Pole Points

The global application also uses the C-grid, but with two simple modifications. First, the poles correspond to $v$ velocity points (Fig. 9). Note that at the poles, $m_x\rightarrow\infty$, so that from Eq. 287, the mass fluxes $V_{i,Ny+1}=V_{i,1}=0$. Finally, the lateral boundary conditions are periodic.

Figure 9: The location of the variables on the staggered (Arakawa) C-grid using a Plate Carrée projection at uppermost grid row bordering the north pole: the reversed stencil applies to the south pole: poles correspond to $V$ ($v$) points. The scalar and vertical velocity (sigma-dot) terms are located at the center of each grid box, while the horizontal velocity components, $u$ and $v$ (and mass fluxes, $U$ and $V$) are offset by half-grid distances.

When averaging scalars in the meridional direction (i.e. to $v$ points on the C-grid), the averaging operator is defined as

$${\overline \phi^y} = {\frac {A_{i,j}\phi_{i,j} + A_{i,j-1}\phi_{i,j-1}}{A_{i,j}+A_{i,j-1}}}$$ (311)

where $\phi$ is some arbitrary scalar located at the mass point on the C-grid, and the grid cell area is defined as

$$A_{i,j} = {\frac {d^2}{m_{x\,i,j}\,m_{y\,i,j}}}$$ (312)

Simple arithmetic averaging is used in the longitudinal ($x$) direction (see Eq. 124).

Finally, for horizontal diffusion considering map factors (second and fourth order options), the maximum diffusivity must be computed using map factors for the $x$ and $y$ directions. For example, using the basic definition from Eq. 103 in the $x$ direction, the maximum diffusivity is defined as

$$\kappa_{x\,{\rm max}}^o = {\frac {d^{o}}{2^{o} \, \Delta t\, m_{x}^o}}$$ (313)

applied to the diffusive fluxes in Section 2.7.5. The same rule applies to the horizontal divergence damping coefficients in Section 2.2.1.