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2.1 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}_v , {\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} $
$ ~^{-i}$ : $ \overline{\Phi}^i = ( \Phi_{i+1/2} + \Phi_{i-1/2} ) / 2 $
$ \delta_x $ : $ \delta_x \Phi = \frac{1}{\Delta x} \delta_i \Phi $

$ \overline{\nabla}$ = horizontal gradient operator : $ \overline{\nabla} \Phi = \{ \delta_x \Phi , \delta_y \Phi \}$
$ \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 \} $
$ \overline{\nabla}^2 $ = horizontal Laplacian operator : $ \overline{\nabla}^2 \Phi = \overline{\nabla}\cdot \overline{\nabla}\Phi $


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