begin{array}{*{20}l}
+      \qquad = \nabla \cdot \left( {{\rm {\bf U}}\,u}	\right)
+      -   \left( \nabla \cdot {\rm {\bf U}} \right) \ u
+      +\frac{1}{e_1 e_2 }\left(
 		-v^2		\frac{\partial e_2 }{\partial i}
 		+uv	\,		\frac{\partial e_1 }{\partial j} 	\right) \\
-\end{array} } 		&
+    \end{array}
+  } 		&
 \end{flalign*}
 as $\nabla \cdot {\rm {\bf U}}\;=0$ (incompressibility) it comes:
 \begin{flalign*}
-&{	\begin{array}{*{20}l}
-\qquad = \nabla \cdot \left( 	{{\rm {\bf U}}\,u}		\right)
-+	\frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i}
-						       -u \; \frac{\partial e_1}{\partial j} 	\right)  \left( -v \right)	
-\end{array} } 		&
+  &{
+    \begin{array}{*{20}l}
+      \qquad = \nabla \cdot \left( 	{{\rm {\bf U}}\,u}		\right)
+      +	\frac{1}{e_1 e_2 }   \left( v \; \frac{\partial e_2}{\partial i}
+      -u \; \frac{\partial e_1}{\partial j} 	\right)  \left( -v \right)
+    \end{array}
+  } 		&
 \end{flalign*}
 
 The flux form of the momentum advection term is therefore given by:
-\begin{multline} \label{eq:PE_flux_form}
-		\left[ 
-  \left( 	{\nabla \times {\rm {\bf U}}} 	\right) \times {\rm {\bf U}}
-+\frac{1}{2}	\nabla \left( 	{{\rm {\bf U}}^2} 	\right)
-		\right]_h 
-\\
-= \nabla \cdot 	\left( {{\begin{array}{*{20}c}	{\rm {\bf U}} \, u 	\hfill \\
-												{\rm {\bf U}} \, v	\hfill \\
-						\end{array} }} 	
-				\right)
-+\frac{1}{e_1 e_2 }		\left( 
-		 v\frac{\partial e_2}{\partial i}
-		-u\frac{\partial e_1}{\partial j} 
-						\right) {\rm {\bf k}} \times {\rm {\bf U}}_h
+\begin{multline}
+  \label{eq:PE_flux_form}
+  \left[
+    \left( 	{\nabla \times {\rm {\bf U}}} 	\right) \times {\rm {\bf U}}
+    +\frac{1}{2}	\nabla \left( 	{{\rm {\bf U}}^2} 	\right)
+  \right]_h \\
+  = \nabla \cdot 	\left( {{
+        \begin{array}{*{20}c}
+          {\rm {\bf U}} \, u 	\hfill \\
+          {\rm {\bf U}} \, v	\hfill \\
+        \end{array}
+      }}
+  \right)
+  +\frac{1}{e_1 e_2 }		\left(
+    v\frac{\partial e_2}{\partial i}
+    -u\frac{\partial e_1}{\partial j}
+  \right) {\rm {\bf k}} \times {\rm {\bf U}}_h
 \end{multline}
 
@@ -546,8 +590,9 @@
 and the second one is due to the curvilinear nature of the coordinate system used.
 The latter is called the \emph{metric} term and can be viewed as a modification of the Coriolis parameter: 
-\begin{equation} \label{eq:PE_cor+metric}
-f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i}
-					         -u \frac{\partial e_1}{\partial j}  \right)
-\end{equation}
+\[
+  % \label{eq:PE_cor+metric}
+  f \to f + \frac{1}{e_1\;e_2}  \left(  v \frac{\partial e_2}{\partial i}
+    -u \frac{\partial e_1}{\partial j}  \right)
+\]
 
 Note that in the case of geographical coordinate,
@@ -555,7 +600,4 @@
 we recover the commonly used modification of the Coriolis parameter $f \to f+(u/a) \tan \varphi$.
 
-
-$\ $\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:
@@ -564,20 +606,23 @@
 $\bullet$ \textbf{Vector invariant form of the momentum equations} :
 
-\begin{subequations} \label{eq:PE_dyn_vect}
-\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}
+\begin{subequations}
+  \label{eq:PE_dyn_vect}
+  \[
+    % \label{eq:PE_dyn_vect_u}
+    \begin{split}
+      \frac{\partial u}{\partial t}
+      = +   \left( {\zeta +f} \right)\,v
+      -   \frac{1}{2\,e_1}           \frac{\partial}{\partial i} \left(  u^2+v^2   \right)
+      -   \frac{1}{e_3    }  w     \frac{\partial u}{\partial k}      &      \\
+      -   \frac{1}{e_1    }            \frac{\partial}{\partial i} \left( \frac{p_s+p_h }{\rho_o}    \right)
+      &+   D_u^{\vect{U}}  +   F_u^{\vect{U}}      \\ \\
+      \frac{\partial v}{\partial t} =
+      -   \left( {\zeta +f} \right)\,u
+      -   \frac{1}{2\,e_2 }        \frac{\partial }{\partial j}\left(  u^2+v^2  \right)
+      -   \frac{1}{e_3     }   w  \frac{\partial v}{\partial k}     &      \\
+      -   \frac{1}{e_2     }        \frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)
+      &+  D_v^{\vect{U}}  +   F_v^{\vect{U}}
+    \end{split}
+  \]
 \end{subequations}
 
