Relating Small-Scale Emission and Concentration Variability in Air Quality Models - Meteorological and Air Quality Models for Urban Areas - page 9

averaged over the grid cell where the emission source is located. The source can
be linear (e.g. roads), areal (fields or urban areas) or point (factory) but after the
averaging procedure, it is considered as a surface source with the same extent as the
grid cell. This means that not only is the surface heterogeneity lost in terms of its
level of variability but also it will not be accounted for in the upper atmospheric lev-
els and the impact of the spatial distribution of emissions on the spatial distribution
of concentrations, is lost. This can represent a serious issue in the case of passive
as well as chemically reactive species, or for the estimation of long- or short-term
exposures.
Since this crucial component can be at smaller scales than the grid size, it could
be accounted for through a sub-grid parameterization. In this paper, we propose a
novel approach to this parameterization, including a formulation for the sub-grid
variability of pollutant concentrations that takes into account the spatial heterogene-
ity of the emissions. The formulation that can be used in mesoscale models relies
on the resolution of a prognostic equation for the sub-grid concentration variance,
i.e. the quantity that accounts for the distribution of concentration within a grid-
cell of a mesoscale model, by using a 1.5 order closure. The parameterization is
implemented in a 3D transport model and tested against large eddy simulations of
convective atmospheric boundary layers.

2.2 Formulation

We propose a parameterization to account for the concentration variability ( c 2 ) that
is based the conservation equation of the concentration variance. The latter reads:

u i c 2
∂

c 2
∂

c 2
∂

− ∂

∂

u i ∂ 2 c u i ∂

c

2 c E

+ =− − ε c + , (2.1)

E

D

t x i ∂ x i x i

S

A T

G

where u i , c , E represent the wind components, the concentration and the emission.
Equation (2.1) accounts for the time evolution of concentration variance ( S ) created
by turbulent motion ( T ) and transported in 3D ( A and G ) space while it is dissipated
( D ). It contains an extra term that accounts for the contribution to the variance pro-
duction originating from the surface spatial variability of the emissions. In order to
solve (2.1) for the variance we need to close some of the terms. While terms G , T
and D can be closed conventionally by using well assessed parameterizations (see
Appendix), for the term E we propose the following expression,

r c 2 1 / 2 E 2 1 / 2 , c E =

(2.2)

where r is the correlation coefficient between the concentration and the emission
variances.