Horizontal discretization

The Arakawa-C grid is used for this model over a limited area domain (LAM), and it is shown in Fig. 5. Note that integer values are used for the indices in order to facilitate compatibility with the model code. The grid has $N_x x N_y$ grid boxes.

Figure 5: The location of the variables on the staggered (Arakawa) C-grid. The scalar and vertical velocity (sigma-dot) terms are located at the center of each grid box, while the horizontal velocity components, $u$ and $v$ are offset by half-grid distances. Mass variables are defined at $N_x x N_y$ grid points, while for $u$ there are $N_{x+1} x N_y$ values and for $v$ there are $N_{x} x N_{y+1}$ values.

The grid distance in Fig. 5 is defined as

$$d = \Delta x = \Delta y$$ (122)

The horizontal mass fluxes are located at the velocity points and are written as

$$U_{i,j} = u_{i,j} \left( \mu_{d\,i-1,j} + \mu_{d\,i,j}\right)/2$$ (123)

$$V_{i,j} = v_{i,j} \left( \mu_{d\,i,j-1} + \mu_{d\,i,j}\right)/2$$ (124)

so that the horizontal divergence is simply expressed as

$$\nabla\cdot{\left( {\bf V}\mu\right)}_{i,j} = \left(U_{i+1,j}-U_{i,j}+V_{i,j+1}-V_{i,j}\right)/d$$ (125)