Update chap_DYN.tex

latex/NEMO/subfiles/chap_DYN.tex

@@ -1101,6 +1101,39 @@ As explained in \autoref{subsec:MB_ldf},
 this formulation (as the gradient of a divergence and curl of the vorticity) preserves symmetry and
 ensures a complete separation between the vorticity and divergence parts of the momentum diffusion.
 
+In v5.0 a symetrical lateral iso-level operator as been introduced : 
+\begin{equation}
+  \label{eq:DYN_ldf_lap_sym}
+  \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) \\ \\
+      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)
+    \end{aligned}
+  \right.
+\end{equation}
+
+Where $\psi$ and $\vartheta$ 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]
+    \end{aligned}
+  \right.
+\end{equation}
+
+
+
+
+
 %% =================================================================================================
 \subsection[Rotated laplacian (\forcode{ln_dynldf_iso})]{Rotated laplacian operator (\protect\np{ln_dynldf_iso}{ln\_dynldf\_iso})}
 \label{subsec:DYN_ldf_iso}