@@ -585,63 +630,72 @@
 \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}
-\frac{\partial u}{\partial t}=
-+   \left( { f + \frac{1}{e_1 \; e_2}
-					\left( 	 v \frac{\partial e_2}{\partial i}
-						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
-- \frac{1}{e_1 \; e_2} 	\left( 
-					\frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
+\begin{subequations}
+  % \label{eq:PE_dyn_flux}
+  \begin{multline*}
+    % \label{eq:PE_dyn_flux_u}
+    \frac{\partial u}{\partial t}=
+    +   \left( { f + \frac{1}{e_1 \; e_2}
+        \left( 	 v \frac{\partial e_2}{\partial i}
+          -u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
+    - \frac{1}{e_1 \; e_2} 	\left(
+      \frac{\partial \left( {e_2 \,u\,u} \right)}{\partial i}
 		+			\frac{\partial \left( {e_1 \,v\,u} \right)}{\partial j}	\right)
-                 - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\
--   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho _o}   \right)
-+   D_u^{\vect{U}} +   F_u^{\vect{U}}
-\end{multline}
-\begin{multline} \label{eq:PE_dyn_flux_v}
-\frac{\partial v}{\partial t}=
--   \left( { f + \frac{1}{e_1 \; e_2}
-					\left( 	 v \frac{\partial e_2}{\partial i}
-						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
- \frac{1}{e_1 \; e_2} 	\left( 
-					\frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
+    - \frac{1}{e_3 }\frac{\partial \left( {         w\,u} \right)}{\partial k}    \\
+    -   \frac{1}{e_1 }\frac{\partial}{\partial i}\left( \frac{p_s+p_h }{\rho_o}   \right)
+    +   D_u^{\vect{U}} +   F_u^{\vect{U}}
+  \end{multline*}
+  \begin{multline*}
+    % \label{eq:PE_dyn_flux_v}
+    \frac{\partial v}{\partial t}=
+    -   \left( { f + \frac{1}{e_1 \; e_2}
+        \left( 	 v \frac{\partial e_2}{\partial i}
+          -u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
+    \frac{1}{e_1 \; e_2} 	\left(
+      \frac{\partial \left( {e_2 \,u\,v} \right)}{\partial i}
 		+			\frac{\partial \left( {e_1 \,v\,v} \right)}{\partial j}	\right)
-                 - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\
--   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}    \right)
-+  D_v^{\vect{U}} +  F_v^{\vect{U}} 
-\end{multline}
+    - \frac{1}{e_3 } \frac{\partial \left( {        w\,v} \right)}{\partial k}    \\
+    -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}    \right)
+    +  D_v^{\vect{U}} +  F_v^{\vect{U}}
+  \end{multline*}
 \end{subequations}
 where $\zeta$, the relative vorticity, is given by \autoref{eq:PE_curl_Uh} and
 $p_s $, the surface pressure, is given by:
-\begin{equation} \label{eq:PE_spg}
-p_s =  \rho \,g \,\eta 
-\end{equation}
+\[
+  % \label{eq:PE_spg}
+  p_s =  \rho \,g \,\eta
+\]
 with $\eta$ is solution of \autoref{eq:PE_ssh}.
 
 The vertical velocity and the hydrostatic pressure are diagnosed from the following equations:
-\begin{equation} \label{eq:w_diag}
-\frac{\partial w}{\partial k}=-\chi \;e_3 
-\end{equation}
-\begin{equation} \label{eq:hp_diag}
-\frac{\partial p_h }{\partial k}=-\rho \;g\;e_3 
-\end{equation}
+\[
+  % \label{eq:w_diag}
+  \frac{\partial w}{\partial k}=-\chi \;e_3
+\]
+\[
+  % \label{eq:hp_diag}
+  \frac{\partial p_h }{\partial k}=-\rho \;g\;e_3
+\]
 where the divergence of the horizontal velocity, $\chi$ is given by \autoref{eq:PE_div_Uh}.
 
 \vspace{+10pt}
 $\bullet$ \textit{tracer equations} :
-\begin{equation} \label{eq:S}
-\frac{\partial T}{\partial t} = 
--\frac{1}{e_1 e_2 }\left[ {	   \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
-						+\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
--\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
-\end{equation}
-\begin{equation} \label{eq:T}
-\frac{\partial S}{\partial t} = 
--\frac{1}{e_1 e_2 }\left[ 	  {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
-						+\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
--\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
-\end{equation}
-\begin{equation} \label{eq:rho}
-\rho =\rho \left( {T,S,z(k)} \right)
-\end{equation}
+\[
+  % \label{eq:S}
+  \frac{\partial T}{\partial t} =
+  -\frac{1}{e_1 e_2 }\left[ {	   \frac{\partial \left( {e_2 T\,u} \right)}{\partial i}
+      +\frac{\partial \left( {e_1 T\,v} \right)}{\partial j}} \right]
+  -\frac{1}{e_3 }\frac{\partial \left( {T\,w} \right)}{\partial k} + D^T + F^T
+\]
+\[
+  % \label{eq:T}
+  \frac{\partial S}{\partial t} =
+  -\frac{1}{e_1 e_2 }\left[ 	  {\frac{\partial \left( {e_2 S\,u} \right)}{\partial i}
+      +\frac{\partial \left( {e_1 S\,v} \right)}{\partial j}} \right]
+  -\frac{1}{e_3 }\frac{\partial \left( {S\,w} \right)}{\partial k} + D^S + F^S
+\]
+\[
+  % \label{eq:rho}
+  \rho =\rho \left( {T,S,z(k)} \right)
+\]
 
 The expression of \textbf{D}$^{U}$, $D^{S}$ and $D^{T}$ depends on the subgrid scale parameterisation used.
@@ -652,5 +706,5 @@
 
 \newpage 
-$\ $\newline    % force a new ligne
+
 % ================================================================
 % Curvilinear generalised vertical coordinate System
@@ -680,6 +734,7 @@
 In fact one is totally free to choose any space and time vertical coordinate by
 introducing an arbitrary vertical coordinate :
-\begin{equation} \label{eq:PE_s}
-s=s(i,j,k,t)
+\begin{equation}
+  \label{eq:PE_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 $k$,
@@ -750,14 +805,16 @@
 Let us define the vertical scale factor by $e_3=\partial_s z$  ($e_3$ is now a function of $(i,j,k,t)$ ),
 and the slopes in the (\textbf{i},\textbf{j}) directions between $s-$ and $z-$surfaces by:
-\begin{equation} \label{eq:PE_sco_slope}
-\sigma _1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s 
-\quad \text{, and } \quad 
-\sigma _2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
+\begin{equation}
+  \label{eq:PE_sco_slope}
+  \sigma_1 =\frac{1}{e_1 }\;\left. {\frac{\partial z}{\partial i}} \right|_s
+  \quad \text{, and } \quad
+  \sigma_2 =\frac{1}{e_2 }\;\left. {\frac{\partial z}{\partial j}} \right|_s
 \end{equation}
 We also introduce  $\omega $, a dia-surface velocity component, defined as the velocity 
 relative to the moving $s$-surfaces and normal to them:
-\begin{equation} \label{eq:PE_sco_w}
-\omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma _1 \,u - \sigma _2 \,v    \\
-\end{equation}
+\[
+  % \label{eq:PE_sco_w}
+  \omega  = w - e_3 \, \frac{\partial s}{\partial t} - \sigma_1 \,u - \sigma_2 \,v    \\
+\]
 
