5. GOVERNING DIFFUSION EQUATION AND

STURCTURE OF CMAQ CTM

The conservation equation for the trace species, Eq. (9),

describes the instantaneous state of the trace concentration

in the atmosphere. Because of the stochastic nature of

atmospheric motions, the equation must be averaged to form

a deterministic conservation equation before it can be solved

numerically.

5.1 Governing Diffusion Equation

The air density and species concentration are decomposed

into mean and turbulent terms (Reynolds decomposition):

where ρ , ρ' are the mean and turbulent components of air

density and ϕ i and ϕ

i ' the same for concentration of trace

species i, respectively. Some of the parameters in the

conservation equations (7) through (9) are nonlinearly

related to each other and, therefore, direct application of

Reynolds decomposition to these parameters will introduce

covariance terms that complicate the turbulence equations.

Instead, we define averaged mixing ratio and its fluctuation

component based on Eqs. (12) and (13):

Similarly, the average contravariant wind components and

their fluctuations are defined as

v

These definitions allow the continuity equation for the

Reynolds averaged variables to keep the original

conservation form as

Note that the metric tensor components (m and Jξ ) that

define the coordinate transformation rules are not turbulence

variables. This means that we can define the coordinates

based on the Reynolds averaged quantities. The vertical

grids are defined incrementally between time steps when a

time-dependent vertical coordinate is used.

Decomposing velocity components using Eqs. (15 a and b),

we obtain a Reynolds averaged trace species conservation

equation, neglecting the molecular diffusion:

n1001 - n1002 - n1003 - n1004 - n1005 - n1006 - n1007 - n1008 - n1009 - n1010 - n1011 - n1012 - n1013 - n1014 - n1015 - n1016 - n1017 - n1018 - n1019 - n1020 - n1021 - n1022 - n1023 - n1024 - n1025 - n1026 - n1027 - n1028 - n1029 - n1030 - n1031 - n1032 - n1033 - n1034 - n1035 - n1036 - n1037 - n1038 - n1039 - n1040 - n1041 - n1042 - n1043 - n1044 - n1045 - n1046 - n1047 - n1048 - n1049 - n1050

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