Update file chap_DYN.tex

latex/NEMO/subfiles/chap_DYN.tex

@@ -392,163 +392,6 @@ 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})}