 The equations solved by the ocean model \autoref{eq:PE} in $s-$coordinate can be written as follows
@@ -766,51 +823,55 @@
  \vspace{0.5cm}
 $\bullet$ Vector invariant form of the momentum equation :
-\begin{multline} \label{eq:PE_sco_u_vector}
-\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 
-	+   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
-\end{multline}
-\begin{multline} \label{eq:PE_sco_v_vector}
-\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   
-	+  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
-\end{multline}
+\begin{multline*}
+  % \label{eq:PE_sco_u_vector}
+  \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
+  +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
+\end{multline*}
+\begin{multline*}
+  % \label{eq:PE_sco_v_vector}
+  \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
+  +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
+\end{multline*}
 
  \vspace{0.5cm}
 $\bullet$ Flux form of the momentum equation :
-\begin{multline} \label{eq:PE_sco_u_flux}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
-   +   \left( { f + \frac{1}{e_1 \; e_2 }
-					\left( 	 v \frac{\partial e_2}{\partial i}
-						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
-   - \frac{1}{e_1 \; e_2 \; e_3 } 	\left( 
-					\frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i}
-		+			\frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}	\right)
-   - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\
-	- \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho _o}    \right)    
-	+  g\frac{\rho }{\rho _o}\sigma _1 
-	+   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
-\end{multline}
-\begin{multline} \label{eq:PE_sco_v_flux}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
-   -   \left( { f + \frac{1}{e_1 \; e_2}
-					\left( 	 v \frac{\partial e_2}{\partial i}
-						-u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
-   - \frac{1}{e_1 \; e_2 \; e_3 } 	\left( 
-					\frac{\partial \left( {e_2 \; e_3  \,u\,v} \right)}{\partial i}
-		+			\frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}	\right)
-                 - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\
-	-   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho _o}  \right) 
-	 +  g\frac{\rho }{\rho _o }\sigma _2   
-	+  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
-\end{multline}
+\begin{multline*}
+  % \label{eq:PE_sco_u_flux}
+  \frac{1}{e_3} \frac{\partial \left(  e_3\,u  \right) }{\partial t}=
+  +   \left( { f + \frac{1}{e_1 \; e_2 }
+      \left( 	 v \frac{\partial e_2}{\partial i}
+        -u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, v    \\
+  - \frac{1}{e_1 \; e_2 \; e_3 } 	\left(
+    \frac{\partial \left( {e_2 \, e_3 \, u\,u} \right)}{\partial i}
+    +			\frac{\partial \left( {e_1 \, e_3 \, v\,u} \right)}{\partial j}	\right)
+  - \frac{1}{e_3 }\frac{\partial \left( { \omega\,u} \right)}{\partial k}    \\
+  - \frac{1}{e_1} \frac{\partial}{\partial i} \left( \frac{p_s + p_h}{\rho_o}    \right)
+  +  g\frac{\rho }{\rho_o}\sigma_1
+  +   D_u^{\vect{U}}  +   F_u^{\vect{U}} \quad
+\end{multline*}
+\begin{multline*}
+  % \label{eq:PE_sco_v_flux}
+  \frac{1}{e_3} \frac{\partial \left(  e_3\,v  \right) }{\partial t}=
+  -   \left( { f + \frac{1}{e_1 \; e_2}
+      \left( 	 v \frac{\partial e_2}{\partial i}
+        -u \frac{\partial e_1}{\partial j} 	\right)} 	\right) \, u   \\
+  - \frac{1}{e_1 \; e_2 \; e_3 } 	\left(
+    \frac{\partial \left( {e_2 \; e_3  \,u\,v} \right)}{\partial i}
+    +			\frac{\partial \left( {e_1 \; e_3  \,v\,v} \right)}{\partial j}	\right)
+  - \frac{1}{e_3 } \frac{\partial \left( { \omega\,v} \right)}{\partial k}    \\
+  -   \frac{1}{e_2 }\frac{\partial }{\partial j}\left( \frac{p_s+p_h }{\rho_o}  \right)
+  +  g\frac{\rho }{\rho_o }\sigma_2
+  +  D_v^{\vect{U}}  +   F_v^{\vect{U}} \quad
+\end{multline*}
 
 where the relative vorticity, \textit{$\zeta $}, the surface pressure gradient,
@@ -818,27 +879,30 @@
 they do not represent exactly the same quantities.
 $\omega$ is provided by the continuity equation (see \autoref{apdx:A}):
-\begin{equation} \label{eq:PE_sco_continuity}
-\frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0   
-\qquad \text{with }\;\;  
-\chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u} 
-\right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial 
-j}} \right]
-\end{equation}
+\[
+  % \label{eq:PE_sco_continuity}
+  \frac{\partial e_3}{\partial t} + e_3 \; \chi + \frac{\partial \omega }{\partial s} = 0
+  \qquad \text{with }\;\;
+  \chi =\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3 \,u}
+        \right)}{\partial i}+\frac{\partial \left( {e_1 e_3 \,v} \right)}{\partial
+        j}} \right]
+\]
 
  \vspace{0.5cm}
 $\bullet$ tracer equations:
-\begin{multline} \label{eq:PE_sco_t}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}=
--\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
-                                           +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\
--\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad
-\end{multline}
-
-\begin{multline} \label{eq:PE_sco_s}
-\frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}=
--\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
-                                           +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\
--\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad
-\end{multline}
+\begin{multline*}
+  % \label{eq:PE_sco_t}
+  \frac{1}{e_3} \frac{\partial \left(  e_3\,T  \right) }{\partial t}=
+  -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,T} \right)}{\partial i}
+      +\frac{\partial \left( {e_1 e_3\,v\,T} \right)}{\partial j}} \right]   \\
+  -\frac{1}{e_3 }\frac{\partial \left( {T\,\omega } \right)}{\partial k}   + D^T + F^S   \qquad
+\end{multline*}
+
+\begin{multline*}
+  % \label{eq:PE_sco_s}
+  \frac{1}{e_3} \frac{\partial \left(  e_3\,S  \right) }{\partial t}=
+  -\frac{1}{e_1 e_2 e_3 }\left[ {\frac{\partial \left( {e_2 e_3\,u\,S} \right)}{\partial i}
+      +\frac{\partial \left( {e_1 e_3\,v\,S} \right)}{\partial j}} \right]    \\
+  -\frac{1}{e_3 }\frac{\partial \left( {S\,\omega } \right)}{\partial k}     + D^S + F^S   \qquad
+\end{multline*}
 
