diff --git a/latex/NEMO/subfiles/chap_DYN.tex b/latex/NEMO/subfiles/chap_DYN.tex
index 97a0a2900716d5ef36bdd1b6f2b9cf8049b727ac..c1895bce8839e6f5bfb9b1c8a889c690af8c68b1 100644
--- a/latex/NEMO/subfiles/chap_DYN.tex
+++ b/latex/NEMO/subfiles/chap_DYN.tex
@@ -50,8 +50,8 @@ The external forcings and parameterisations require complex inputs
that are carried out in modules SBC, LDF and ZDF and are described in
\autoref{chap:SBC}, \autoref{chap:LDF} and \autoref{chap:ZDF}, respectively.
-In the present chapter we also describe the diagnostic equations used to compute the horizontal divergence,
-curl of the velocities (\emph{divcur} module) and the vertical velocity (\emph{wzvmod} module).
+In the present chapter we also describe the diagnostic equations used to compute
+the horizontal divergence of the velocities (\emph{divhor} module) and the vertical velocity (\emph{sshwzv} module).
The different options available to the user are managed by namelist variables.
For term \textit{ttt} in the momentum equations, the logical namelist variables are \textit{ln\_dynttt\_xxx},
@@ -68,38 +68,56 @@ can be derived from the 3D terms.
Does MISC correspond to "extracting tendency terms" or "vorticity balance"?}
%% =================================================================================================
-%\section{Sea surface height and diagnostic variables ($\eta$, $\zeta$, $\chi$, $w$)}
+%\section{Continuity equation and diagnostic variables ($\chi$, $w$)}
%\label{sec:DYN_divcur_wzv}
%% =================================================================================================
-\section[Horizontal divergence (\textit{sshwzv.F90}, $\eta$, $w$ )]{Horizontal divergence (\protect\mdl{sshwzv})}
-\label{sec:DYN_sshwzv}
+\section[Continuity equation (\textit{sshwzv.F90}, $w$ )]{Continuity equation (\protect\mdl{sshwzv})}
+\label{sec:DYN_wzv}
-The sea surface height is given by:
-\begin{equation}
- \label{eq:DYN_spg_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 $\rho_w$=1,026~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.
-In the presence of ice-shelf (\np[=.true.]{ln_isf}{ln\_isf})
-\textit{emp} can be written as the evaporation minus precipitation, minus the ice-shelf water flux.
-In the leapfrog case, the sea-surface height is evaluated using exactly the same time stepping scheme as
-the tracer equation \autoref{eq:TRA_nxt}:
-a leapfrog scheme in combination with an Asselin time filter,
-\ie\ the velocity appearing in \autoref{eq:DYN_spg_ssh} is centred in time (\textit{now} velocity).
-This is of paramount importance.
-Replacing $T$ by the number $1$ in the tracer equation and summing over the water column must lead to
-the sea surface height equation otherwise tracer content will not be conserved
-\citep{griffies.pacanowski.ea_MWR01, leclair.madec_OM09}.
+The evolution of sea surface height ($\eta$) and vertical velocity ($w$) in ocean modeling
+is fundamentally governed by the continuity equation, which ensures the conservation of volume.
+Through this equation, we derive expressions for $\eta$ and $w$ that are essential for maintaining
+the balance between horizontal and vertical transport in the ocean’s volume-conserving framework.
+Both are deduced from the horizontal divergence $\ksi$ and require the horizontal divergence calculation.
+Because $\eta$ evolution is related to the external mode it is described in \autoref{sec:D2D}
+while $w$ evolution is described here after.
+
+%% =================================================================================================
+\subsection[Horizontal divergence (\textit{divhor.F90}, $\chi$ )]{Horizontal divergence (\protect\mdl{divhor})}
+\label{subsec:DYN_divhor}
+
+The horizontal divergence is defined at a $T$-point.
+It is given by:
+\[
+ % \label{eq:DYN_divcur_div}
+ \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
+ \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
+ +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
+\]
+
+\noindent Besides the velocity three other sources may be added to the horizontal divergence :
+
+(1) When \np[=.true.]{ln_rnf}{ln\_rnf} (see \autoref{sec:SBC_rnf}),
+ the divergence caused by river runoff is included.
+
+(2) When \np[=.true.]{ln_isf}{ln\_isf} (see \autoref{sec:SBC_isf}),
+explicit or parameterised contributions from ice-shelf cavities are taken into account.
