From ea600b6cf18409728fe642ad893267843abd9553 Mon Sep 17 00:00:00 2001
From: Sibylle Techene <sibylle.techene@locean-ipsl.upmc.fr>
Date: Tue, 19 Nov 2024 11:06:54 +0000
Subject: [PATCH] Update file chap_DYN.tex
---
latex/NEMO/subfiles/chap_DYN.tex | 39 +++++++++++++++++++++++---------
1 file changed, 28 insertions(+), 11 deletions(-)
diff --git a/latex/NEMO/subfiles/chap_DYN.tex b/latex/NEMO/subfiles/chap_DYN.tex
index b0a735d..3240876 100644
--- a/latex/NEMO/subfiles/chap_DYN.tex
+++ b/latex/NEMO/subfiles/chap_DYN.tex
@@ -124,11 +124,13 @@ taking into account the change of the thickness of the levels:
\begin{equation}
\label{eq:DYN_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)
+ - \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}
@@ -417,14 +419,6 @@ It is given by:
- \overline u ^{j+1/2}\delta_{j+1/2} \left[ {e_{1u} } \right] \right)
\end{aligned*}
-\vskip 0.5cm
-
-\noindent Any of the (\autoref{eq:DYN_vor_ens}), (\autoref{eq:DYN_vor_ene}), (\autoref{eq:DYN_vor_enT} described hereafter ) 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.
-
% 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})}
@@ -445,7 +439,15 @@ It is given by:
\end{equation}
-This term is evaluated using either a leapfrog scheme or a RK3 scheme.
+\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.
@@ -454,7 +456,22 @@ 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,
--
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