 The equation of state has the same expression as in $z$-coordinate,
@@ -889,15 +953,16 @@
 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} 
+\[
+  % \label{eq:z-star}
+  H +  \textit{z*} = (H + z) / r \quad \text{and} \ \delta \textit{z*} = \delta z / r \quad \text{with} \ r = \frac{H+\eta} {H}
+\] 
 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*} 
+\[ 
+  \frac{\partial r}{\partial t} = \nabla_{\textit{z*}} \cdot \left( r \; \rm{\bf U}_h \right)
+  \left( r \; w\textit{*} \right) = 0 
+\] 
 %}
 
@@ -906,7 +971,8 @@
 
 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
-\begin{equation} \label{eq:PE_}
-	z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
-\end{equation}
+\[
+  % \label{eq:PE_}
+  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\]
 
 This coordinate is closely related to the "eta" coordinate used in many atmospheric models
@@ -937,5 +1003,5 @@
 
 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. 
+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.
@@ -951,5 +1017,6 @@
 
 
-\newpage 
+\newpage
+
 % -------------------------------------------------------------------------------------------------------------
 % Terrain following  coordinate System
@@ -994,7 +1061,8 @@
 The horizontal pressure force in $s$-coordinate consists of two terms (see \autoref{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 
+\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}
 
@@ -1041,5 +1109,6 @@
 
 
-\newpage 
+\newpage
+
 % -------------------------------------------------------------------------------------------------------------
 % Curvilinear z-tilde coordinate System
@@ -1055,4 +1124,5 @@
 
 \newpage 
+
 % ================================================================
 % Subgrid Scale Physics
@@ -1097,11 +1167,12 @@
 while an accurate consideration of the details of turbulent motions is simply impractical.
 The resulting vertical momentum and tracer diffusive operators are of second order:
-\begin{equation} \label{eq:PE_zdf}
-   \begin{split}
-{\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\         
-D^{vT}                         &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,
-\quad
-D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right)
-   \end{split}
+\begin{equation}
+  \label{eq:PE_zdf}
+  \begin{split}
+    {\vect{D}}^{v \vect{U}} &=\frac{\partial }{\partial z}\left( {A^{vm}\frac{\partial {\vect{U}}_h }{\partial z}} \right) \ , \\
+    D^{vT}                         &= \frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial T}{\partial z}} \right) \ ,
+    \quad
+    D^{vS}=\frac{\partial }{\partial z}\left( {A^{vT}\frac{\partial S}{\partial z}} \right)
+  \end{split}
 \end{equation}
 where $A^{vm}$ and $A^{vT}$ are the vertical eddy viscosity and diffusivity coefficients, respectively.
@@ -1173,11 +1244,14 @@
 
 The lateral Laplacian tracer diffusive operator is defined by (see \autoref{apdx:B}):
-\begin{equation} \label{eq:PE_iso_tensor}
-D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad 
-\mbox{with}\quad \;\;\Re =\left( {{\begin{array}{*{20}c}
- 1 \hfill & 0 \hfill & {-r_1 } \hfill \\
- 0 \hfill & 1 \hfill & {-r_2 } \hfill \\
- {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
-\end{array} }} \right)
+\begin{equation}
+  \label{eq:PE_iso_tensor}
+  D^{lT}=\nabla {\rm {\bf .}}\left( {A^{lT}\;\Re \;\nabla T} \right) \qquad
+  \mbox{with}\quad \;\;\Re =\left( {{
+        \begin{array}{*{20}c}
+          1 \hfill & 0 \hfill & {-r_1 } \hfill \\
+          0 \hfill & 1 \hfill & {-r_2 } \hfill \\
+          {-r_1 } \hfill & {-r_2 } \hfill & {r_1 ^2+r_2 ^2} \hfill \\
+        \end{array}
+      }} \right)
 \end{equation}
 where $r_1 \;\mbox{and}\;r_2 $ are the slopes between the surface along which the diffusive operator acts and
@@ -1197,12 +1271,13 @@
 For \textit{geopotential} diffusion,
 $r_1$ and $r_2 $ are the slopes between the geopotential and computational surfaces:
-they are equal to $\sigma _1$ and $\sigma _2$, respectively (see \autoref{eq:PE_sco_slope}).
+they are equal to $\sigma_1$ and $\sigma_2$, respectively (see \autoref{eq:PE_sco_slope}).
 
