Update section 3 of chap_DYN.tex it concerns up3 mostly

latex/NEMO/subfiles/chap_DYN.tex

@@ -165,6 +165,10 @@ Note also that the $k$-axis is re-orientated downwards in the \fortran\ code com
 the indexing used in the semi-discrete equations such as \autoref{eq:DYN_wzv}
 (see \autoref{subsec:DOM_Num_Index_vertical}).
 
+When \np[=.true.]{ln_zad_Aimp}{ln\_zad\_Aimp},
+a proportion of the vertical advection can be treated implicitly 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{Coriolis and advection: vector invariant form}
 \label{sec:DYN_adv_cor_vect}
@@ -208,29 +212,6 @@ In the case of ENS, ENE or MIX schemes the land sea mask may be slightly modifie
 vorticity term with analytical equations (\np[=.true.]{ln_dynvor_con}{ln\_dynvor\_con}).
 The vorticity terms are all computed in dedicated routines that can be found in the \mdl{dynvor} module.
 
-%                 energy conserving scheme at T-point
-%% =================================================================================================
-\subsubsection[Energy conserving scheme  (\forcode{ln_dynvor_enT})]{Energy conserving scheme (\protect\np{ln_dynvor_enT}{ln\_dynvor\_enT})}
-\label{subsec:DYN_vor_enT}
-
-The kinetic energy conserving scheme (ENT scheme) conserves the global kinetic energy but not the global enstrophy.
-It is given by:
-\begin{equation}
-  \label{eq:DYN_vor_enT}
-  \left\{
-    \begin{aligned}
-      &+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
-            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{v}^{\,j}}^{\,i+1/2} \\
-      &-\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
-            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{u}^{\,i}}^{\,j+1/2}
-    \end{aligned}
-  \right.
-\end{equation}
-
-
-
-
-
 %                 enstrophy conserving scheme
 %% =================================================================================================
 \subsubsection[Enstrophy conserving scheme (\forcode{ln_dynvor_ens})]{Enstrophy conserving scheme (\protect\np{ln_dynvor_ens}{ln\_dynvor\_ens})}
@@ -420,12 +401,6 @@ the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
     \end{aligned}
   \right.
 \]
-When \np[=.true.]{ln_dynzad_zts}{ln\_dynzad\_zts},
-a split-explicit time stepping with 5 sub-timesteps is used on the vertical advection term.
-This option can be useful when the value of the timestep is limited by vertical advection \citep{lemarie.debreu.ea_OM15}.
-Note that in this case,
-a similar split-explicit time stepping should be used on vertical advection of tracer to ensure a better stability,
-an option which is only available with a TVD scheme (see \np{ln_traadv_tvd_zts}{ln\_traadv\_tvd\_zts} in \autoref{subsec:TRA_adv_tvd}).
 
 %% =================================================================================================
 \section{Coriolis and advection: flux form}
@@ -454,9 +429,31 @@ It is given by:
       -  \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right]  }  \ \right)
 \end{multline*}
 
-Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}) and (\autoref{eq:DYN_vor_een}) schemes can be used to
+%                 energy conserving scheme at T-point
+%% =================================================================================================
+\subsubsection[Energy conserving scheme  (\forcode{ln_dynvor_enT})]{Energy conserving scheme (\protect\np{ln_dynvor_enT}{ln\_dynvor\_enT})}
+\label{subsec:DYN_vor_enT}
+
+The kinetic energy conserving scheme at T-point (ENT scheme) conserves the global kinetic energy but not the global enstrophy.
+It is given by:
+\begin{equation}
+  \label{eq:DYN_vor_enT}
+  \left\{
+    \begin{aligned}
+      &+\frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
+            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{v}^{\,j}}^{\,i+1/2} \\
+      &-\frac{1}{e_{1v}\,e_{2v}\,e_{3v}} \, \overline{ \left( f^T + \overline{\overline{ \zeta }}^{\,i,j} \right)
+            e_{1t}\,e_{2t}\,e_{3t} \,  \overline{u}^{\,i}}^{\,j+1/2}
+    \end{aligned}
+  \right.
+\end{equation}
+
+
+Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}), (\autoref{eq:DYN_vor_enT}) and (\autoref{eq:DYN_vor_een}) 
+schemes can be used to
 compute the product of the Coriolis parameter and the vorticity.
-However, the energy-conserving scheme (\autoref{eq:DYN_vor_een}) has exclusively been used to date.
+However, the energy-conserving schemes (\autoref{eq:DYN_vor_een} and \autoref{eq:DYN_vor_enT}) 
+have exclusively been used to date.
 This term is evaluated using either a leapfrog scheme or a RK3 scheme.
 In the leapfrog case it is centred in time (\textit{now} velocity).
 In the RK3 case it is forward in time (\textit{before} velocity) at stage 1, 
@@ -486,9 +483,9 @@ The discrete expression of the advection term is given by:
 
