Environmental Engineering Reference
In-Depth Information
Unfortunately such resolution is 2-3 orders of magnitude larger than the spa-
tial scale of heterogeneities in urban areas (the size of the streets or buildings can
be considered of the order of 1-10 m). Even if the computer power is increasing
very quickly, the gap to bridge is quite significant, before we will be able to run a
mesoscale model with a spatial resolution of few meters over a typical mesoscale
domain. A quick estimation can be done. Let assume that the spatial resolution
must be increased by a factor 10
2-3
in both horizontal direction, and a factor 10 in
the vertical. The total number of point will increase by a factor 10
5-7
. Moreover, an
increase of resolution will imply a reduction of the time step by the same amount to
fulfil the CFL (Courant Friedich Levy) condition. It can be estimated that the time
step will need to be decreased by a factor 100. In total, so, we can expect that the
CPU time of the simulation will increase by factor 10
7-9
compared to the CPU time
of a standard today's mesoscale simulation. Assuming that the computational power
will keep increasing by a factor of ten every five years, (as it has happened in the
last decades, Foster, 1994), it is possible that 35-45 years will be needed before to
reach enough CPU power to run a mesoscale model with a resolution of few meters.
Even in the case of a reduction of model domain size (for example to the city size),
still we can expect that 20-25 years will be needed. Moreover, the increased CPU
time may be used not only to increase the resolution, but also for other needs, for
example, to make:
•
longer runs (multi-year);
•
multiple runs with different input parameters, to span their uncertainty (ensemble
approaches);
•
using of more complex and sophisticated physical parameterizations;
•
coupling of the model with other models (hydrological, building energy, etc.).
Due to these considerations, it seems reasonable to make an effort to improve the
techniques used to parameterize urban impact on the spatially averaged variables
computed by mesoscale models (Urban Canopy Parameterizations, UCP).
4.2 Averaging Schemes in Mesoscale Models
There are two ways to define the averaging operators used in mesoscale models.
The first (Pielke, 1984) is to consider the averaging operator as a simple spatial
average over the grid cell and time average over the time step, or:
x
+
x
/
2
y
+
y
/
2
z
+
z
/
2
t
+
t
t
ψ
(
x
,
y
,
z
,
t
)
dxdydzdt
x
−
x
/
2
y
−
y
/
2
z
−
z
/
2
ψ =
(4.1)
x
y
z
t
With this approach all the features smaller than the grid cell (no matter if turbu-
lent or not) are parameterized, and all the features larger than the grid cell (no matter
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