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