+
+(3) When \np[=.true.]{lk_asminc .AND. ln_sshinc .AND. ln_asmiau}{lk\_asminc .AND. ln\_sshinc .AND. ln\_asmiau}
+(see \autoref{sec:ASM_IAU}),
+the contribution from IAU weighted ssh increments is included.
+
+\vskip 0.5cm
+
+\noindent In a leapfrog time-stepping scheme, the divergence at \textit{now} time step is used
+to calculate both nonlinear advection and vertical velocity.
+In an RK3 time-stepping scheme, the divergence at \textit{before} time step is applied
+during the first stage, while the \textit{now} time step divergence is used
+for calculating nonlinear advection and vertical velocity in the following stages.
+
+%% =================================================================================================
+\subsection[Vertical velocity (\textit{sshwzv.F90})]{Vertical velocity (\protect\mdl{sshwzv})}
+\label{subsec:DYN_wzv}
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:
@@ -114,10 +132,6 @@ taking into account the change of the thickness of the levels:
\right.
\end{equation}
-In the case of a non-linear free surface (\key{qco}), the top vertical velocity is $-\textit{emp}/\rho_w$,
-as changes in the divergence of the barotropic transport are absorbed into the change of the level thicknesses,
-re-orientated downward.
-\cmtgm{not sure of this... to be modified with the change in emp setting}
In the case of a linear free surface(\key{linssh}), the time derivative in \autoref{eq:DYN_wzv} disappears.
The upper boundary condition applies at a fixed level $z=0$.
The top vertical velocity is thus equal to the divergence of the barotropic transport
@@ -134,41 +148,6 @@ When \np[=.true.]{ln_zad_Aimp}{ln\_zad\_Aimp},
a proportion of the vertical advection can be treated implicitly (see \autoref{sec:DYN_zdf}) depending on the Courant number.
This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
-
-%% =================================================================================================
-\section[Horizontal divergence and relative vorticity (\textit{divhor.F90}, $\zeta$, $\chi$ )]{Horizontal divergence and relative vorticity (\protect\mdl{divcur})}
-\label{subsec:DYN_divcur}
-
-The vorticity is defined at an $f$-point (\ie\ corner point) as follows (used in ):
-\begin{equation}
- \label{eq:DYN_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:
-\[
- % \label{eq:DYN_divcur_div}
- \chi =\frac{1}{e_{1t}\,e_{2t}\,e_{3t} }
- \left( {\delta_i \left[ {e_{2u}\,e_{3u}\,u} \right]
- +\delta_j \left[ {e_{1v}\,e_{3v}\,v} \right]} \right)
-\]
-
-Note that although the vorticity has the same discrete expression in $z$- and $s$-coordinates,
-its physical meaning is not identical.
-$\zeta$ is a pseudo vorticity along $s$-surfaces
-(only pseudo because $(u,v)$ are still defined along geopotential surfaces,
-but are not necessarily defined at the same depth).
-
-The vorticity and divergence at the \textit{before} step are used in the computation of
-the horizontal diffusion of momentum.
-Note that because they have been calculated prior to the Asselin filtering of the \textit{before} velocities,
-the \textit{before} vorticity and divergence arrays must be included in the restart file to
-ensure perfect restartability.
-The vorticity and divergence at the \textit{now} time step are used for the computation of
-the nonlinear advection and of the vertical velocity respectively.
-
%% =================================================================================================
\section{Coriolis and advection: vector invariant form}
\label{sec:DYN_adv_cor_vect}
@@ -179,10 +158,17 @@ the nonlinear advection and of the vertical velocity respectively.
\label{lst:namdyn_adv}
\end{listing}
-The vector invariant form of the momentum equations is the one most often used in
-applications of the \NEMO\ ocean model.
+The vector invariant form of the momentum equation is most commonly used in coarse-resolution (1°) applications of the \NEMO\ ocean model.
The flux form option (see next section) has been present since version $2$.
-Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and
+By structuring the equations in vector invariant form, the dynamics are expressed in terms of
+intrinsic geometric properties like gradients, curls, and divergences.
+This ensures that the physics remain consistent and interpretable regardless of
+the underlying curvilinear grid or coordinate system.
+It highlights key physical terms like the kinetic energy advection and the relative vorticity.
+
+\vskip 0.5cm
+
+\noindent Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables Coriolis and
momentum advection terms are evaluated either using a leapfrog scheme or a RK3 scheme.
In the leapfrog case the velocity appearing in these expressions is centred in time (\textit{now} velocity).
In the RK3 case the velocity appearing in these expressions is forward in time (\textit{before} velocity) at stage 1,