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suspended particulates will also exhibit a broadband size spectrum closely related
to the atmospheric eddy energy spectrum.
It is now established (Lovejoy and Schertzer
2010
) that atmospheric flows ex-
hibit self-similar fractal fluctuations generic to dynamical systems in nature such
as fluid flows, heart-beat patterns, population dynamics, spread of forest fires, etc.
Power spectra of fractal fluctuations exhibit inverse power law of form −
f
−α
, where
α is a constant indicating long-range space-time correlations or persistence. Inverse
power law for power spectrum indicates scale invariance, i.e., the eddy energies at
two different scales (space-time) are related to each other by a scale factor (α in
this case) alone independent of the intrinsic properties such as physical, chemical,
electrical, etc., of the dynamical system.
A general systems theory for turbulent fluid flows predicts that the eddy energy
spectrum, i.e., the variance (square of eddy amplitude) spectrum is the same as the
probability distribution
P
of the eddy amplitudes, i.e., the vertical velocity
W
val-
ues. Such a result that the additive amplitudes of eddies, when squared, represent
the probabilities is exhibited by the subatomic dynamics of quantum systems such
as the electron or photon. Therefore, the unpredictable or irregular fractal space-
time fluctuations generic to dynamical systems in nature, such as atmospheric flows
is a signature of quantum-like chaos. The general systems theory for turbulent fluid
flows predicts (Selvam and Fadnavis
1998
; Selvam
1990
,
2005
,
2007
,
2009
,
2011
,
2012a
,
b
,
2013
) that the atmospheric eddy energy spectrum represented by the prob-
ability distribution
P
follows inverse power law form incorporating the
golden mean
τ and the normalized deviation σ for values of σ ≥ 1 and σ ≤ − 1 as given below:
τ
4
(1.15)
P
=
.
The vertical velocity
W
spectrum will, therefore, be represented by the probability
distribution
P
for values of σ≥ 1 and σ≤ − 1 given in Eq. (1.15) since fractal fluctua-
tions exhibit quantum-like chaos as explained above.
W
==τ
4
(1.16)
.
Values of the normalized deviation σ in the range − 1 < σ < 1 refer to regions of pri-
mary 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 distribution
P
of fractal fluctuations (see Sect. 1.6.3 below).
1.6.3
Primary Eddy Growth Region Fractal Fluctuation
Probability Distribution
Normalized deviation σ ranging from − 1 to + 1 corresponds to the primary eddy
growth region. In this region, the probability
P
is shown to be equal to
P
τ
4
k
=
(see
below) where
k
is the fractional volume dilution by eddy mixing (Eq. 1.2).
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