 For \textit{isoneutral} diffusion $r_1$ and $r_2$ are the slopes between the isoneutral and computational surfaces.
 Therefore, they are different quantities, but have similar expressions in $z$- and $s$-coordinates.
 In $z$-coordinates:
-\begin{equation} \label{eq:PE_iso_slopes}
-r_1 =\frac{e_3 }{e_1 }	\left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad
-r_2 =\frac{e_3 }{e_2 }	\left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \,
+\begin{equation}
+  \label{eq:PE_iso_slopes}
+  r_1 =\frac{e_3 }{e_1 }	\left( \pd[\rho]{i} \right) \left( \pd[\rho]{k} \right)^{-1} \, \quad
+  r_2 =\frac{e_3 }{e_2 }	\left( \pd[\rho]{j} \right) \left( \pd[\rho]{k} \right)^{-1} \,
 \end{equation}
 while in $s$-coordinates $\pd[]{k}$ is replaced by $\pd[]{s}$.
@@ -1211,29 +1286,33 @@
 When the \textit{eddy induced velocity} parametrisation (eiv) \citep{Gent1990} is used,
 an additional tracer advection is introduced in combination with the isoneutral diffusion of tracers:
-\begin{equation} \label{eq:PE_iso+eiv}
-D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
-           +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
-\end{equation}
+\[
+  % \label{eq:PE_iso+eiv}
+  D^{lT}=\nabla \cdot \left( {A^{lT}\;\Re \;\nabla T} \right)
+  +\nabla \cdot \left( {{\vect{U}}^\ast \,T} \right)
+\]
 where ${\vect{U}}^\ast =\left( {u^\ast ,v^\ast ,w^\ast } \right)$ is a non-divergent,
 eddy-induced transport velocity. This velocity field is defined by:
-\begin{equation} \label{eq:PE_eiv}
-   \begin{split}
- u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\ 
- v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\ 
- w^\ast &=  -\frac{1}{e_1 e_2 }\left[ 
-                      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
-                     +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
-   \end{split}
-\end{equation}
+\[
+  % \label{eq:PE_eiv}
+  \begin{split}
+    u^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_1 } \right] \\
+    v^\ast  &= +\frac{1}{e_3       }\frac{\partial }{\partial k}\left[ {A^{eiv}\;\tilde{r}_2 } \right] \\
+    w^\ast &=  -\frac{1}{e_1 e_2 }\left[
+      \frac{\partial }{\partial i}\left( {A^{eiv}\;e_2\,\tilde{r}_1 } \right)
+      +\frac{\partial }{\partial j}\left( {A^{eiv}\;e_1\,\tilde{r}_2 } \right)      \right]
+  \end{split}
+\]
 where $A^{eiv}$ is the eddy induced velocity coefficient
 (or equivalently the isoneutral thickness diffusivity coefficient),
 and $\tilde{r}_1$ and $\tilde{r}_2$ are the slopes between isoneutral and \emph{geopotential} surfaces.
 Their values are thus independent of the vertical coordinate, but their expression depends on the coordinate: 
-\begin{align} \label{eq:PE_slopes_eiv}
-\tilde{r}_n = \begin{cases}
-   r_n            &      \text{in $z$-coordinate}    \\
-   r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}  
-              \end{cases}
-\quad \text{where } n=1,2
+\begin{align}
+  \label{eq:PE_slopes_eiv}
+  \tilde{r}_n =
+  \begin{cases}
+    r_n            &      \text{in $z$-coordinate}    \\
+    r_n + \sigma_n &      \text{in \textit{z*} and $s$-coordinates}
+  \end{cases}
+                     \quad \text{where } n=1,2
 \end{align}
 
@@ -1246,8 +1325,9 @@
 
 The lateral bilaplacian tracer diffusive operator is defined by:
-\begin{equation} \label{eq:PE_bilapT}
-D^{lT}= - \Delta \left( \;\Delta T \right) 
-\qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
-\end{equation}
+\[
+  % \label{eq:PE_bilapT}
+  D^{lT}= - \Delta \left( \;\Delta T \right)
+  \qquad \text{where} \;\; \Delta \bullet = \nabla \left( {\sqrt{B^{lT}\,}\;\Re \;\nabla \bullet} \right)
+\]
 It is the Laplacian operator given by \autoref{eq:PE_iso_tensor} applied twice with
 the harmonic eddy diffusion coefficient set to the square root of the biharmonic one. 
@@ -1258,17 +1338,20 @@
 The Laplacian momentum diffusive operator along $z$- or $s$-surfaces is found by
 applying \autoref{eq:PE_lap_vector} to the horizontal velocity vector (see \autoref{apdx:B}):
-\begin{equation} \label{eq:PE_lapU}
-\begin{split}
-{\rm {\bf D}}^{l{\rm {\bf U}}} 
-&= \quad \  \nabla _h \left( {A^{lm}\chi } \right)
-   \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\
-&=   \left(      \begin{aligned}
-             \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i} 
-         &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\
-             \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}   
-         &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
-        \end{aligned}    \right)
-\end{split}
-\end{equation}
+\[
+  % \label{eq:PE_lapU}
+  \begin{split}
+    {\rm {\bf D}}^{l{\rm {\bf U}}}
+    &= \quad \  \nabla _h \left( {A^{lm}\chi } \right)
+    \ - \ \nabla _h \times \left( {A^{lm}\,\zeta \;{\rm {\bf k}}} \right)     \\
+    &=   \left(
+      \begin{aligned}
+        \frac{1}{e_1      } \frac{\partial \left( A^{lm} \chi          \right)}{\partial i}
+        &-\frac{1}{e_2 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial j}  \\
+        \frac{1}{e_2      }\frac{\partial \left( {A^{lm} \chi         } \right)}{\partial j}
+        &+\frac{1}{e_1 e_3}\frac{\partial \left( {A^{lm} \;e_3 \zeta} \right)}{\partial i}
+      \end{aligned}
+    \right)
+  \end{split}
+\]
 
 Such a formulation ensures a complete separation between the vorticity and horizontal divergence fields
@@ -1278,10 +1361,11 @@
 ($i.e.$ geopotential diffusion in $s-$coordinates or isoneutral diffusion in both $z$- and $s$-coordinates),
 the $u$ and $v-$fields are considered as independent scalar fields, so that the diffusive operator is given by:
-\begin{equation} \label{eq:PE_lapU_iso}
-\begin{split}
- D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\ 
- D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right)
-\end{split}
-\end{equation}
+\[
+  % \label{eq:PE_lapU_iso}
+  \begin{split}
+    D_u^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla u} \right) \\
+    D_v^{l{\rm {\bf U}}} &= \nabla .\left( {A^{lm} \;\Re \;\nabla v} \right)
+  \end{split}
+\]
 where $\Re$ is given by \autoref{eq:PE_iso_tensor}.
 It is the same expression as those used for diffusive operator on tracers.
@@ -1297,4 +1381,6 @@
 Nevertheless it is currently not available in the iso-neutral case.
 
+\biblio
+
 \end{document}
 
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_model_basics_zstar.tex	(revision 10419)
@@ -1,6 +1,7 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 % ================================================================
-% Chapter 1 ——— Model Basics
+% Chapter 1 Model Basics
 % ================================================================
 % ================================================================
@@ -20,12 +21,12 @@
 
 \colorbox{yellow}{  end of to be updated}
-\newline
 
 % from MOM4p1 documentation
 
 To overcome problems with vanishing surface and/or bottom cells, we consider the zstar coordinate 
-\begin{equation} \label{eq:PE_}
-	z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
-\end{equation}
+\[
+  % \label{eq:PE_}
+  z^\star = H \left( \frac{z-\eta}{H+\eta} \right)
+\]
 
 This coordinate is closely related to the "eta" coordinate used in many atmospheric models
@@ -65,11 +66,8 @@
 the surface height, again so long as $\eta > -H$. 
 
