Proc. IEEE, NAECON 91, Dayton,
U. S. A. May 20-24.
Abstract
The observed long-range
spatiotemporal correlations of real world dynamical systems is governed
by quantumlike mechanics with inherent non-local connections. In summary,
microscopic scale local fluctuations form a unified self-organized adaptive
network manifested as the macro-scale dynamical system with implicit ordered
energy flow between the larger and smaller scales. Such a concept of ladder
networks may find applications in the design of artificial intelligence
systems.
Introduction
Long-range spatiotemporal
correlations manifested as the self-similar fractal geometry to the spatial
pattern concomitant with inverse power law form for the power spectrum
of temporal fluctuations are ubiquitous to real world dynamical systems
and such non-local connections are now identified as signatures of self-organized
criticality (Bak, Tang and Wiesenfeld, 1988) or deterministic chaos. The
physics of deterministic chaos is not yet identified. A striking example
of macro-scale dynamical system exhibiting the signatures of deterministic
chaos is the planetary atmospheric boundary layer atmospheric flow
structure where, the co-operative existence of fluctuations ranging in
size from the planetary scale of thousands of kilometers to the turbulence
scale of a few millimeters gives rise to coherent weather systems with
long-range spatiotemporal correlations such as the El-Nino/Southern Oscillation
cycle of period 2-7 years marked by episodes of abnormal warming off the
coast of Peru associated with devastating changes in the global climate
pattern (Mary Selvam, 1990). The recently identified self-similar fractal
geometry to the global cloud cover pattern and the inverse power law form
for the atmospheric eddy energy spectrum (Lovejoy and Schertzer, 1986)
are signatures of deterministic chaos in real world atmospheric flows.
A cell dynamical system model for atmospheric flows (Mary Selvam, 1990)
applicable to real world dynamical systems shows that quantumlike mechanics
govern atmospheric flow structure and is manifested as the observed long-range
spatiotemporal correlations. In summary, the following model predictions
are applicable to all real world dynamical systems: (1) the energy flow
structure in macro-scale dynamical systems consists of a nested continuum
of vortex roll (large eddy) circulations with overall logarithmic spiral
envelope enclosing internal circulations tracing the quasiperiodic Penrose
tiling pattern such that short-range energy circulation balance requirements
impose long-range orientational order in the spatial pattern, (2) The model
envisages the co-operative existence of a continuum of fluctuations with
ordered energy flow between the larger and smaller scales resulting in
the mixing of the environment into the macro-scale dynamical system. (3)
The universal constant k for deterministic chaos is identified as
the steady state fractional volume dilution of the macro-scale dynamical
system by inherent small-scale spatiotemporal fluctuations. The value of
k
is equal to 1/t2
( @ 0.382) where t
is the golden mean [(1+Ö5)/2].
(4)
The universal Feigenbaum's constants (Feigenbaum, 1980) a and d
are functions of k . (5) The steady state ordered emergence of fractal
(broken) Euclidean geometry of the macro-scale dynamical system is quantified
by the universal algorithm
2a2 = pd
where
the energy 2a2 released from the medium of propagation
during stretching provides the spin angular momentum pd
for a dominant cycle of evolution of the logarithmic spiral pattern for
energy propagation, e.g. the latent heat of condensation in rising air
parcels generates the strikingly spiral shaped hurricane cloud patterns.
(6) The dynamical system is the macro-scale manifestation of the internal
microscopic domain energy circulation networks and results in quantumlike
mechanics for the macro-scale evolution. The apparent paradox of wave-particle
duality of quantum mechanics for the sub-atomic domain is physically consistent
in the context of atmospheric flows where formation of clouds occurs in
updrafts with simultaneous dissipation of clouds in adjacent downdrafts
giving rise to the observed discrete cellular geometry to individual clouds
and to cloud ensembles. The bimodal (formation and dissipation) of the
phenomenological form for energy manifestation is intrinsic to the bidirectional
energy flow of eddy circulations. Quantumlike mechanics is therefore manifested
as the commonplace occurrence of real world dynamical systems as discrete
entities. In summary, quantumlike mechanics govern self-organized adaptive
networks such as the ladder networks, i.e. networks formed by numerous
repetitions of an elementary cell, (e.g., neural networks) with inherent
quasiperiodic Penrose tiling pattern for the network with long-range spatiotemporal
correlations. Recent studies on ladder networks on electrical circuits
with equal magnitudes for longitudinal and transverse impedances indicate
that electrical characteristics are strictly and surprisingly related to
the Fibonacci numbers (D' Amico, M.Faccio and G. Ferri, 1990). The Fibonacci
numbers are incorporated in the quasiperiodic Penrose tiling pattern formed
by the ladder network.
