The coordinate system for MM5 is (x, y, sigma). The x and y are a regular lattice of equally spaced points (delta-x = delta-y = horizontal grid spacing in kilometers) forming rows and columns. Sigma is a terrain-following vertical coordinate that is a function of the pressure at the point on the grid (in hydrostatic runs) or the reference state pressure (in non-hydrostatic runs), the surface pressure at the grid point, and the pressure at the top of the model. Sigma varies from 1 at the surface to 0 at the top of the model. The influence of the terrain on the sigma structure diminishes with height, so that the sigma surfaces near the top of the model are nearly parallel.
See Figure 5.1 for a sample sigma structure of 16 vertical layers (17 full-sigma levels). Dashed lines denote half-sigma levels, and solid lines denote full-sigma levels. Vertical velocity (ω) is calculated on full-sigma levels. All other prognostic variables (a) are calculated on half-sigma levels. The figure (based on Dudhia et al., [1998]) is not drawn to scale. The vertical grid in MM5 is staggered, and all of the prognostic meteorological variables, except for vertical motion, are calculated on half-sigma surfaces. Each half-sigma is located in space approximately half the distance between two full-sigma levels.
The horizontal grid in MM5 has an Arakawa-B staggering of the velocity vectors with respect to the scalars [Arakawa and Lamb, 1977]. The momentum variables (u- and v-components of wind and the Coriolis force) are on “dot” points, while all other variables (e.g., mass and moisture variables) are on “cross” points. The dot points form the regular lattice for the simulation domain, while the cross points are offset by 0.5 grid point in both the x and y directions. Refer to Figure 5.2 for a sample Arakawa-B staggering. Note that the interpolation of the variables to the staggered grid is done automatically within the INTERP program. The smaller inner box is a representative mesh staggering for a 3:1 coarse-grid distance to fine-grid distance ratio.
