s_algorithm/text/notation.tex

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\section{Notation} 

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.
\\ $A_c,A_w,A_s,A_{\zeta}$ : 
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+1/2} - \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}
\\
\\ $\overline{\nabla}$ = horizontal gradient operator :  
$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
\label{eq:d_grad}
\\ $\overline{\nabla} \cdot$ = horizontal divergence operator :  
$\overline{\nabla}\cdot \vec{\mathrm{f}}  = 
\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}
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3
4	\section{Notation}
5
6	Because of the particularity of the vertical direction in stratified fluid
7	context, in this chapter, the vector notations are mostly used
8	for the horizontal component:
9	the horizontal part of a vector is simply written
10	$\vec{\bf v}$ (instead of ${\bf v_h}$ or $\vec{\mathbf{v}}_{h}$ in chaper 1)
11	and a 3.D vector is simply written $\vec{v}$ (instead of $\vec{\mathbf{v}}$
12	in chapter 1).
13
14	The notations we use to describe the discrete formulation
15	of the model are summarized hereafter:\\
16	general notation:
17	\\ $\Delta x, \Delta y, \Delta r$ grid spacing in X,Y,R directions.
18	\\ $A_c,A_w,A_s,A_{\zeta}$ :
19	horizontal area of a grid cell surrounding $\theta,u,v,\zeta$ point.
20	\\ ${\cal V}_u , {\cal V}_v , {\cal V}_w , {\cal V}_\theta$ :
21	Volume of the grid box surrounding $u,v,w,\theta$ point;
22	\\ $i,j,k$ : current index relative to X,Y,R directions;
23	\\basic operator:
24	\\ $\delta_i $ : $\delta_i \Phi = \Phi_{i+1/2} - \Phi_{i-1/2} $
25	\label{eq:delta_i}
26	\\ $~^{-i}$ : $\overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2 $
27	\label{eq:bar_i}
28	\\ $\delta_x $ : $\delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $
29	\label{eq:delta_x}
30	\\
31	\\ $\overline{\nabla}$ = horizontal gradient operator :
32	$\overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
33	\label{eq:d_grad}
34	\\ $\overline{\nabla} \cdot$ = horizontal divergence operator :
35	$\overline{\nabla}\cdot \vec{\mathrm{f}} =
36	\frac{1}{\cal A} \{ \delta_i \Delta y \, \mathrm{f}_x
37	+ \delta_j \Delta x \, \mathrm{f}_y \} $
38	\label{eq:d_div}
39	\\ $\overline{\nabla}^2 $ = horizontal Laplacian operator :
40	$ \overline{\nabla}^2 \Phi =
41	\overline{\nabla}\cdot \overline{\nabla}\Phi $
42	\label{eq:d_lap}