 Two advection schemes are available:
 a $2^{nd}$ order centered finite difference scheme, CEN2,
-or a $3^{rd}$ order upstream biased scheme, UBS.
+or a $3^{rd}$ order upstream biased scheme, UP3.
 The latter is described in \citet{shchepetkin.mcwilliams_OM05}.
-The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_ubs}{ln\_dynadv\_ubs}.
+The schemes are selected using the namelist logicals \np{ln_dynadv_cen2}{ln\_dynadv\_cen2} and \np{ln_dynadv_up3}{ln\_dynadv\_up3}.
 In flux form, the schemes differ by the choice of a space and time interpolation to define the value of
 $u$ and $v$ at the centre of each face of $u$- and $v$-cells, \ie\ at the $T$-, $f$-,
 and $uw$-points for $u$ and at the $f$-, $T$- and $vw$-points for $v$.
@@ -515,17 +512,17 @@ produce a sensible solution.
 The associated time-stepping is performed using a leapfrog scheme in conjunction with an Asselin time-filter,
 so $u$ and $v$ are the \emph{now} velocities.
 
-%                 UBS scheme
+%                 UP3 scheme
 %% =================================================================================================
-\subsubsection[UBS: Upstream Biased Scheme (\forcode{ln_dynadv_ubs})]{UBS: Upstream Biased Scheme (\protect\np{ln_dynadv_ubs}{ln\_dynadv\_ubs})}
-\label{subsec:DYN_adv_ubs}
+\subsubsection[UP3: Upstream Biased Scheme (\forcode{ln_dynadv_up3})]{UP3: Upstream Biased Scheme (\protect\np{ln_dynadv_up3}{ln\_dynadv\_up3})}
+\label{subsec:DYN_adv_up3}
 
-The UBS advection scheme is an upstream biased third order scheme based on
+The UP3 advection scheme is an upstream biased third order scheme based on
 an upstream-biased parabolic interpolation.
-For example, the evaluation of $u_T^{ubs} $ is done as follows:
+For example, the evaluation of $u_T^{up3} $ is done as follows:
 \begin{equation}
-  \label{eq:DYN_adv_ubs}
-  u_T^{ubs} =\overline u ^i-\;\frac{1}{6}
+  \label{eq:DYN_adv_up3}
+  u_T^{up3} =\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 $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
@@ -542,28 +539,21 @@ As the scheme already includes a diffusion component, it can be used without exp
 (\ie\ \np[=]{ln_dynldf_lap}{ln\_dynldf\_lap}\np[=.false.]{ln_dynldf_bilap}{ln\_dynldf\_bilap}),
 and it is recommended to do so.
 
-The UBS scheme is not used in all directions.
-In the vertical, the centred $2^{nd}$ order evaluation of the advection is preferred, \ie\ $u_{uw}^{ubs}$ and
-$u_{vw}^{ubs}$ in \autoref{eq:DYN_adv_cen2} are used.
-UBS is diffusive and is associated with vertical mixing of momentum. \cmtgm{ gm  pursue the
+The UP3 scheme is used in all directions.
+UP3 is diffusive and is associated with vertical mixing of momentum. \cmtgm{ gm  pursue the
 sentence:Since vertical mixing of momentum is a source term of the TKE equation...  }
 
-For stability reasons, the first term in (\autoref{eq:DYN_adv_ubs}),
+For stability reasons, the first term in (\autoref{eq:DYN_adv_up3}),
 which corresponds to a second order centred scheme, is evaluated using the \textit{now} velocity (centred in time),
 while the second term, which is the diffusion part of the scheme,
 is evaluated using the \textit{before} velocity (forward in time).
 This is discussed by \citet{webb.de-cuevas.ea_JAOT98} in the context of the Quick advection scheme.
 
-Note that the UBS and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
+Note that the UP3 and QUICK (Quadratic Upstream Interpolation for Convective Kinematics) schemes only differ by
 one coefficient.
-Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_ubs}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
+Replacing $1/6$ by $1/8$ in (\autoref{eq:DYN_adv_up3}) leads to the QUICK advection scheme \citep{webb.de-cuevas.ea_JAOT98}.
 This option is not available through a namelist parameter, since the $1/6$ coefficient is hard coded.
-Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_ubs} module and obtain a QUICK scheme.
-
-Note also that in the current version of \mdl{dynadv\_ubs},
-there is also the possibility of using a $4^{th}$ order evaluation of the advective velocity as in ROMS.
-This is an error and should be suppressed soon.
-\cmtgm{action :  this have to be done}
+Nevertheless it is quite easy to make the substitution in the \mdl{dynadv\_up3} module and obtain a QUICK scheme.
 
 %% =================================================================================================
 \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}