Geoscience Reference
In-Depth Information
d
N
r
=
3
2
2
(1.28)
NWz
.
d
(ln)
an
Substituting for
W
from Eqs. (1.16) and (1.20) in terms of the universal probability
density
P
for fractal fluctuations
d
N
r
=
3
2
2
NPz
.
(1.29)
d
(ln)
an
The above equation is for the scale length
z
. The volume across unit cross-section
associated with scale length
z
is equal to
z
. The particle radius corresponding to this
volume is equal to
z
1/3
.
The above Eq. (1.28) is for the scale length
z
and the corresponding radius equal
to
z
1/3
. The Eq. (1.28) normalized for scale length and associated drop radius is
given as follows:
2
d
N
r
3
2
NPz
3
2
2
3
=
=
NPz
.
(1.30)
d
(ln)
1
3
zz
×
an
The general systems theory predicts that fractal fluctuations may be resolved into
an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern
for the internal structure such that the successive eddy lengths follow the Fibonacci
mathematical number series (Selvam and Fadnavis
1998
; Selvam
1990
,
2005
,
2007
,
2009
,
2011
,
2012a
,
b
,
2013
). The eddy length scale ratio
z
for length step σ is, there-
fore, a function of the golden mean τ given as follows:
(1.31)
z
= τ
σ
.
Expressing the scale length
z
in terms of the golden mean τ in Eq. (1.29)
dN
dr
NP
=
3
2
2
3
σ
τ
.
(1.32)
(ln)
an
In Eq. (1.32),
N
is the steady-state aerosol concentration at level
z
. The normalized
aerosol concentration any level
z
is given as follows:
1
dN
dr
P
3
2
τ
σ
2
3
=
.
(1.33)
N
(ln)
an
The fractal fluctuations probability density is
P
= τ
4
(Eq. 1.16) for values of the
normalized deviation σ ≥ 1 and σ ≤ − 1 on either side of σ = 0 as explained earlier
(Sects. 1.6.2 and 1.6.3). Values of the normalized deviation − 1 ≤ σ ≤ 1 refer to re-
gions of primary eddy growth where the fractional volume dilution
k
(Eq. 1.2) by
eddy mixing process has to be taken into account for determining the probability
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