where ϕ

i is the trace species concentration (mass per unit

volume), and the Q-terms represent sources and sinks of

each conservative property. Although the source term for

air density (Qρ ) should be zero in an ideal case, it is

retained here to capture the possible density error originating

from numerical procedures in a meteorological model.

Because the error influences computations of other

parameters such as vertical velocity and mass conservation,

it is important to minimize propagation of the error in the

system. Eqs. (3), (4), (7), (8) and (9) with additional

diagnostic relations form a governing set of equations for

the fully compressible atmosphere.

The generalized coordinate system allows transformations

among various horizontal map projections (e.g., spherical,

rectangular, Lambert, Mercator, and polar stereographic),

and various vertical coordinates (e.g., pressure or geometric

height). For most urban and regional applications, the

equations account for the choice of horizontal map

projection with the map scale factor by simply changing a

few scaling parameters defining the domain boundary, map

origin, and orientation. The dynamics used in

meteorological models are often linked to the choice of the

vertical coordinate. In those vertical coordinates that

depend on atmospheric pressure, the coordinate values

decrease with height. To simplify implementation of the

generalized coordinate in CMAQ, without a loss of

generality, we redefine the terrain-following vertical

coordinate s with a positive definite coordinate ξ = xˆ 3 as




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