Changeset 994 for trunk/DOC/TexFiles/Chapters/Annex_C.tex

Timestamp:

2008-05-28T11:01:09+02:00 (16 years ago)

Author:

gm

Message:

trunk - add steven correction + several other things + rename BETA into TexFiles?

Location:

trunk/DOC/TexFiles

Files:

• . (moved) (moved from trunk/DOC/BETA)
• Chapters/Annex_C.tex (copied) (copied from trunk/DOC/BETA/Chapters/Annex_C.tex) (10 diffs)

trunk/DOC/TexFiles/Chapters/Annex_C.tex

@@ -9 +9 @@
 I'm writting this appendix. It will be available in a forthcoming release of the documentation
 
-\gmcomment{
+%\gmcomment{
 
 % ================================================================
@@ -160 +160 @@
 and introducing the horizontal divergence $\chi $, it becomes:
 \begin{align*}
-\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left(  \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_1T\,e_2T\,e_3T\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 
+\equiv& \sum\limits_{i,j,k} - \frac{1} {2} \left(  \frac{\zeta} {e_{3f}} \right)^2 \; \overline{\overline{ e_{1T}\,e_{2T}\,e_{3T}\, \chi}}^{\,i+1/2,j+1/2} \;\;\equiv 0 
 \qquad \qquad \qquad \qquad \qquad \qquad \qquad &&&&\\
 \end{align*}
@@ -169 +169 @@
 the two components of the vorticity term is optionally offered (ENE scheme) :
 \begin{equation*}
-\frac{1} {e_3 }\nabla \times 
-   \left( 
-   \zeta \;{\textbf{k}}\times {\textbf {U}}_h 
-   \right)
+   - \zeta \;{\textbf{k}}\times {\textbf {U}}_h 
 \equiv 
    \left( {{
@@ -192 +189 @@
 enstrophy on the relative vorticity term and energy on the Coriolis term.
 \begin{flalign*}
-&\int\limits_D \textbf{U}_h \times   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
+&\int\limits_D - \textbf{U}_h \cdot   \left(  \zeta \;\textbf{k} \times \textbf{U}_h  \right)  \;  dv &&  \\
 \equiv& \sum\limits_{i,j,k}   \biggl\{    
       \overline {\left(  \frac{\zeta} {e_{3f}}      \right) 
@@ -348 +345 @@
       &&& \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
    + \delta_k         \left[   \overline {e_{1w}\,e_{2w}\,w}^{\,i+12}\;\overline u^{\,k+1/2}  \right]
-      \; \biggr\}    &&& \displaybreak[0] \\
-%
-\equiv& \sum\limits_{i,j,k}
+      \; \biggr\} \; u    &&& \displaybreak[0] \\
+%
+\equiv& - \sum\limits_{i,j,k}
    \biggl\{ 
       \overline {e_{2u}\,e_{3u}\,u}^{\,i}\;        \overline u^{\,i}       \delta_i 
@@ -359 +356 @@
        + \overline {e_{1w}\,e_{2w}\,w}^{\,i+1/2}\; \overline u^{\,k+1/2}   \delta_{k+1/2}    \left[ u \right]     \biggr\}     && \displaybreak[0] \\
 %
-\equiv& \sum\limits_{i,j,k}
+\equiv& - \sum\limits_{i,j,k}
    \biggl\{ 
         \overline {e_{2u}\,e_{3u}\,u}^{\,i}        \delta_i       \left[ u^2 \right] 
@@ -1224 +1221 @@
 constraint of conservation of tracers:
 \begin{flalign*}
-&\int\limits_D  T\;\nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
+&\int\limits_D  \nabla  \cdot \left( A\;\nabla T \right)\;dv  &&&\\ 
 \\
 &\equiv \sum\limits_{i,j,k} 
@@ -1248 +1245 @@
 \end{flalign*}
 
+In fact, this property is simply resulting from the flux form of the operator. 
 
 % -------------------------------------------------------------------------------------------------------------
@@ -1257 +1255 @@
 constraint of dissipation of tracer variance:
 \begin{flalign*}
-\int\limits_D T\;\nabla \cdot \left( A\;\nabla T \right)\;dv   &&&\\ 
-\end{flalign*}
-\begin{flalign*}
-\equiv \sum\limits_{i,j,k} T
-   \biggl\{    \biggr.
-   \delta_i 
-      \left[ 
-      A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} 
-         \left[ T \right] 
-      \right]
-   + \delta_j 
-      &\left[ 
-      A_v^{\,lT} \frac{e_{1v}\,e_{3v}} {e_{2v}} \delta_{j+1/2} 
-         \left[ T \right] 
-      \right] 
-      && \\ 
-    \biggl. 
-    + \delta_k 
-      &\left[ 
-      A_w^{\,vT} \frac{e_{1T}\,e_{2T}} {e_{3T}} \delta_{k+1/2} 
-         \left[ T \right] 
-      \right]
-   \biggr\} 
-   &&\\ 
+\int\limits_D T\;\nabla & \cdot \left( A\;\nabla T \right)\;dv &&&\\ 
+&\equiv   \sum\limits_{i,j,k} \; T
+\biggl\{  \biggr.
+     \delta_i \left[ A_u^{\,lT} \frac{e_{2u}\,e_{3u}} {e_{1u}} \delta_{i+1/2} \left[T\right] \right]
+& + \delta_j \left[ A_v^{\,lT} \frac{e_{1v} \,e_{3v}} {e_{2v}} \delta_{j+1/2} \left[T\right] \right] 
+      \quad&& \\ 
+ \biggl. 
+&&+ \delta_k \left[A_w^{\,vT}\frac{e_{1T}\,e_{2T}} {e_{3T}}\delta_{k+1/2}\left[T\right]\right]
+\biggr\} && 
 \end{flalign*}
 \begin{flalign*}
@@ -1313 +1296 @@
 
 %%%%  end of appendix in gm comment
-}
+%}