Diffusion and filters

The diffusion-damping terms, $D_\phi$, in Eq.s 30-34 and Eq.s 49-50 are defined as

$$D_u = D_{h\,u}^o + D_{z\,u}^o + D_{d\,u}^o + {\cal{D}}_{h\,u}$$ (76)

$$D_v = D_{h\,v}^o + D_{z\,v}^o + D_{d\,v}^o + {\cal{D}}_{h\,v}$$ (77)

$$D_w = D_{h\,w}^o + {\cal{D}}_{z,w}$$ (78)

$$D_T = D_{h\,T}^o + D_{z\,T}^o + {\cal{D}}_{h\,T}$$ (79)

$$D_r = D_{h\,r}^o + D_{z\,r}^o + {\cal{D}}_{h\,r_q}$$ (80)

$$D_e = D_{h\,e}^o + D_{z\,e}^o$$ (81)

where $D_h$ and $D_z$ correspond to the horizontal and vertical components of the diffusion, respectively. The divergence damping terms (also horizontal) are represented by $D_d$. The order of the operators is given by the superscript $o$ (and options exist for order 2, 4 or 6). The ${\cal{D}}$ terms represent Raleigh relaxation type terms, and ${\cal{D}}_h$ are those which are non-zero in the lateral sponge zones associated with incorporating the lateral boundary conditions. Note that only ${\cal{D}}_{h\,r_q}$ is non-zero since only the water vapor is nudged in the lateral sponge zones.