Revert "Update file chap_DYN.tex"

This reverts commit 2d52af2e.
which temporarily deleted a section for debugging purposes

latex/NEMO/subfiles/chap_DYN.tex

@@ -392,6 +392,163 @@ the change of KE due to the gradient of KE (see \autoref{apdx:INVARIANTS}).
   \right.
 \]
 
+%% =================================================================================================
+\section{Coriolis and advection: flux form}
+\label{sec:DYN_adv_cor_flux}
+
+Options are defined through the \nam{dyn_adv}{dyn\_adv} namelist variables.
+In the flux form (as in the vector invariant form),
+the Coriolis and momentum advection terms are evaluated using either 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, 
+it is is centred in time (\textit{now} velocity) at stage 2 and 3.
+At the lateral boundaries either free slip,
+no slip or partial slip boundary conditions are applied following \autoref{chap:LBC}.
+
+%% =================================================================================================
+\subsection[Coriolis plus curvature metric terms (\textit{dynvor.F90})]{Coriolis plus curvature metric terms (\protect\mdl{dynvor})}
+\label{subsec:DYN_cor_flux}
+
+In flux form, the vorticity term reduces to a Coriolis term in which the Coriolis parameter has been modified to account for the "metric" term.
+This altered Coriolis parameter is thus discretised at $f$-points.
+It is given by:
+\begin{aligned*}
+  % \label{eq:DYN_cor_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{aligned*}
+
+%                 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}
+
+
+\noindent 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 schemes (\autoref{eq:DYN_vor_een} and \autoref{eq:DYN_vor_enT})
+have exclusively been used to date.
+
+\vskip 0.5cm
+
+\noindent 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, 
+it is is centred in time (\textit{now} velocity) at stage 2 and 3. 
+
+%% =================================================================================================
+\subsection[Flux form advection term (\textit{dynadv.F90})]{Flux form advection term (\protect\mdl{dynadv})}
+\label{subsec:DYN_adv_flux}
+
+The discrete expression of the advection term is given by:
+\[
+  % \label{eq:DYN_adv}
+  \left\{
+    \begin{aligned}
+      \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} 
+      \left( \delta_{i+1/2} \left[ \overline{e_{2u}\,e_{3u}\;u }^{i} \ u_t \right]
+        + & \delta_{j} \left[ \overline{e_{1u}\,e_{3u}\;v }^{i+1/2} \ u_f \right] \right.  \\
+       \left. + & \delta_{k} \left[ \overline{e_{1w}\,e_{2w}\;w}^{i+1/2} \ u_{uw} \right] \right)   \\[10pt]
+      \frac{1}{e_{1v}\,e_{2v}\,e_{3v}}
+      \left( \delta_{i} \left[ \overline{e_{2u}\,e_{3u }\;u }^{j+1/2} \ v_f \right]
+        + & \delta_{j+1/2} \left[ \overline{e_{1u}\,e_{3u }\;v }^{i} \ v_t \right] \right.  \\
+       \left. + & \delta_{k} \left[ \overline{e_{1w}\,e_{2w}\;w}^{j+1/2} \ v_{vw} \right] \right) \\
+    \end{aligned}
+  \right.
+\]
+
+Two advection schemes are available:
+a $2^{nd}$ order centered finite difference scheme, CEN2,
+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_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 $v$.
+
+%                 2nd order centred scheme
+%% =================================================================================================
+\subsubsection[CEN2: $2^{nd}$ order centred scheme (\forcode{ln_dynadv_cen2})]{CEN2: $2^{nd}$ order centred scheme (\protect\np{ln_dynadv_cen2}{ln\_dynadv\_cen2})}
+\label{subsec:DYN_adv_cen2}
+
+In the centered $2^{nd}$ order formulation, the velocity is evaluated as the mean of the two neighbouring points:
+\begin{equation}
+  \label{eq:DYN_adv_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}
+
+The scheme is non diffusive (\ie\ conserves the kinetic energy) but dispersive (\ie\ it may create false extrema).
+It is therefore notoriously noisy and must be used in conjunction with an explicit diffusion operator to
+produce a sensible solution.
+
+%                 UP3 scheme
+%% =================================================================================================
+\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 UP3 advection scheme is an upstream biased third order scheme based on
+an upstream-biased parabolic interpolation.
+For example, the evaluation of $u_T^{up3} $ is done as follows:
+\begin{equation}
+  \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 $\ \overline{e_{2u}\,e_{3u} \ u}^i  < 0$ }
+  \end{cases}
+\end{equation}
+where $u"_{i+1/2} =\delta_{i+1/2} \left[ {\delta_i \left[ u \right]} \right]$.
+This results in a dissipatively dominant (\ie\ hyper-diffusive) truncation error
+\citep{shchepetkin.mcwilliams_OM05}.
+The overall performance of the advection scheme is similar to that reported in \citet{farrow.stevens_JPO95}.
+It is a relatively good compromise between accuracy and smoothness.
+It is not a \emph{positive} scheme, meaning that false extrema are permitted.
+But the amplitudes of the false extrema are significantly reduced over those in the centred second order method.
+As the scheme already includes a diffusion component, it can be used without explicit lateral diffusion on momentum
+(\ie\ \np[=.true.]{ln_dynldf_OFF}{ln\_dynldf\_OFF}),
+and it is recommended to do so.
+
+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...  }
+
+In a leapfrog environment, 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).
+In an RK3 environment, the first term in (\autoref{eq:DYN_adv_up3}),
+which corresponds to a second order centred scheme, is evaluated using the \textit{before} velocity at stage 1 
+and using the \textit{before} velocity (centred in time) at stage 2 and 3,
+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 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_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\_up3} module and obtain a QUICK scheme.
 
 %% =================================================================================================
 \section[Hydrostatic pressure gradient (\textit{dynhpg.F90})]{Hydrostatic pressure gradient (\protect\mdl{dynhpg})}