1.
Introduction
2.
Nonlinear Dynamics and Chaos
2.1 Fractal
Geometry
2.1.1
Fractals in Pure Mathematics
2.1.2
Fractal Time Signals and Power Laws
2.1.3
Self-Organized Criticality: Space-Time Fractals
1. Introduction
Atmospheric flows exhibit irregular
(chaotic) space-time fluctuations on all scales ranging from climate (kilometers-years)
to turbulence (millimeters-second) and is a representative example of turbulent
fluid flows. Dynamical systems in nature, i.e., systems that change with
time, such as fluid flows, heartbeat patterns, spread of infectious diseases,
etc., exhibit nonlinear (unpredictable) fluctuations. Conventional mathematical
and statistical theories deal only with linear systems and the exact quantification
and description of nonlinear fluctuations was not possible till the identification
in the 1970s by Mandelbrot (1977;1983 References
) of the universal symmetry of selfsimilarity, i.e. fractal geometry
underlying the seemingly irregular fluctuations in space and time (Schroeder,
1991; Stanley, 1995 References ). The
study of selfsimilar space-time fluctuations generic to dynamical systems,
now (since 1980s), belongs to the newly emerging multidisciplinary science
of nonlinear dynamics and chaos (Gleick,1987 References
).
Selfsimilar fluctuations
in space and time imply long-range spatiotemporal correlations and are
recently identified as signatures of self-organized criticality (Bak
et.
al.,
1988 References ). Self-organized criticality
in atmospheric flows is manifested as the fractal geometry to the
global cloud cover pattern concomitant with inverse power law form for
power spectra of temporal fluctuations documented and discussed in detail
by Lovejoy and his group (Lovejoy, S.,1982; Lovejoy and Schertzer, 1986a,
b; Schertzer and Lovejoy;1991,1994;Tessier et. al., 1993,1996
and all the references therein References
). Standard meteorological theory cannot explain satisfactorily the observed
self-organized
criticality in atmospheric flows (Tessier et.
al., 1993,1996).
Also, traditional mathematical models for simulation and description of
irregular fluctuations in general, and atmospheric flows in particular
are nonlinear and finite precision computer solutions are unrealistic (chaotic)
because of deterministic chaos (Gleick,1987). In this paper, an
alternative nondeterministic cell dynamical system model for atmospheric
flows developed by Mary Selvam (1990 References
) is summarized. The model predicts the observed self-organized criticality
as intrinsic to quantumlike mechanics governing flow dynamics. The model
concepts enable universal quantification for the observed nonlinear variability
in terms of the statistical normal distribution. The model predictions
are in agreement with several standard long-term climatological data sets
for meteorological parameters.The implications of model concepts for long-term
climate change prediction are discussed.
The paper is organized
as follows: Section 2 gives a detailed summary of general concepts
in the newly emerging science of nonlinear dynamics and chaos and
applications for quantifying the observed atmospheric flow patterns. Limitations
of current concepts in standard meteorological theory and deterministic
chaos in model solutions are discussed in Section 3. An alternative
nondeterministic cell dynamical system model for atmospheric flows is summarized
in Section 4. Details of climatological data sets used and analyses techniques
are presented in Section 5. Section 6 contains results and discussions.
Possible applications of model concepts for prediction of climate variability
and climate change are given in Section 7. Conclusions regarding validity
of model concepts and predictions are given in Section 8.
2.
Nonlinear Dynamics and Chaos: A Multidisciplinary Science
The new science of nonlinear dynamics and
chaos (Gleick, 1987 References ) deals
with unified concepts for fundamental aspects intrinsic to the complex
(nonlinear) and apparently random (chaotic) space-time structures found
in nature. Scientific community at large will derive immense benefit in
terms of new insights and powerful analytical techniques in this multidisciplinary
approach to quantify basic similarities in form and function in disparate
contexts ranging from the microscopic to the macroscopic scale.
The apparently random,
noisy or irregular space-time signals (patterns) of a dynamical system,
however, exhibit qualitative similarity in pattern geometry on all scales
and are therefore correlated. In general, the spatiotemporal evolution
of dynamical systems trace a zigzag (jagged) pattern of alternating increase
and decrease, associated with bifurcation or branching on all scales of
space and time, generating wrinkled or folded surfaces in three dimensions.
Representative examples for time series of some meteorological parameters
used in the present study are shown in Figure 1.
FIGURE 1
Time series data of some of the meteorological parameters
used in the present study are shown as representative examples for irregular
(zigzag) fluctuations generic to dynamical systems in nature.
Physical, chemical,
biological and other dynamical systems exhibit similar universal irregular
space-time fluctuations. A fascinating aspect of patterns in nature is
that many of them have a universal character (Dennin et. al.,
1996 References ).
2.1
Fractal Geometry
Irregular space-time fluctuations associated
with basic bifurcation or branching geometry of wrinkles or folds on all
scales is associated with the symmetry of selfsimilarity under scales transformation
or just selfsimilarity (Liu, 1992 References
). A symmetry principle is simply a statement that something looks the
same from certain different points of view. Such symmetries are often called
principles of invariance (Weinberg, 1993 References
). The fundamental similarity or universality in the basic geometric structure,
namely irregularity, was identified as fractal in the late 1970s
by Mandelbrot (1977,1983 References ).
Fractal geometry is ubiquitous in nature, the fine structure on all scales
being the optimum design for sustenance and growth of large scale complex
systems comprised of an integrated network of sub-units. The branching
architecture of river tributaries, bronchial tree, tree branches, lightning
discharge, etc., serve to collect/disperse fluids over a maximum surface
area within a minimum volume. Fine scale fluctuations help efficient mixing
of fluids such as pollution dispersion in the atmosphere. The basic similarity
in the branching form underlying the individual leaf and the tree as a
whole was identified more than three centuries ago in botany (Arber,1950
References
). The importance of scaling concepts were recognized nearly a century
ago in biology and botany where the dependence of a property y
on size x is usually expressed by the allometric equation
y=AxB
where A and B are constants (Thompson,1963;
Strathmann, 1990; Jean, 1994; Stanley, Amaral, Buldyrev, Goldberger et.
al.,
1996 References ). This type of scaling
implies a hierarchy of substructures and was used by D'Arcy Thompson
for scaling anatomical structures, for example, how proportions tend to
vary as an animal grows in size (West, 1990a).
D'Arcy Thompson (1963,
first published in 1917) in his book On Growth and Form has
dealt extensively with similitude principle for biological modelling. Rapid
advances have been made in recent years in the fields of biology and medicine
in the application of scaling (fractal) concepts for description and quantification
of physiological systems and their functions (Goldberger, Rigney and West,
1990; West, 1990a, b; Deering and West,1992; Skinner,1994; Stanley, Amaral,
Buldyrev, Goldberger et.
al., 1996 References
). In meteorological theory, the concept of selfsimilar fluctuations was
introduced in the description of turbulent flows by Richardson (1965,
originally published in 1922), Kolmogorov(1941,1962), Mandelbrot (1975)
(Kadanoff 1996) and others (see Monin and Yaglom,1975 for a review References
).