-
-
 %%%
 %  essai update time splitting...
 %%%
 
-
 % ================================================================
 % Surface Pressure Gradient and Sea Surface Height
@@ -79,5 +77,5 @@
 %-----------------------------------------nam_dynspg----------------------------------------------------
 
-\nlst{nam_dynspg} 
+%\nlst{nam_dynspg} 
 %------------------------------------------------------------------------------------------------------------
 Options are defined through the \ngn{nam\_dynspg} namelist variables.
@@ -105,26 +103,29 @@
 describe the external gravity waves (typically a few ten 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)} 
+\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 (evaporation minus precipitation, and minus river runoffs
 (if the later are introduced as a surface freshwater flux, see \autoref{chap:SBC}) expressed in $Kg.m^{-2}.s^{-1}$,
-and $\rho _w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
+and $\rho_w =1,000\,Kg.m^{-3}$ is the volumic mass of pure water.
 The sea-surface height is evaluated using a leapfrog scheme in combination with an Asselin time filter,
 i.e. the velocity appearing in (\autoref{eq:dynspg_ssh}) is centred in time (\textit{now} velocity). 
 
 The surface pressure gradient, also evaluated using a leap-frog scheme, is then simply given by:
-\begin{equation} \label{eq:dynspg_exp}
-\left\{ \begin{aligned}
- - \frac{1}    						{e_{1u}} \;	\delta _{i+1/2} \left[  \,\eta\,  \right] 	\\
- \\
- - \frac{1}    						{e_{2v}} \;	\delta _{j+1/2} \left[  \,\eta\,  \right]  
-\end{aligned} \right.
+\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]
+    \end{aligned}
+  \right.
 \end{equation} 
 
-Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho _o$ factor is omitted in
+Consistent with the linearization, a $\left. \rho \right|_{k=1} / \rho_o$ factor is omitted in
 (\autoref{eq:dynspg_exp}). 
 
@@ -144,23 +145,26 @@
 
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
-\begin{figure}[!t] 	\begin{center}
-\includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
-\caption{	 \protect\label{fig:DYN_dynspg_ts}
-  Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
-  after \citet{Griffies2004}.
-  Time increases to the right.
-  Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
-  The curved line represents a leap-frog time step,
-  and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
-  The vertically integrated forcing \textbf{M}(t) computed at
-  baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
-  While keeping the total depth, tracer, and freshwater forcing fields fixed,
-  a leap-frog integration carries the surface height and vertically integrated velocity from
-  t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
-  Time averaging the barotropic fields over the N+1 time steps (endpoints included)
-  centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
-  A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
-  the time averaged vertically integrated velocity taken from baroclinic time step t. }
-\end{center}
+\begin{figure}[!t]
+  \begin{center}
+    \includegraphics[width=0.90\textwidth]{Fig_DYN_dynspg_ts}
+    \caption{
+      \protect\label{fig:DYN_dynspg_ts}
+      Schematic of the split-explicit time stepping scheme for the barotropic and baroclinic modes,
+      after \citet{Griffies2004}.
+      Time increases to the right.
+      Baroclinic time steps are denoted by $t-\Delta t$, $t, t+\Delta t$, and $t+2\Delta t$.
+      The curved line represents a leap-frog time step,
+      and the smaller barotropic time steps $N \Delta t=2\Delta t$ are denoted by the zig-zag line.
+      The vertically integrated forcing \textbf{M}(t) computed at
+      baroclinic time step t represents the interaction between the barotropic and baroclinic motions.
+      While keeping the total depth, tracer, and freshwater forcing fields fixed,
+      a leap-frog integration carries the surface height and vertically integrated velocity from
+      t to $t+2 \Delta t$ using N barotropic time steps of length $\Delta t$.
+      Time averaging the barotropic fields over the N+1 time steps (endpoints included)
+      centers the vertically integrated velocity at the baroclinic timestep $t+\Delta t$.
+      A baroclinic leap-frog time step carries the surface height to $t+\Delta t$ using the convergence of
+      the time averaged vertically integrated velocity taken from baroclinic time step t.
+    }
+  \end{center}
 \end{figure}
 %>   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >   >
@@ -178,13 +182,15 @@
 the small barotropic time step $\Delta t$.
 We have
-\begin{equation} \label{eq:DYN_spg_ts_eta}
-\eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
-	= 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
-\end{equation}
-\begin{multline} \label{eq:DYN_spg_ts_u}
-\textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
-	= 2\Delta t \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}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \eta^{(b)}(\tau,t_{n+1}) - \eta^{(b)}(\tau,t_{n+1}) (\tau,t_{n-1})
+  = 2 \Delta t \left[-\nabla \cdot \textbf{U}^{(b)}(\tau,t_n) + \text{EMP}_w(\tau) \right] 
+\]
+\begin{multline*}
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}^{(b)}(\tau,t_{n+1}) - \textbf{U}^{(b)}(\tau,t_{n-1})  \\
+  = 2\Delta t \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*}
 \
 
