thermodynamics occurs in the form of the pressure gradient


The foundation of the CMAQ’s multiscale capability is the

use of a consistent governing set of equations for

meteorological and air quality modeling applications as

proposed by Byun [28]. We can relate the generalized

meteorological curvilinear coordinates (xˆ 1, xˆ 2 , xˆ 3 ,tˆ ) in a

conformal map projection to the rotated earth-tangential

coordinates (x, y, z, t) as:

xˆ 1 = mx

xˆ 2 = my

xˆ 3 = s

tˆ = t

x = m1xˆ 1

y = m1xˆ 2

z = h(xˆ 1, xˆ 2, xˆ 3,tˆ ) = hAGL(xˆ 1, xˆ 2, xˆ 3,tˆ ) + zsfc (xˆ 1, xˆ 2 )

t = tˆ

where m is the map scale factor, s is the generalized

meteorological vertical coordinate, zsfc is the topographic

height, h is the geometric height, and hAGL represents height

above the ground level (AGL). Here, s is a symbol for the

generalized vertical coordinate and z is a specific Cartesian

coordinate representing vertical displacement, i.e., height.

In the generalized coordinate system, the contravariant and

covariant wind components are represented with vˆ j and vˆ k .

The square root of the determinant of the metric (Jacobian)


where Vz = s / m is the horizontal wind vector represented

in the Cartesian coordinate system, and F3 is the forcing

term for the w-component. The contravariant vertical

velocity component is related to the Cartesian vertical

velocity with:

With the ideal gas law, the thermodynamic variables (i.e.,

temperature, entropy, pressure gradients, and density) are

diagnostically related (Byun [28]). The entropy per unit

volume (entropy density), ζ is defined as

where T is temperature, Cvd is the specific heat capacity for

dry air at constant volume, and Rd is the dry air gas

constant. Too is temperature of the reference atmosphere at

the reference pressure poo = 105 Pascal. The atmospheric

pressure is treated as a thermodynamic variable that is fully

defined by the density and entropy of the atmosphere. The

conservation equations for air density, entropy density, and

tracer concentrations are:



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