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Next: 2.2 Time-stepping Up: 2. Discretization and Algorithm Previous: 2. Discretization and Algorithm   Contents

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}_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} $
$ ~^{-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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Copyright 2006 Massachusetts Institute of Technology Last update 2018-01-23