@@ -197,39 +203,43 @@
 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{eq:DYN_spg_ts_t}
-t_n=\tau+n\Delta t   
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_t}
+  t_n=\tau+n\Delta t   
+\]
 with $n$ an integer.
 The density scaled surface pressure is evaluated via 
-\begin{equation} \label{eq: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}
+\[
+  % \label{eq:DYN_spg_ts_ps}
+  p_s^{(b)}(\tau,t_{n}) =
+  \begin{cases}
+    g \;\eta_s^{(b)}(\tau,t_{n}) \;\rho(\tau)_{k=1}) / \rho_o  &      \text{non-linear case} \\
+    g \;\eta_s^{(b)}(\tau,t_{n})  &      \text{linear case}
+  \end{cases}
+\]
 To get started, we assume the following initial conditions 
-\begin{equation} \label{eq: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}) + \Delta t \ \text{RHS}_{n=0} 
-\end{split}
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_eta}
+  \begin{split}
+    \eta^{(b)}(\tau,t_{n=0}) &= \overline{\eta^{(b)}(\tau)} \\
+    \eta^{(b)}(\tau,t_{n=1}) &= \eta^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+  \end{split}
+\]
 with 
-\begin{equation} \label{eq:DYN_spg_ts_etaF}
- \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_etaF}
+  \overline{\eta^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N \eta^{(b)}(\tau-\Delta t,t_{n})
+\]
 the time averaged surface height taken from the previous barotropic cycle.
 Likewise,
-\begin{equation} \label{eq: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}) + \Delta t \ \text{RHS}_{n=0}  	
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}^{(b)}(\tau,t_{n=0}) = \overline{\textbf{U}^{(b)}(\tau)}	\\ \\
+  \textbf{U}(\tau,t_{n=1}) = \textbf{U}^{(b)}(\tau,t_{n=0}) + \Delta t \ \text{RHS}_{n=0}
+\]
 with 
-\begin{equation} \label{eq:DYN_spg_ts_u}
- \overline{\textbf{U}^{(b)}(\tau)} 
- 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \overline{\textbf{U}^{(b)}(\tau)} = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau-\Delta t,t_{n})
+\]
 the time averaged vertically integrated transport.
 Notably, there is no Robert-Asselin time filter used in the barotropic portion of the integration. 
@@ -237,12 +247,14 @@
 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{eq:DYN_spg_ts_u}
-\textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)} 
- 	= \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_u}
+  \textbf{U}(\tau+\Delta t) = \overline{\textbf{U}^{(b)}(\tau+\Delta t)}
+  = \frac{1}{N+1} \sum\limits_{n=0}^N\textbf{U}^{(b)}(\tau,t_{n})
+\]
 The surface height on the new baroclinic time step is then determined via
 a baroclinic leap-frog using the following form 
-\begin{equation} \label{eq:DYN_spg_ts_ssh}
-\eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]  
+\begin{equation}
+  \label{eq:DYN_spg_ts_ssh}
+  \eta(\tau+\Delta) - \eta^{F}(\tau-\Delta) = 2\Delta t \ \left[ - \nabla \cdot \textbf{U}(\tau) + \text{EMP}_w \right]
 \end{equation}
 
@@ -257,14 +269,16 @@
 reasonably good maintenance of tracer conservation properties (see ??) 
 
-\begin{equation} \label{eq:DYN_spg_ts_sshf}
-\eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)} 
+\begin{equation}
+  \label{eq:DYN_spg_ts_sshf}
+  \eta^{F}(\tau-\Delta) =  \overline{\eta^{(b)}(\tau)}
 \end{equation}
 Another approach tried was 
 
-\begin{equation} \label{eq:DYN_spg_ts_sshf2}
-\eta^{F}(\tau-\Delta) = \eta(\tau) 
-	+ (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
-				    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
-\end{equation}
+\[
+  % \label{eq:DYN_spg_ts_sshf2}
+  \eta^{F}(\tau-\Delta) = \eta(\tau)
+  + (\alpha/2) \left[\overline{\eta^{(b)}}(\tau+\Delta t)
+    + \overline{\eta^{(b)}}(\tau-\Delta t) -2 \;\eta(\tau) \right]
+\]
 
 which is useful since it isolates all the time filtering aspects into the term multiplied by $\alpha$.
@@ -274,8 +288,4 @@
 be more conservative, and so is recommended. 
 
-
-
-
-
 %-------------------------------------------------------------
 % Filtered formulation 
@@ -304,4 +314,5 @@
 In particular, this means that in filtered case, the matrix to be inverted has to be recomputed at each time-step.
 
+\biblio
 
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/chap_time_domain.tex	(revision 10419)
@@ -1,15 +1,16 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
 % ================================================================
-% Chapter 2 ——— Time Domain (step.F90)
+% Chapter 2 Time Domain (step.F90)
 % ================================================================
 \chapter{Time Domain (STP) }
 \label{chap: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,
@@ -17,8 +18,5 @@
 %%%%
 
-
 \newpage
-$\ $\newline    % force a new ligne
-
 
 Having defined the continuous equations in \autoref{chap:PE}, we need now to choose a time discretization,
@@ -28,5 +26,4 @@
 the consequences for the order in which the equations are solved.
 
-$\ $\newline    % force a new ligne
 % ================================================================
 % Time Discretisation
@@ -36,6 +33,7 @@
 
 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}
+\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 $S$;
@@ -85,6 +83,7 @@
 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]
+\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.
@@ -111,17 +110,19 @@
 For a tendancy $D_x$, representing a diffusion term or a restoring term to a tracer climatology
 (when present, see \autoref{sec:TRA_dmp}), a forward time differencing scheme is used :
-\begin{equation} \label{eq:STP_euler}
+\[
+  % \label{eq:STP_euler}
    x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \ {D_x}^{t-\rdt}
-\end{equation} 
+\] 
 
 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.
+\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.
@@ -138,18 +139,20 @@
 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}
+\[
+  % \label{eq:STP_ts}
+  \begin{split}
+    & x_\ast ^{t-\rdt} = x^{t-\rdt}   \\
+    & x_\ast ^{t-\rdt+L\frac{2\rdt}{N}}=x_\ast ^{t-\rdt+\left( {L-1}
+      \right)\frac{2\rdt}{N}}+\frac{2\rdt}{N}\;\text{DF}^{t-\rdt+\left( {L-1} \right)\frac{2\rdt}{N}}
+    \quad \text{for $L=1$ to $N$}      \\
+    & x^{t+\rdt} = x_\ast^{t+\rdt}
+  \end{split}
+\]
 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}
+\begin{equation}
+  \label{eq:STP_imp}
+  x^{t+\rdt} = x^{t-\rdt} + 2 \, \rdt \  \text{RHS}_x^{t+\rdt}
 \end{equation} 
 
@@ -161,19 +164,21 @@
 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}
+\[
+  % \label{eq:STP_imp_zdf}
+  \frac{T(k)^{t+1}-T(k)^{t-1}}{2\;\rdt}\equiv \text{RHS}+\frac{1}{e_{3t} }\delta
+  _k \left[ {\frac{A_w^{vT} }{e_{3w} }\delta_{k+1/2} \left[ {T^{t+1}} \right]}
+  \right]
+\]
 where RHS is the right hand side of the equation except for the vertical diffusion term.
 We rewrite \autoref{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)
+\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)    
+\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*}
 
