60
Air Pollution and Turbulence: Modeling and Applications
and Q(k,s) given by
0
( , )
( , )
( , )d
t
E k s
Q k s
E k s s
= ∫
(3.101b)
Since, to our knowledge, there are no conclusive observations of the CBL turbulence
decay process, we compare our theoretical expressions with LES data (Nieuwstadt
and Brost, 1986). The velocity variance of the turbulent fl ow is calculated from the
following equation:
2
0
( ; )
( , ; )d
i
i
t z
F k t z k
∞
σ
=
∫
(3.102)
Figure 3.3 shows the temporal evolution of the vertical velocity variance decay
across the CBL (0.2 < z/z i < 0.8), calculated from Equations 3.96, 3.99, 3.101, and
3.102 made dimensionless by
2
*
w .
Figure 3.3 shows that our theoretical model (solid line) agrees very well with the
results simulated by LES (crosses; Nieuwstadt and Brost, 1986).
The eddy diffusivities calculated by Batchelor (1949) can be identifi ed with
those from advection–diffusion equation when a homogeneous turbulent fl ow
0.1
0.01
1/2 (σ
w
2
/w
*
2
)
1E-3
0.1
1
t *
10
FIGURE 3.3 Temporal evolution of the vertical velocity variance decay across the CBL
(0.2 < z/z i < 0.8), calculated from Equations 3.96, 3.99, 3.101, and 3.102 made dimensionless
by
2
*
w .
© 2010 by Taylor and Francis Group, LLC