Continuous periodogram
analysis of the time series of 115 years (1871-1985) summer monsoon (June-September)
rainfall over the Indian region show that the power spectra of the temporal
fluctuations are the same as the normal distribution with the square of
the eddy amplitude representing the eddy probability density corresponding
to the normalized standard deviation t equal to [(log l/
log
l50)
-1] where l
is the period length in years and l50
represents the period up to which the cumulative percentage contribution
to total variance is equal to 50. The above result, namely that power spectra
of the temporal fluctuations of rainfall follow the universal and unique
inverse power law form of the statistical normal distribution implies quantumlike
mechanics for the dynamics of atmospheric flows and is also a signature
of deterministic chaos.
Cell Dynamical System Model
The mean flow in the planetary
atmospheric boundary layer (ABL) possesses an upward momentum flux of surface
frictional origin. This turbulence scale upward momentum flux is progressively
amplified by the exponential decrease of atmospheric density with height
coupled with buoyant energy generation in microscale fractional condensation
by deliquescence on hygroscopic nuclei even in an unsaturated environment.
The incessant upward momentum flux generates helical vortex roll (large
eddy) circulations manifested as cloud rows/streets and meso-scale (~100
kms) cloud clusters (MCC) in global cloud cover pattern. Townsend (1956)
has shown that the spatial integration of inherent turbulent eddies gives
rise to large eddy circulations. The root mean square (r.m.s) circulation
speed W of the large eddy of radius R is therefore expressed
in terms of the r.m.s. circulation speed w* of dominant turbulent
eddy of radius r as follows.
(1)
The growth of large eddy circulations from
turbulence scale buoyant energy generation therefore occurs in unit length
step increments in unit intervals of time, the turbulence scale yardsticks
for length and time being used. Such a concept of large eddies as the macroscale
envelope of a self-sustaining network of small scale circulations is analogous
to the concept of 'cellular automata' computational technique where the
macroscale dynamical system is assumed to consist of identical unit cells
with arbitrary rules for evolution of the ensemble (Oona and Puri, 1988).
The cellular automata computational technique described in this paper for
growth of large eddies from microscopic domain turbulent fluctuations is
based on the governing Equation 1 which is physically consistent and mathematically
rigorous. Further, the growth of large eddies by successive length step
increments equal to the turbulent eddy length scale doubling is identified
as the universal period doubling route to chaos. Such a concept envisages
the growth of an eddy continuum starting from the turbulence scale with
the power spectrum of the temporal fluctuations following the inverse power
law form which is a signature of deterministic chaos. Equation 1 therefore
implies a two-way ordered energy flow between the larger and smaller scales
and is a statement of the law of conservation of energy for the dynamical
system, namely atmospheric flows. In summary, the spatio-temporal growth
of dynamical systems in general occurs by the propagation of inherent small
scale fluctuations which are sustained by energy released from the medium
of propagation during stretching. The energy circulation pattern in a dynamical
system consists of a continuum of vortices within vortices. Equation 1
is hereby identified as the universal algorithm for deterministic chaos
in real world dynamical systems. Computations show that the successive
values of the circulation speed W and radius R of the growing
internal circulation follow the Fibonacci mathematical number series. Therefore
short-range circulation balance requirements generate successively larger
circulation patterns with the precise geometry governed by the Fibonacci
mathematical number series. The successively larger eddy radii may be subdivided
again in the golden mean ratio. The internal structure of large eddy circulations
is therefore made up of balanced small scale circulations tracing out the
well known quasiperiodic Penrose tiling pattern identified as the quasicrystalline
structure in condensed matter physics. The growing large eddy carries the
turbulent eddies as internal circulations which contribute to their (large
eddies) further growth. The turbulent eddy fluctuations mix environmental
mass into the large eddy volume. The steady state fractional volume dilution
k
of the large eddy volume by environmental mixing is given as
(2)
Since the steady state fractional volume dilution
of large eddy by inherent turbulent eddy fluctuations during successive
length step increments is equal to 0.382 , i.e. less than
half, the overall Euclidean geometrical shape of the large eddy is retained,
e.g. cloud billows which resemble spheres. The variable k is hereby
identified as the universal constant for deterministic chaos in
dynamical systems and in the following it is shown that the universal Feigenbaum's
constants are functions of k. The r.m.s. circulation speed W
of
the large eddy which grows from constant turbulence scale acceleration
w*
is given as
(3)
where z is the length scale ratio equal
to R/r. The strange attractor design of energy circulation pattern
in real world dynamical systems therefore consists of a nested continuum
of logarithmic spiral circulations with quasiperiodic Penrose tiling pattern
for the internal structure.
