--- manual/s_algorithm/text/notation.tex	2001/08/08 16:15:21	1.1
+++ manual/s_algorithm/text/notation.tex	2011/05/04 22:42:48	1.8
@@ -1,27 +1,42 @@
-% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/notation.tex,v 1.1 2001/08/08 16:15:21 adcroft Exp $
+% $Header: /home/ubuntu/mnt/e9_copy/manual/s_algorithm/text/notation.tex,v 1.8 2011/05/04 22:42:48 jmc Exp $
 % $Name:  $
 
-\section{Notations} 
+\section{Notation} 
 
-The notations we use to discribe the discrete formulation 
-of the model are summarised hereafter:\\
+Because of the particularity of the vertical direction in stratified fluid
+context, in this chapter, the vector notations are mostly used
+for the horizontal component:
+the horizontal part of a vector is simply written 
+$\vec{\bf v}$ (instead of ${\bf v_h}$ or $\vec{\mathbf{v}}_{h}$ in chaper 1)
+and a 3.D vector is simply written $\vec{v}$ (instead of $\vec{\mathbf{v}}$ 
+in chapter 1).
+
+The notations we use to describe the discrete formulation 
+of the model are summarized hereafter:\\
 general notation:
 \\ $\Delta x, \Delta y, \Delta r$ grid spacing in X,Y,R directions.
-\\ $A_o$ : Area of the face orthogonal to "o" direction (o=u,v,w ...).
-\\ ${\cal V}_u , {\cal V}_v , {\cal V}_v , {\cal V}_\theta$ :
+\\ $A_c,A_w,A_s,A_{\zeta}$ : 
+horizontal area of a grid cell surrounding $\theta,u,v,\zeta$ point.
+\\ ${\cal V}_u , {\cal V}_v , {\cal V}_w , {\cal V}_\theta$ :
 Volume of the grid box surrounding $u,v,w,\theta$ point;
 \\ $i,j,k$ : current index relative to X,Y,R directions;
 \\basic operator:
-\\ $\delta_i $ : $\delta_i \Phi = \Phi_{i+1} - \Phi_i $
-\\ $\overline{~}i$ : $\overline{\Phi}^i = ( \Phi_{i+1} + \Phi_i ) / 2 $ 
+\\ $\delta_i $ : $\delta_i \Phi = \Phi_{i+1/2} - \Phi_{i-1/2} $
+\label{eq:delta_i}
+\\ $~^{-i}$ : $\overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2 $ 
+\label{eq:bar_i}
 \\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
+\label{eq:delta_x}
 \\
-\\ $\overline{\nabla}$ = gradient operator :  
+\\ $\overline{\nabla}$ = horizontal gradient operator :  
 $\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
-\\ $\overline{\nabla} \cdot$ = divergence operator :  
+\label{eq:d_grad}
+\\ $\overline{\nabla} \cdot$ = horizontal divergence operator :  
 $\overline{\nabla}\cdot \vec{\mathrm{f}}  = 
-\frac{1}{\cal V} \{ \delta_i A_x \mathrm{f}_x 
-                  + \delta_j A_y \mathrm{f}_y \} $
-\\ $\overline{\nabla}^2 $ = Laplacien operator :
+\frac{1}{\cal A} \{ \delta_i \Delta y \, \mathrm{f}_x 
+                  + \delta_j \Delta x \, \mathrm{f}_y \} $
+\label{eq:d_div}
+\\ $\overline{\nabla}^2 $ = horizontal Laplacian operator :
 $ \overline{\nabla}^2 \Phi = 
    \overline{\nabla}\cdot \overline{\nabla}\Phi $
+\label{eq:d_lap}