Geoscience Reference
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density
P
of fractal fluctuations. Therefore, the probability density
P
in the primary
eddy growth region (σ ≥ 1 and σ ≤ − 1) is given using the computed value of
k
as
P
= τ
4
(Eq. 1.20).
The normalized radius
r
an
is given in terms of σ and the golden mean τ from
Eqs. (1.25) and (1.31) as follows:
k
3
2
23 23
ln
z
=
ln
r
(1.34)
an
/
τ
σ
/
r
=
z
=
.
an
The normalized aerosol size spectrum is obtained by plotting a graph of normalized
aerosol concentration
1
2
3
σ
dN
dr
P
3
2
=
τ
(Eq. 1.33) versus the normalized aero-
N
(ln)
an
sol radius
r
an
= τ
2/
(Eq. 1.34). The normalized aerosol size spectrum is derived di-
rectly from the universal probability density
P
distribution characteristics of fractal
fluctuations (Eqs. 1.16 and 1.20) and is independent of the height
z
of measurement
and is universal for aerosols in turbulent atmospheric flows. The aerosol size spec-
trum is computed starting from the minimum size, the corresponding probability
density
P
(Eq. 1.33) refers to the cumulative probability density starting from 1 and
is computed as equal to
P
=−
−
τ
σ
.
1
The universal normalized aerosol size spectrum represented by
1
N
N
lnr
d
an
)
ver-
d(
sus
r
an
is shown in Fig.
1.9
.
1.6.5
Large Eddy Growth Time
The time Γ taken for the steady-state aerosol concentration
f
to be established at
the normalized height
z
is equal to the time taken for the large eddy to grow to the
height
z
and is computed as follows.
The time required for the large eddy of radius
R
to grow from the primary turbu-
lence scale radius
r
*
is computed as follows.
The scale ratio
z
R
r
=
.
*
Therefore, for constant turbulence radius
r
*
dR
r
dz
=
.
*
The incremental growth d
R
of large eddy radius is equal to
dR rdz
=
*
.
The time period d
t
for the incremental cloud growth is expressed as follows:
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