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numerical models of dynamical systems is not yet identified. Finite precision com-
puter realizations of mathematical models (nonlinear) of dynamical systems do not
give realistic solutions because of propagation of round-off error into mainstream
computation (Selvam
1993
,
2007
; Sivak et al.
2013
, Lawrence Berkeley National
Laboratory
2013
). During the past three decades, Lovejoy and his group (Lovejoy
and Schertzer
2010
) have done extensive observational and theoretical studies of
fractal nature of atmospheric flows and emphasize the urgent need to formulate and
incorporate quantitative theoretical concepts of fractals in mainstream classical me-
teorological theory. The empirical analyses summarized by Lovejoy and Schertzer
(
2010
) show that the statistical properties such as the mean and variance of atmo-
spheric parameters (temperature, pressure, etc.) are scale dependent and exhibit a
power law relationship with a long fat tail over the space-time scales of measure-
ment. The physics of the widely documented fractal fluctuations in dynamical sys-
tems is not yet identified. The traditional statistical normal (Gaussian) probability
distribution is not applicable for statistical analysis of fractal space-time data sets
because of the following reasons: (i) Gaussian distribution assumes independent
(uncorrelated) data points while fractal fluctuations exhibit long-range correlations;
and (ii) the probability distribution of fractal fluctuations exhibit a long fat tail, i.e.,
extreme events are of more common occurrence than given by the classical theory
(Selvam
2009
; Lovejoy and Schertzer
2010
).
Numerical computations such as addition, multiplication, etc., have inherent
round-off errors. Iterative computations magnify these round-off errors because of
feedback with amplification. Propagation of round-off errors into the main stream
computation results in
deterministic chaos
in computer realizations of nonlinear
mathematical models used for simulation of real-world dynamical systems. The
round-off error growth structures generate the beautiful fractal patterns (Fig.
1.1
).
The complex structures found in nature are all
fractals
, i.e., possesses self-similar
geometry. Simple iterative growth processes may underlie the complex patterns
found in nature. The study of
fractals
belongs to the field of
nonlinear dynamics
and chaos
, a multidisciplinary area of intensive research in all fields of science.
Numerical integration schemes incorporate iterative computations.
Self-similar spatial structures imply long-range spatial correlations or nonlocal
connections. Global cloud cover pattern exhibits
fractal
geometry. The existence of
long-range spatial correlations such as the
El-Nino
impact on global climate is now
accepted. The global atmosphere acts as a unified whole, where, local perturbations
produce a global response.
A general systems theory model for fractal fluctuations (Selvam
2005
,
2007
,
1990
,
2009
,
2011
,
2012a
,
b
,
2013
,
2014
; Selvam and Fadnavis
1998
) predicts that
the amplitude probability distribution as well as the power (variance) spectrum of
fractal fluctuations follow the universal inverse power law τ
−4σ
where τ is the golden
mean (≈ 1.618) and σ the normalized standard deviation. The atmospheric aerosol
size spectrum is derived in terms of the universal inverse power law characteriz-
ing atmospheric eddy energy spectrum. A universal (scale independent) spectrum
is derived for homogeneous (same density) suspended atmospheric particulate size
distribution expressed as a function of the golden mean τ (≈ 1.618), the total number
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