@@ -199,5 +204,6 @@
   \begin{center}
     \includegraphics[width=0.7\textwidth]{Fig_TimeStepping_flowchart}
-    \caption{ 	\protect\label{fig:TimeStep_flowchart}
+    \caption{
+      \protect\label{fig:TimeStep_flowchart}
       Sketch of the leapfrog time stepping sequence in \NEMO from \citet{Leclair_Madec_OM09}.
       The use of a semi-implicit computation of the hydrostatic pressure gradient requires the tracer equation to
@@ -208,5 +214,6 @@
       (see \autoref{sec:DYN_spg}).
     }
-\end{center}   \end{figure}
+  \end{center}
+\end{figure}
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 %}
@@ -226,8 +233,8 @@
 and the time filter is given by \autoref{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] 
+\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}
@@ -259,5 +266,6 @@
   \begin{center}
     \includegraphics[width=0.90\textwidth]{Fig_MLF_forcing}
-    \caption{ 	\protect\label{fig:MLF_forcing}
+    \caption{
+      \protect\label{fig:MLF_forcing}
       Illustration of forcing integration methods.
       (top) ''Traditional'' formulation:
@@ -284,7 +292,8 @@
 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}
+\[
+  % \label{eq:DOM_euler}
+  x^1 = x^0 + \rdt \ \text{RHS}^0
+\]
 This is done simply by keeping the leapfrog environment ($i.e.$ the \autoref{eq:STP} three level time stepping) but
 setting all $x^0$ (\textit{before}) and $x^{1}$ (\textit{now}) fields equal at the first time step and
@@ -301,5 +310,5 @@
 (see \autoref{subsec:DYN_hpg_imp}), an extra three-dimensional field has to
 be added to the restart file to ensure an exact restartability.
-This is done optionally via the  \np{nn\_dynhpg\_rst} namelist parameter,
+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).
@@ -307,5 +316,5 @@
 Note the size of the time step used, $\rdt$, is also saved in the restart file.
 When restarting, if the the time step has been changed, a restart using an Euler time stepping scheme is imposed. 
-Options are defined through the  \ngn{namrun} namelist variables.
+Options are defined through the \ngn{namrun} namelist variables.
 %%%
 \gmcomment{
@@ -351,45 +360,45 @@
 
 \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}    \\
-%
+  &\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}  \\
-%
+  \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}  &
+  \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*}
 
@@ -397,3 +406,5 @@
 }
 %%
+\biblio
+
 \end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/foreword.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/foreword.tex	(revision 10419)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/foreword.tex	(revision 10419)
@@ -0,0 +1,61 @@
+\documentclass[../main/NEMO_manual]{subfiles}
+
+\begin{document}
+
+% ================================================================
+% Abstract
+% ================================================================
+
+\chapter*{Abstract}
+
+\vspace{-40pt}
+
+{\small
+  The ocean engine of NEMO (Nucleus for European Modelling of the Ocean) is a primitive equation model adapted to
+  regional and global ocean circulation problems.
+  It is intended to be a flexible tool for studying the ocean and its interactions with
+  the others components of the earth climate system over a wide range of space and time scales.
+  Prognostic variables are the three-dimensional velocity field, a non-linear sea surface height,
+  the \textit{Conservative} Temperature and the \textit{Absolute} Salinity.
+  In the horizontal direction, the model uses a curvilinear orthogonal grid and in the vertical direction,
+  a full or partial step $z$-coordinate, or $s$-coordinate, or a mixture of the two.
+  The distribution of variables is a three-dimensional Arakawa C-type grid.
+  Various physical choices are available to describe ocean physics, including TKE, and GLS vertical physics.
+  Within NEMO, the ocean is interfaced with a sea-ice model (LIM or CICE),
+  passive tracer and biogeochemical models (TOP) and,
+  via the OASIS coupler, with several atmospheric general circulation models.
+  It also support two-way grid embedding via the AGRIF software.
+} 
+
+% ================================================================
+% Disclaimer
+% ================================================================
+\chapter*{Disclaimer}
+
+Like all components of NEMO,
+the ocean component is developed under the \href{http://www.cecill.info/}{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}
+NEMO reference in papers and other publications is as follows:
+\vspace{0.5cm}
+
+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.\\
+
+
+\vspace{0.5cm}
+Additional information can be found on \href{http://www.nemo-ocean.eu/}{www.nemo-ocean.eu}.
+\vspace{0.5cm}
+
+\biblio
+
+\end{document}
Index: NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/introduction.tex
===================================================================
--- NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/introduction.tex	(revision 10377)
+++ NEMO/branches/2018/dev_r10164_HPC09_ESIWACE_PREP_MERGE/doc/latex/NEMO/subfiles/introduction.tex	(revision 10419)
@@ -1,3 +1,4 @@
-\documentclass[../tex_main/NEMO_manual]{subfiles}
+\documentclass[../main/NEMO_manual]{subfiles}
+
 \begin{document}
 
@@ -9,9 +10,9 @@
 
 The Nucleus for European Modelling of the Ocean (\NEMO) is a framework of ocean related engines,
-namely OPA\footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the ocean dynamics and thermodynamics,
-LIM\footnote{LIM = Louvain la-neuve Ice Model} for the sea-ice dynamics and thermodynamics,
-TOP\footnote{TOP = Tracer in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources 
-minus sinks (LOBSTER\footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for Ecosystem and Resources},
-PISCES\footnote{PISCES = Pelagic Interactions Scheme for Carbon and Ecosystem Studies})).
+namely OPA \footnote{OPA = Oc\'{e}an PArall\'{e}lis\'{e}} for the ocean dynamics and thermodynamics,
+LIM \footnote{LIM = Louvain la-neuve Ice Model} for the sea-ice dynamics and thermodynamics,
+TOP \footnote{TOP = Tracer in the Ocean Paradigm} for the biogeochemistry (both transport (TRP) and sources 
+minus sinks (LOBSTER \footnote{LOBSTER = Lodyc Ocean Biogeochemical SysTem for Ecosystem and Resources},
+PISCES \footnote{PISCES = Pelagic Interactions Scheme for Carbon and Ecosystem Studies})).
 It is intended to be a flexible tool for studying the ocean and its interactions with the other components of
 the earth climate system (atmosphere, sea-ice, biogeochemical tracers, ...) over
@@ -316,3 +317,5 @@
 
 
+\biblio
+
 \end{document}
