Update file chap_DYN.tex

latex/NEMO/subfiles/chap_DYN.tex

@@ -1107,25 +1107,25 @@ In v5.0 a symetrical lateral iso-level operator has been introduced :
   \left\{
     \begin{aligned}
       D_u^{l{\mathrm {\mathbf U}}} &= \frac{1}{e_{1u}\,e_{2u}\,e_{3u} } \left( 
-      \frac{1}{e_{2u}} \delta_{i+1/2} \left[ e_{2t}\,e_{2t}\,e_{3t}\, A_T^{lm} \,\vartheta \right] 
-      - \frac{1}{e_{1u}} \delta_{j+1/2} \left[ e_{1f}\,e_{1f}\,e_{3f}\,A_F^{lm} \psi \right] \right) \\ \\
+      \frac{1}{e_{2u}} \delta_{i+1/2} \left[ e_{2t}\,e_{2t}\,e_{3t}\, A_T^{lm} \,\epsilon_T \right] 
+      - \frac{1}{e_{1u}} \delta_{j+1/2} \left[ e_{1f}\,e_{1f}\,e_{3f}\,A_F^{lm} \epsilon_F \right] \right) \\ \\
       D_v^{l{\mathrm {\mathbf U}}} &= \frac{1}{e_{1v}\,e_{2v}\,e_{3v} } \left( 
-      \frac{1}{e_{2v}} \delta_{j+1/2} \left[ e_{2f}\,e_{2f}\,e_{3f}\, A_F^{lm} \,\psi \right] 
-      - \frac{1}{e_{1v}} \delta_{i+1/2} \left[ e_{1t}\,e_{1t}\,e_{3t}\,A_T^{lm} \vartheta \right] \right)
+      \frac{1}{e_{2v}} \delta_{j+1/2} \left[ e_{2f}\,e_{2f}\,e_{3f}\, A_F^{lm} \,\epsilon_F \right] 
+      - \frac{1}{e_{1v}} \delta_{i+1/2} \left[ e_{1t}\,e_{1t}\,e_{3t}\,A_T^{lm} \epsilon_T \right] \right)
     \end{aligned}
   \right.
 \end{equation}
 
-Where $\psi$ and $\vartheta$ are respectively the shearing stress component (F-point) and the 
+Where $\epsilon_F$ and $\epsilon_T$ are respectively the shearing stress component (F-point) and the 
 tension stress component (T-point)  defined as : 
 \begin{equation}
   \label{eq:DYN_ldf_lap_sheten}
   \left\{
     \begin{aligned}
-      \psi &= \frac{e_{1f}}{e_{2f}} \delta_{j+1/2} \left[ \frac{u}{e_{1u}} \right] + \frac{e_{2f}}{e_{1f}} \delta_{i+1/2} \left[
-        \frac{v}{e_{2v}} \right] \\ \\
-      \vartheta &= \frac{e_{2t}}{e_{1t}} \delta_{i} \left[ \frac{u}{e_{2u}} \right] - \frac{e_{1t}}{e_{2t}} \delta_j \left[
-        \frac{v}{e_{1v}} \right]
+      \epsilon_F &= \frac{e_{1f}}{e_{2f}}\; \delta_{j+1/2} \left[ \frac{u}{e_{1u}} \right] + \frac{e_{2f}}{e_{1f}}\; 
+        \delta_{i+1/2} \left[ \frac{v}{e_{2v}} \right] \\ \\
+      \epsilon_T &= \frac{e_{2t}}{e_{1t}}\; \delta_{i} \left[ \frac{u}{e_{2u}} \right] - \frac{e_{1t}}{e_{2t}}\; 
+        \delta_j \left[ \frac{v}{e_{1v}} \right]
     \end{aligned}
   \right.
 \end{equation}
@@ -1243,31 +1243,29 @@ The turbulent flux of momentum at the bottom of the ocean is specified through a
 
 When activated (\np[=.true.]{ln_zad_Aimp}{ln\_zad\_Aimp}) vertical advection of momentum 
 can be done partly implicitely.
-Variables appearing in these expressions are implicit in time (\textit{after}) 
-and vertical derivatives are done with an $1^{st}$ order upstream scheme.
+Appart from $w_i$ variables appearing in these expressions are implicit in time (\textit{after}) 
+and vertical derivatives are done with a $1^{st}$ order upstream scheme.
 
-In vector form case
-\begin{equation}
-  \label{eq:DYN_zimp_vec}
+In vector form case :
+\[
   \left\{
     \begin{aligned}
-      &\frac{1}{e_{3uw}}\; \delta_{k} \left[ \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,i+1/2,k+1/2} u \right] \\ \\
-      &\frac{1}{e_{3vw}}\; \delta_{k} \left[ \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,j+1/2,k+1/2} v \right]
+      w_i \, \delta_{k} [u] &= \frac{1}{e_{1u}\,e_{2u}} \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,i,k+1/2} \, \delta_{k} \left[\frac{u}{e_{3uw}}\right]^{up1}, \\[10pt]
+      w_i \, \delta_{k} [v] &= \frac{1}{e_{1v}\,e_{2v}} \overline{\overline{ e_{1t}\,e_{2t}\,w_i }}^{\,j,k+1/2} \, \delta_{k} \left[\frac{v}{e_{3vw}}\right]^{up1}.
     \end{aligned}
   \right.
-\end{equation}
+\]
 
-In flux form case
-\begin{equation}
-  \label{eq:DYN_zimp_flu}
+
+In flux form case :
+\[
   \left\{
     \begin{aligned}
-      &\frac{1}{e_{1u}\,e_{2u}\,e_{3u}}\; \delta_{k} \left[ \overline{ e_{1t}\,e_{2t}\,w_i }^{\,i+1/2} \, u \right] \\ \\
-      &\frac{1}{e_{1v}\,e_{2v}\,e_{3v}}\; \delta_{k} \left[ \overline{ e_{1t}\,e_{2t}\,w_i }^{\,j+1/2} \, v \right]
+      \delta_{k} [w_i u] &= \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \left( \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,i} \, u \right)^{k,up1} - \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,i} \, u \right)^{k+1,up1}\right), \\[10pt]
+      \delta_{k} [w_i v] &= \frac{1}{e_{1u}\,e_{2u}\,e_{3u}} \left(\left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,j} \, v \right)^{k,up1} - \left( \overline{ e_{1t}\,e_{2t}\,w_i }^{\,j} \, v \right)^{k+1,up1}\right).
     \end{aligned}
   \right.
-\end{equation}
-
+\]
 
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 \section{External forcings}