The variables W and W2
represent respectively the standard deviation and variance since W
represents the instantaneous spatial average of the perturbation speed
of the large eddy circulation. Therefore, for a constant turbulence scale
acceleration w* , the ratio of the r.m.s. circulation
speeds W1 and W2 of large eddies of
radii R1 and R2 respectively represent
the ratio of the standard deviations of large eddy fluctuations. From Equation
3
(4)
Starting with reference level standard deviation
s
equal to W1 , the successive dominant eddy growths have
standard deviations W2 equal to s
, 2s , 3s
, etc. from Equation 1 where z1 = z2n
and n = 1, 2, 3, ...n for successive period doubling
growth sequence since the successive R values follow the Fibonacci
number series as shown earlier.
Universal Feigenbaum's constants
The universal period doubling
route to chaos has been extensively studied by Feigenbaum (1980) who found
that two universal constants a and d describe the approach
to turbulence independent of the details of the nonlinear equations describing
the physical system. Experimental studies have identified the universal
constants a and d in real world dynamical systems also (Gleick
1987). Delbourgo's (1986) computations show that the universal constants
a
and d follow the relation 2a2 = 3d over
a wide domain. The physical concepts of the large eddy growth by period
doubling process enables to derive Feigenbaum's universal constants a
and d and their mutual relationship as functions inherent to the
scale invariant eddy energy structure of the dynamical system as follows.
The function a may be defined as
(5)
The constant a is equal to 1/k
where k represents the steady state fractional volume dilution of
large eddy by turbulent eddy fluctuations across unit cross-section on
the large eddy envelope. Therefore a represents the steady state
fractional mass dispersion by dilution and a2 represents
the corresponding fractional energy flux into the environment. Let d
represent the ratio of the spin angular momenta of the total mass of the
large and turbulent eddies
(6)
Therefore 2a2
= 3d
(7)
Equation 7 is in agreement with Delbourgo's
(1986) results. Further, the universal algorithm for deterministic chaos
at Equation 1 can now be reformulated in terms of a and d
to give 2a2 = 3d . Therefore the universal algorithm
at Equation 1 is a statement of the law of conservation of energy for the
period doubling growth sequence. The variable 2a2 represents
the total bidirectional eddy energy flow into the environment and is equal
to the spin angular momentum of the large eddy. The property of inertia
enables propagation of turbulence scale perturbation in the medium where
translational kinetic energy is generated during dilation by release of
intrinsic latent energy potential of the medium, e.g. the buoyant energy
generation by water vapour condensation in the updraft regions of the atmospheric
boundary layer.
Deterministic chaos and quantumlike mechanics
in atmospheric flows
Since the large eddy is
the sum total of the smaller scales, the large eddy energy content is equal
to the sum of all its individual component eddy energies and therefore
the kinetic energy distribution is normal and the kinetic energy of any
component eddy expressed as a fraction of the energy content of the largest
eddy in the hierarchy will represent the cumulative normal probability
density distribution, i.e. the eddy energy probability density distribution
is equal to the square of the eddy amplitude. Therefore, the atmospheric
eddy continuum energy structure follows laws similar to quantum mechanical
laws for subatomic dynamics. Therefore, the eddy continuum energy spectrum,
i.e. the variance versus frequency values which are conventionally plotted
on a log-log scale will represent the normal probability density distribution
on a logarithmic scale versus the normalized standard deviation t
as shown in the following. In the case of the eddy continuum energy spectrum,
t
will be obtained directly from Equation 4 as
The above described analogy
of quantumlike mechanics for atmospheric flows is similar to the concept
of a sub-quantum level of fluctuations whose space-time organization gives
rise to the observed manifestation of subatomic phenomena, i.e. quantum
systems as order out of chaos phenomena (Grossing 1989). Further, numerical
simulations and analytic solutions show that the power spectral density
of the stochastic system and the deterministic chaotic system are indistinguishable
(Stone 1990) in agreement with the above result, namely normal distribution
characteristics for the power spectra of strange attractors of dynamical
systems.
Data and analysis
The summer monsoon (June-September)
rainfall for 29 meteorological sub-divisions for 115 years (1871-1985)
was taken from Parthasarathy et al. (1987). The data was subjected
to a quasi-continuous periodogram spectral analyses (Jenkinson 1977). The
cumulative percentage contribution to total variance (P), the cumulative
percentage normal probability density and the corresponding normalized
standard deviation t values are plotted in Figure 1. It is seen
that the cumulative percentage contribution to total variance closely follows
the cumulative normal probability density distribution. The "goodness of
fit" was tested using the chi-square test. The horizontal lines in the
figure indicate the values of P above which the fit is good at 95% level
of significance.
Figure 1
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