How to Use Computational Fluid Dynamics Models for Urban Canopy Parameterizations - Meteorological and Air Quality Models for Urban Areas - page 26

if turbulent or not) are resolved. The advantage of this approach is that it is relatively
easy to understand. The disadvantage is that the turbulence closures should depend
on the resolution. Moreover, when the resolution reaches few hundreds of meters,
the largest turbulent eddies may be explicitly resolved by the model. If part of the
turbulent motions is resolved explicitly, due to their stochastic nature, only one of
the many possible realizations is represented by model's solution. In such situa-
tions, a time average of the results may be needed in order to recover some useful
statistical information (Calmet et al., 2007). However, the determination of the aver-
aging time, in particular, in complex situations as urban areas, is not always easy to
identify.
A second approach consists in performing at first a Reynolds decomposition of
the atmospheric variables in mean (deterministic) and turbulent (stochastic) parts,
where the mean can be defined using a probability density function f.

∞

ψ = ψ

( ψ )

ψ

f d

(4.2)

−∞

ψ = ψ − ψ

With this method all the turbulent features are not resolved and need to be param-
eterized, while only the mean deterministic fields are explicitly resolved. Then, due
to the spatial resolution of the mesoscale model a spatial average over the volume
of the cell V is needed:

V ψ

dV

ψ =

(4.3)

V

The consequences of this procedure are in the arising of an extra term in the con-
servation equation (dispersive flux), representing the flux due to mean deterministic
structures smaller than the grid cell. This term is usually neglected in mesoscale
models, but it may be important over heterogeneous surface as urban areas. The
advantage of this approach is that the model outputs are mean values, which is use-
ful information in many applications. Note, also, that, despite in the majority of
publications on model formulations the definition (1) is used for the averaging oper-
ator, the turbulence closures adopted in the models are largely based on definition
(2) (or ensemble averaging; see, for example, Mellor and Yamada's papers).
In any case, no matter which definition is chosen, it is clear that model results
are spatial averages, and should be compared with spatially averaged variables. The
problem is that in urban areas the heterogeneity is so important that a point mea-
surement cannot be representative of a spatial average. The ergodic assumption, in
fact, usually done over flat and homogeneous surfaces, saying that the spatial aver-
age is equal to a time average in one point, is clearly not valid in urban areas, in
particular within the urban canopy. One solution could be to have a measurement
network very dense in order to be able performing such spatial averaging. This can
be technically difficult and very costly. Another option is to use CFD models.